%%% -*-BibTeX-*- %%% ==================================================================== %%% BibTeX-file{ %%% author = "Nelson H. F. Beebe", %%% version = "1.29", %%% date = "06 February 2018", %%% time = "10:28:41 MST", %%% filename = "kepler.bib", %%% address = "University of Utah %%% Department of Mathematics, 110 LCB %%% 155 S 1400 E RM 233 %%% Salt Lake City, UT 84112-0090 %%% USA", %%% telephone = "+1 801 581 5254", %%% FAX = "+1 801 581 4148", %%% URL = "http://www.math.utah.edu/~beebe", %%% checksum = "07911 7596 31732 325398", %%% email = "beebe at math.utah.edu, beebe at acm.org, %%% beebe at computer.org (Internet)", %%% codetable = "ISO/ASCII", %%% keywords = "Archimedean solid packing; BibTeX; %%% bibliography; David Hilbert's 18th Problem %%% (1900); Johannes Kepler (1571--1630); Kepler %%% Conjecture; Kepler optimal sphere-packing %%% problem; Kepler Theorem; kissing numbers; %%% Platonic solid packing; Project Flyspeck; %%% sphere packing", %%% license = "public domain", %%% supported = "no", %%% docstring = "This is a bibliography about the Kepler %%% Conjecture on optimal sphere packing, and %%% recent advances in the general solid-body %%% packing problem. More information about the %%% Kepler Conjecture and its credible, but %%% controversial, proof are given below. %%% %%% At version 1.29, the year coverage looked %%% like this: %%% %%% 1611 ( 1) 1747 ( 0) 1883 ( 0) %%% 1628 ( 0) 1764 ( 0) 1900 ( 1) %%% 1629 ( 0) 1765 ( 0) 1901 ( 1) %%% 1630 ( 0) 1766 ( 0) 1902 ( 1) %%% 1632 ( 0) 1768 ( 0) 1904 ( 1) %%% 1653 ( 0) 1789 ( 0) 1925 ( 1) %%% 1680 ( 0) 1816 ( 0) 1952 ( 1) %%% 1681 ( 0) 1817 ( 0) 1953 ( 2) %%% 1686 ( 0) 1822 ( 0) 1958 ( 1) %%% 1689 ( 0) 1825 ( 0) 1961 ( 1) %%% 1690 ( 0) 1826 ( 0) 1962 ( 1) %%% 1692 ( 0) 1828 ( 0) 1964 ( 2) %%% 1694 ( 0) 1830 ( 0) 1966 ( 1) %%% 1695 ( 0) 1831 ( 0) 1967 ( 3) %%% 1696 ( 0) 1832 ( 0) 1968 ( 1) %%% 1697 ( 0) 1833 ( 0) 1969 ( 1) %%% 1698 ( 0) 1834 ( 0) 1970 ( 2) %%% 1699 ( 0) 1835 ( 0) 1971 ( 1) %%% 1700 ( 0) 1836 ( 0) 1972 ( 1) %%% 1703 ( 0) 1839 ( 0) 1975 ( 1) %%% 1707 ( 0) 1843 ( 0) 1979 ( 4) %%% 1709 ( 0) 1845 ( 0) 1981 ( 2) %%% 1710 ( 0) 1846 ( 0) 1982 ( 1) %%% 1711 ( 0) 1847 ( 0) 1983 ( 2) %%% 1712 ( 0) 1848 ( 0) 1984 ( 1) %%% 1713 ( 0) 1849 ( 0) 1985 ( 3) %%% 1716 ( 0) 1852 ( 0) 1988 ( 1) %%% 1717 ( 0) 1853 ( 0) 1989 ( 2) %%% 1718 ( 0) 1854 ( 0) 1990 ( 1) %%% 1719 ( 0) 1855 ( 0) 1991 ( 4) %%% 1720 ( 0) 1856 ( 0) 1992 ( 4) %%% 1721 ( 0) 1857 ( 0) 1993 ( 11) %%% 1722 ( 0) 1858 ( 0) 1994 ( 3) %%% 1723 ( 0) 1859 ( 0) 1995 ( 4) %%% 1724 ( 0) 1860 ( 0) 1996 ( 2) %%% 1725 ( 0) 1861 ( 0) 1997 ( 7) %%% 1726 ( 0) 1862 ( 0) 1998 ( 6) %%% 1727 ( 0) 1863 ( 0) 1999 ( 10) %%% 1728 ( 0) 1864 ( 0) 2000 ( 14) %%% 1729 ( 0) 1865 ( 0) 2001 ( 13) %%% 1730 ( 0) 1866 ( 0) 2002 ( 10) %%% 1731 ( 0) 1867 ( 0) 2003 ( 8) %%% 1732 ( 0) 1868 ( 0) 2004 ( 7) %%% 1733 ( 0) 1869 ( 0) 2005 ( 11) %%% 1734 ( 0) 1870 ( 0) 2006 ( 19) %%% 1735 ( 0) 1871 ( 0) 2007 ( 13) %%% 1736 ( 0) 1872 ( 0) 2008 ( 10) %%% 1737 ( 0) 1873 ( 0) 2009 ( 11) %%% 1738 ( 0) 1874 ( 0) 2010 ( 17) %%% 1739 ( 0) 1875 ( 0) 2011 ( 10) %%% 1740 ( 0) 1876 ( 0) 2012 ( 8) %%% 1741 ( 0) 1877 ( 0) 2013 ( 10) %%% 1742 ( 0) 1878 ( 0) 2014 ( 8) %%% 1743 ( 0) 1879 ( 0) 2015 ( 2) %%% 1744 ( 0) 1880 ( 0) 2016 ( 8) %%% 1745 ( 0) 1881 ( 0) 2017 ( 2) %%% %%% Article: 181 %%% Book: 35 %%% InCollection: 17 %%% InProceedings: 19 %%% Misc: 2 %%% PhdThesis: 3 %%% Proceedings: 14 %%% TechReport: 2 %%% Unpublished: 1 %%% %%% Total entries: 274 %%% %%% A brief statement of the Kepler Conjecture %%% (from page xvi of entry Conway:1999:SPL) is %%% ``the so-called Kepler Conjecture [is that] %%% no packing of three-dimensional spheres can %%% have a greater density than that of the %%% face-centered cubic lattice.'' %%% %%% Proof of the Kepler Conjecture is problem %%% number 15 (later 18) of David Hilbert's 1900 %%% list of 20 (later revised to 23) important %%% mathematical problems for the 20th Century %%% (see entries Hilbert:1900:MPV, %%% Hilbert:1901:MPV, Hilbert:1902:MP, and %%% Hilbert:2000:MP). [Some of those problems, %%% though not that of the Kepler Conjecture, are %%% included in the Clay Mathematics Institute %%% Millennium Problem list for which validated %%% solutions win a prize of one million US %%% dollars.] %%% %%% Thomas C. Hales announced a proof of the %%% Kepler Conjecture on 9 August 1998, so it %%% should now be called the Kepler Theorem, but %%% the proof was not published until 2005. See %%% entry Morgan:2005:BRB for the reasons for the %%% long delay in publication, and entry %%% Hales:2000:CH for the prize-winning paper %%% with a largely nonmathematical description of %%% the nature of the problem. %%% %%% Hales' complicated proof is detailed in a %%% book (entry Lagarias:2011:KCH), and discussed %%% in nonmathematical form in two other books %%% (entries Casti:2001:MMF and Szpiro:2003:KCH). %%% The proof was also published in seven papers %%% in a special journal issue (entries %%% Hales:2006:SPIb, Toth:2006:GEF, %%% Ferguson:2006:SPV, Hales:2006:SPV, %%% Hales:2006:FKC, Hales:2006:HOK, %%% Hales:2006:SPIa). %%% %%% Because of the great difficulty of the proof %%% of the Kepler Conjecture, and the inability %%% of other mathematicians to verify the proof, %%% Hales began Project Flyspeck (see entry %%% Hales:2008:FP) to produce a verification of %%% the proof by a computer. Successful %%% completion of that project was announced on %%% various e-mail lists on 10-Aug-2014. Hales %%% reported: %%% %%% ``The verification of all nonlinear %%% inequalities in HOL Light on the %%% Microsoft Azure cloud took approximately %%% 5000 processor-hours. Almost all %%% verifications were made in parallel with %%% 32 cores, hence the real time is about %%% 5000 / 32 = 156.25 hours. Nonlinear %%% inequalities were verified with compiled %%% versions of HOL Light and the %%% verification tool developed in Solovyev's %%% 2012 thesis.'' %%% %%% The final report of that project appears in %%% entry Hales:2015:FPK. Entry Adams:2014:FF %%% discusses the proof, and the independent %%% validation of the proof. %%% %%% The checksum field above contains a CRC-16 %%% checksum as the first value, followed by the %%% equivalent of the standard UNIX wc (word %%% count) utility output of lines, words, and %%% characters. This is produced by Robert %%% Solovay's checksum utility.", %%% } %%% ====================================================================

@Preamble{ "\ifx \undefined \booktitle \def \booktitle#1{{{\em #1}}} \fi" # "\ifx \undefined \mathbb \def \mathbb #1{{\bf #1}}\fi" }

%%% ==================================================================== %%% Journal abbreviations:

@String{j-AM-SCI= "American Scientist"} @String{j-AMER-MATH-MONTHLY= "American Mathematical Monthly"} @String{j-ANN-MATH-2= "Annals of Mathematics (2)"} @String{j-ANN-MATH-ARTIF-INTELL= "Annals of Mathematics and Artificial Intelligence"} @String{j-ANN-INST-STAT-MATH-TOKYO= "Annals of the Institute of Statistical Mathematics (Tokyo)"} @String{j-APPL-MATH-COMP= "Applied Mathematics and Computation"} @String{j-BELL-SYST-TECH-J= "The Bell System Technical Journal"} @String{j-BULL-AMS= "Bulletin of the American Mathematical Society"} @String{j-CACM= "Communications of the ACM"} @String{j-CAN-J-MATH= "Canadian Journal of Mathematics = Journal canadien de math{\'e}matiques"} @String{j-CAN-MATH-BULL= "Bulletin canadien de math\-{\'e}\-mat\-iques = Canadian Mathematical Bulletin"} @String{j-COMBINATORICA= "Combinatorica"} @String{j-COMP-GEOM= "Computational Geometry. Theory and Applications"} @String{j-COMP-J= "The Computer Journal"} @String{j-COMP-OP-RES= "Computers and Operations Research"} @String{j-COMP-PHYS-COMM= "Computer Physics Communications"} @String{j-COMPUT-MATH-APPL= "Computers and Mathematics with Applications"} @String{j-DISCRETE-COMPUT-GEOM= "Discrete and Computational Geometry"} @String{j-DISCRETE-MATH= "Discrete Mathematics"} @String{j-DOC-MATH= "Documenta mathematica"} @String{j-ELECT-J-COMB= "Electronic Journal of Combinatorics"} @String{j-EUR-J-COMB= "European Journal of Combinatorics"} @String{j-EXP-MATH= "Experimental Mathematics"} @String{j-FORMA= "Forma (Tokyo)"} @String{j-IEEE-TRANS-INF-THEORY= "IEEE Transactions on Information Theory"} @String{j-INT-J-MATH= "International Journal of Mathematics"} @String{j-INT-J-NUMER-METHODS-ENG= "International Journal for Numerical Methods in Engineering"} @String{j-J-AM-MATH-SOC= "Journal of the American Mathematical Society"} @String{j-J-APPL-PROBAB= "Journal of Applied Probability"} @String{j-J-AUTOM-REASON= "Journal of Automated Reasoning"} @String{j-J-CHEM-PHYS= "Journal of Chemical Physics"} @String{j-J-COMB-OPTIM= "Journal of Combinatorial Optimization"} @String{j-J-COMB-THEORY-A= "Journal of Combinatorial Theory (Series A)"} @String{j-J-COMPUT-APPL-MATH= "Journal of Computational and Applied Mathematics"} @String{j-J-COMPUT-PHYS= "Journal of Computational Physics"} @String{j-J-MATH-CHEM= "Journal of Mathematical Chemistry"} @String{j-J-MATH-PHYS= "Journal of Mathematical Physics"} @String{j-J-PHYS-CHEM-B= "Journal of Physical Chemistry. B. Condensed matter, materials, surfaces, interfaces \& biophysical"} @String{j-J-THEOR-BIOL= "Journal of Theoretical Biology"} @String{j-LECT-NOTES-COMP-SCI= "Lecture Notes in Computer Science"} @String{j-LECT-NOTES-MATH= "Lecture Notes in Mathematics"} @String{j-MATH-ANN= "Mathematische Annalen"} @String{j-MATH-GAZ= "Mathematical Gazette"} @String{j-MATH-INTEL= "The Mathematical Intelligencer"} @String{j-MATH-MAG= "Mathematics Magazine"} @String{j-MATH-Z= "Mathematische Zeitschrift"} @String{j-MATHEMATIKA= "Mathematika"} @String{j-MDMV= "Mitteilungen der Deutschen Mathematiker-Vereinigung"} @String{j-MONAT-MATH= "Monatshefte f{\"u}r Mathematik"} @String{j-MONTE-CARLO-METHODS-APPL= "Monte Carlo Methods and Applications"} @String{j-NAMS= "Notices of the American Mathematical Society"} @String{j-NATURE= "Nature"} @String{j-NORMAT= "Normat: Nordisk matematisk tidskrift"} @String{j-PERIOD-MATH-HUNGAR= "Periodica Mathematica Hungarica"} @String{j-PHIL-MAGAZINE= "Philosophical Magazine"} @String{j-PHYS-LET-A= "Physics Letters A"} @String{j-PHYS-REV-A= "Physical Review A (Atomic, Molecular, and Optical Physics)"} @String{j-PHYS-REV-B-CMMP= "Physical Review B: Condensed Matter and Materials Physics"} @String{j-PHYS-REV-E= "Physical Review E (Statistical physics, plasmas, fluids, and related interdisciplinary topics)"} @String{j-PHYS-REV-LET= "Physical Review Letters"} @String{j-PROC-NATL-ACAD-SCI-USA= "Proceedings of the National Academy of Sciences of the United States of America"} @String{j-RELIABLE-COMPUTING= "Reliable Computing = Nadezhnye vychisleniia"} @String{j-RES-DEV= "Research \& Development"} @String{j-REV-MOD-PHYS= "Reviews of Modern Physics"} @String{j-SCI-AMER= "Scientific American"} @String{j-SIAM-J-DISCR-MATH= "SIAM Journal on Discrete Mathematics"} @String{j-SIAM-REVIEW= "SIAM Review"} @String{j-THEOR-COMP-SCI= "Theoretical Computer Science"} @String{j-TOG= "ACM Transactions on Graphics"}

%%% ==================================================================== %%% Publisher abbreviations:

@String{pub-ACM= "ACM Press"} @String{pub-ACM:adr= "New York, NY 10036, USA"} @String{pub-CAMBRIDGE= "Cambridge University Press"} @String{pub-CAMBRIDGE:adr= "Cambridge, UK"} @String{pub-CLARENDON= "Clarendon Press"} @String{pub-CLARENDON:adr= "Oxford, UK"} @String{pub-ELS= "Elsevier Science Publishers B.V."} @String{pub-ELS:adr= "Amsterdam, The Netherlands"} @String{pub-IEEE= "IEEE Computer Society Press"} @String{pub-IEEE:adr= "1109 Spring Street, Suite 300, Silver Spring, MD 20910, USA"} @String{pub-IOP= "Institute of Physics Publishing"} @String{pub-IOP:adr= "Bristol, UK and Philadelphia, PA, USA"} @String{pub-MATH-ASSOC-AMER= "Mathematical Association of America"} @String{pub-MATH-ASSOC-AMER:adr= "Washington, DC, USA"} @String{pub-NAP= "National Academy Press"} @String{pub-NAP:adr= "Washington, DC, USA"} @String{pub-NORTH-HOLLAND= "North-Holland Publishing Co."} @String{pub-NORTH-HOLLAND:adr= "Amsterdam, The Netherlands"} @String{pub-NORTON= "W. W. Norton \& Co."} @String{pub-NORTON:adr= "New York, NY, USA"} @String{pub-OXFORD= "Oxford University Press"} @String{pub-OXFORD:adr= "Walton Street, Oxford OX2 6DP, UK"} @String{pub-PRINCETON= "Princeton University Press"} @String{pub-PRINCETON:adr= "Princeton, NJ, USA"} @String{pub-SIAM= "Society for Industrial and Applied Mathematics"} @String{pub-SIAM:adr= "Philadelphia, PA, USA"} @String{pub-SV= "Spring{\-}er-Ver{\-}lag"} @String{pub-SV:adr= "Berlin, Germany~/ Heidelberg, Germany~/ London, UK~/ etc."} @String{pub-U-CAL= "University of California Press"} @String{pub-U-CAL:adr= "Berkeley, CA, USA"} @String{pub-VIEWEG= "Friedrich Vieweg und Sohn"} @String{pub-VIEWEG:adr= "Braunschweig, Germany"} @String{pub-WILEY= "Wiley"} @String{pub-WILEY:adr= "New York, NY, USA"} @String{pub-WORLD-SCI= "World Scientific Publishing Co. Pte. Ltd."} @String{pub-WORLD-SCI:adr= "P. O. Box 128, Farrer Road, Singapore 9128"}

%%% ==================================================================== %%% Series abbreviations:

@String{ser-LNAI= "Lecture Notes in Artificial Intelligence"} @String{ser-LNCS= "Lecture Notes in Computer Science"}

%%% ==================================================================== %%% Bibliography entries, sorted by increasing year and then by citation %%% label within each year, with `bibsort -byyear':

@Book{Kepler:1611:SSN, author = "Johannes Kepler", title = "Strena, Seu, de Nive Sexangula", publisher = "Godfrey Tampach", address = "Frankfurt-am-Main, Germany", year = "1611", bibdate = "Wed Jan 09 08:37:25 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", note = "This book contains the original formulation of the now-famous Kepler Sphere Packing Conjecture. English translation in \cite{Kepler:1966:SCS}.", acknowledgement = ack-nhfb, } @Article{Hilbert:1900:MPV, author = "David Hilbert", title = "{Mathematische Probleme. Vortrag, gehalten auf dem internationalen Mathematiker-Kongre{\ss} zu Paris 1900}. ({German}) [Mathematical Problems: Lecture held at the {International Mathematicians Congress in Paris 1900}]", journal = "G{\"o}tt. Nachr.", volume = "??", number = "??", pages = "253--297", month = "????", year = "1900", bibdate = "Wed Jan 09 12:29:52 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", note = "This is the paper in which Hilbert lays out his now-famous list of 20 (revised to 23 in 1901) challenging problems for mathematicians to address in the 20th Century. Reprinted in \cite{Hilbert:1901:MPV}. English translation in \cite{Hilbert:1902:MP,Hilbert:2000:MP}", ZMnumber = "JFM 31.0068.03", acknowledgement = ack-nhfb, keywords = "Hilbert's 20 Problems", language = "German", remark = "The problems are (in German): \\ 1. Cantor's Problem von der M{\"a}chtigkeit des Continuums \\ 2. Die Widerspruchslosigkeit der arithmetischen Axiome \\ 3. Die Volumengleichheit zweier Tetraeder von gleicher Grundfl{\"a}che und H{\"o}he (Unm{\"o}glichkeit des Beweises durch Zerlegung in congruente Teile) \\ 4. Problem von der Geraden als k{\"u}rzester Verbindung zweier Punkte \\ 5. Mathematische Behandlung der Axiome der Physik \\ 6. Irrationalit{\"a}t und Transcendenz bestimmter Zahlen (z. B. $ \pi $, $ \sqrt {2} $, and $e$ (??)) \\ 7. Primzahlenprobleme \\ 8. Beweis des allgemeinsten Reciprocit{\"a}tsgesetzes im beliebigen Zahlk{\"o}rper \\ 9. Entscheidung der L{\"o}sbarkeit einer diophantischen Gleichung \\ 10. Ausdehnung des Kronecker'schen Satzes {\"u}ber Abel'sche K{\"o}rper auf einen beliebigen algebraischen Rationalit{\"a}tsbereich \\ 11. Unm{\"o}glichkeit der L{\"o}sung der allgemeinen Gleichung \\ 7. Grades mittels Functionen von nur zwei Argumenten \\ 12. Strenge Begr{\"u}ndung von Schubert's Abz{\"a}hlungscalc{\"u}l \\ 13. Problem der Topologie algebraischer Curven und Fl{\"a}chen \\ 14. Darstellung definiter Formen durch Quadrate \\ 15. Aufbau des Raumes aus congruenten Polyedern \\ 16. Sind die L{\"o}sungen regul{\"a}rer Variationsprobleme stets notwendig analytisch? \\ 17. Allgemeines Randwertproblem \\ 18. Beweis der Existenz linearer Differentialgleichungen mit vorgeschriebener Monodromiegruppe \\ 19. Uniformisirung analytischer Beziehungen mittels automorpher Functionen \\ 20. Weiterf{\"u}hrung der Methoden der Variationsrechnung \\ (Rough) English translations of the problem titles are: \\ 1. Cantor's problem of the thickness of continua \\ 2. The consistency of the axioms of arithmetic \\ 3. The volume equality of two tetrahedra of equal base and height (impossibility of proof by breaking it down into congruent parts) \\ 4. Problem of the straight line as a short connection of two points \\ 5. Mathematical treatment of the axioms of physics \\ 6. Irrationality and transcendence of certain numbers (such as $ \pi $, $ \sqrt {2}$, and $e$ (??)) \\ 7 Prime problems \\ 8. Evidence of the most common reciprocity theorem in the numbers \\ 9. Decision of the solvability of a Diophantine equation \\ 10. Expansion of the Kronecker set over Abelian bodies on any algebraic rational area \\ 11. Inability to solve the general equation of 7th degree by means of functions of two arguments \\ 12. Rigorous justification of Schubert's counting calculus \\ 13. Issue of the topology of algebraic curves and surfaces \\ 14. Representation of definite forms by squares \\ 15. Construction of space from congruent polyhedra \\ 16. Are the regular variation problems solutions always necessary analytic? \\ 17. General boundary value problem \\ 18. Evidence of the existence of linear differential equations with prescribed Monodromiegruppe \\ 19. Uniformization of analytic relations by means of automorphic functions \\ 20. Continuation of the methods of the calculus of variations", } @Article{Hilbert:1901:MPV, author = "David Hilbert", title = "{Mathematische Probleme. Vortrag, gehalten auf dem internationalen Mathematiker-Kongre{\ss} zu Paris 1900}. ({German}) [Mathematical Problems: Lecture held at the {International Mathematicians Congress in Paris 1900}]", journal = "Arch. d. Math. u. Phys", volume = "3", number = "1", pages = "44--63, 213--237", month = "????", year = "1901", DOI = "", bibdate = "Wed Jan 09 09:37:38 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "", ZMnumber = "JFM 32.0084.05", acknowledgement = ack-nhfb, ajournal = "Arch. Math. Phys.", keywords = "Hilbert's 23 Problems", language = "German", } @Article{Hilbert:1902:MP, author = "David Hilbert", title = "Mathematical Problems", journal = j-BULL-AMS, volume = "8", number = "10", pages = "437--479", month = "????", year = "1902", CODEN = "BAMOAD", DOI = "https://doi.org/10.1090/S0002-9904-1902-00923-3", ISSN = "0002-9904 (print), 1936-881X (electronic)", ISSN-L = "0002-9904", MRnumber = "1557926", bibdate = "Wed Jan 09 17:25:01 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", note = "English translation of Hilbert's famous list of 23 important problems in mathematics for the 20th Century.", URL = "http://www.ams.org/journals/bull/1902-08-10/S0002-9904-1902-00923-3/S0002-9904-1902-00923-3.pdf", acknowledgement = ack-nhfb, fjournal = "Bulletin of the American Mathematical Society", journal-URL = "http://www.ams.org/journals/bull/all_issues.html", keywords = "Hilbert's 23 Problems", } @Article{Minkowski:1904:DGL, author = "H. Minkowski", title = "{Dichteste gitterf{\"o}rmige Lagerung kongruenter K{\"o}rper}. ({German}) [{Closest} lattice packing of congruent bodies]", journal = "Nachr. K. Ges. Wiss. G{\"o}ttingen", volume = "??", number = "??", pages = "311--355", month = "????", year = "1904", DOI = "", bibdate = "Wed Jan 09 09:43:51 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", note = "The `proof' of a packing theorem in this paper was later shown to be wrong; see \cite[pages 1543--1544]{Lagarias:2012:MPR}", URL = "", acknowledgement = ack-nhfb, language = "German", } @Article{Struik:1925:PIL, author = "D. J. Struik", title = "Het Probleem {``De impletione loci''}. ({Dutch}) [{The} problem {``On Local Filling''}]", journal = "Nieuw Archief voor Wiskunde, Series 2", volume = "15", number = "??", pages = "121--137", year = "1925--1928", DOI = "", bibdate = "Wed Jan 09 09:56:24 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "", acknowledgement = ack-nhfb, language = "Dutch", } @Book{Weyl:1952:S, author = "Hermann Weyl", title = "Symmetry", publisher = pub-PRINCETON, address = pub-PRINCETON:adr, pages = "168", year = "1952", ISBN = "0-691-08045-3", ISBN-13 = "978-0-691-08045-1", LCCN = "", bibdate = "Wed Jan 09 15:33:17 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, author-dates = "1885--1955", } @Article{Schutte:1953:PDK, author = "K. Sch{\"u}tte and B. L. van der Waerden", title = "{Das Problem der dreizehn Kugeln}. ({German}) [{The} problem of thirteen bullets]", journal = j-MATH-ANN, volume = "125", number = "??", pages = "325--334", month = "????", year = "1953", CODEN = "MAANA3", DOI = "https://doi.org/10.1007/BF01343127", ISSN = "0025-5831 (print), 1432-1807 (electronic)", ISSN-L = "0025-5831", MRclass = "52.0X", MRnumber = "0053537 (14,787e)", MRreviewer = "H. S. M. Coxeter", bibdate = "Wed Jan 09 09:48:48 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://link.springer.com/article/10.1007/BF01343127", acknowledgement = ack-nhfb, fjournal = "Mathematische Annalen", language = "German", } @Book{Toth:1953:LEK, author = "L. Fejes T{\'o}th", title = "{Lagerungen in der Ebene, auf der Kugeln und im Raum}. ({German}) [{Ball} deposits in the plane and in space]", volume = "65", publisher = pub-SV, address = pub-SV:adr, pages = "x + 197", year = "1953", LCCN = "", MRclass = "52.0X", MRnumber = "0057566 (15,248b)", MRreviewer = "W. O. Moser", bibdate = "Wed Jan 09 09:16:25 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", note = "See also second edition \cite{Toth:1972:LEK}.", series = "{Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Ber{\"u}cksichtigung der Anwendungsgebiete}", acknowledgement = ack-nhfb, language = "German", } @Article{Renyi:1958:ODP, author = "A. R{\'e}nyi", title = "On a one-dimensional problem concerning random space filling", journal = "Magyar Tud. Akad. Mat. Kutat{\'o} Int. K{\"o}zl.", volume = "3", number = "1--2", pages = "109--127", month = "????", year = "1958", MRclass = "60.00", MRnumber = "0104284 (21 \#3039)", MRreviewer = "P. Erd{\H{o}}s", bibdate = "Wed Feb 29 21:10:40 2012", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib", acknowledgement = ack-nhfb, xxjournal = "Publ. Math. Inst. Hung. Acad. Sci.", } @Article{Higuti:1961:SSR, author = "Isao Higuti", title = "A statistical study of random packing of unequal spheres", journal = j-ANN-INST-STAT-MATH-TOKYO, volume = "12", number = "3", pages = "257--271", month = oct, year = "1961", CODEN = "AISXAD", DOI = "https://doi.org/10.1007/BF01728935", ISSN = "0020-3157 (print), 1572-9052 (electronic)", ISSN-L = "0020-3157", bibdate = "Sat Jan 31 16:59:46 MST 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/anninststatmath.bib; http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://link.springer.com/article/10.1007/BF01728935", acknowledgement = ack-nhfb, fjournal = "Annals of the Institute of Statistical Mathematics", journal-URL = "http://link.springer.com/journal/10463", } @Article{Groemer:1962:BDG, author = "H. Groemer", title = "{{\"U}ber die dichteste gitterf{\"o}rmige Lagerung kongruenter Tetraeder}. ({German}) [On the densest lattice storage of congruent tetrahedra]", journal = j-MONAT-MATH, volume = "66", number = "??", pages = "12--15", month = "????", year = "1962", CODEN = "MNMTA2", ISSN = "0026-9255 (print), 1436-5081 (electronic)", ISSN-L = "0026-9255", MRclass = "52.45 (10.25)", MRnumber = "0139077 (25 \#2516)", MRreviewer = "T. Estermann", bibdate = "Wed Jan 09 09:25:04 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "", acknowledgement = ack-nhfb, fjournal = "Monatshefte f{\"u}r Mathematik", language = "German", } @Article{Leech:1964:SSP, author = "John Leech", title = "Some sphere packings in higher space", journal = j-CAN-J-MATH, volume = "16", number = "??", pages = "657--682", month = "????", year = "1964", CODEN = "CJMAAB", DOI = "https://doi.org/10.4153/CJM-1964-065-1", ISSN = "0008-414X (print), 1496-4279 (electronic)", ISSN-L = "0008-414X", bibdate = "Sat Sep 10 15:38:33 MDT 2011", bibsource = "http://cms.math.ca/cjm/v16/; http://www.math.utah.edu/pub/tex/bib/canjmath1960.bib; http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Canadian Journal of Mathematics = Journal canadien de math{\'e}matiques", journal-URL = "http://cms.math.ca/cjm/", } @Book{Rogers:1964:PC, author = "C. A. (Claude Ambrose) Rogers", title = "Packing and covering", volume = "54", publisher = pub-CAMBRIDGE, address = pub-CAMBRIDGE:adr, pages = "viii + 111", year = "1964", LCCN = "QA248 .R64", MRclass = "52.45", MRnumber = "0172183 (30 \#2405)", MRreviewer = "R. P. Bambah", bibdate = "Wed Jan 9 09:48:56 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib; z3950.loc.gov:7090/Voyager", series = "Cambridge tracts in mathematics and mathematical physics", acknowledgement = ack-nhfb, subject = "Generalized spaces", } @Book{Kepler:1966:SCS, editor = "Johannes Kepler", title = "The six-cornered snowflake", publisher = pub-CLARENDON, address = pub-CLARENDON:adr, pages = "xvi + 75", year = "1966", ISBN = "020", ISBN-13 = "020", LCCN = "QC929.S7 K413", bibdate = "Wed Jan 9 08:38:36 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib; z3950.loc.gov:7090/Voyager", note = "Translated from the Latin by Colin Hardie, with essays by L. L. Whyte and B. F. J. Mason. Text in Latin with English translation on facing page. Notes and commentaries in English. Originally published in \cite{Kepler:1611:SSN}.", price = "21/-", acknowledgement = ack-nhfb, author-dates = "1571--1630", subject = "Snowflakes; Early works to 1800", } @Article{Leech:1967:FDN, author = "John Leech", title = "Five dimensional non-lattice sphere packings", journal = j-CAN-MATH-BULL, volume = "10", number = "??", pages = "387--394", month = "????", year = "1967", CODEN = "CMBUA3", DOI = "https://doi.org/10.4153/CMB-1967-037-5", ISSN = "0008-4395 (print), 1496-4287 (electronic)", ISSN-L = "0008-4395", bibdate = "Thu Sep 8 10:03:24 MDT 2011", bibsource = "http://cms.math.ca/cmb/v10/; http://www.math.utah.edu/pub/tex/bib/canmathbull.bib; http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Canadian mathematical bulletin = Bulletin canadien de math{\'e}matiques", journal-URL = "http://cms.math.ca/cmb/", } @Article{Leech:1967:NSP, author = "John Leech", title = "Notes on sphere packings", journal = j-CAN-J-MATH, volume = "19", number = "??", pages = "251--267", month = "????", year = "1967", CODEN = "CJMAAB", DOI = "https://doi.org/10.4153/CJM-1967-017-0", ISSN = "0008-414X (print), 1496-4279 (electronic)", ISSN-L = "0008-414X", bibdate = "Sat Sep 10 15:38:36 MDT 2011", bibsource = "http://cms.math.ca/cjm/v19/; http://www.math.utah.edu/pub/tex/bib/canjmath1960.bib; http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Canadian Journal of Mathematics = Journal canadien de math{\'e}matiques", journal-URL = "http://cms.math.ca/cjm/", } @InProceedings{Solomon:1967:RPD, author = "H. Solomon", editor = "Lucien M. {Le Cam} and Jerzy Neyman", booktitle = "{Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability. Volume III. Physical Sciences}", title = "Random packing density", volume = "3", publisher = pub-U-CAL, address = pub-U-CAL:adr, pages = "119--134", year = "1967", LCCN = "QA276 .B4 v.3", bibdate = "Wed Feb 29 21:12:50 2012", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib", note = "Held at the Statistical Laboratory, University of California, June 21--July 18, 1965 and December 27, 1965--January 7, 1966.", URL = "http://projecteuclid.org/euclid.bsmsp/1200513624", acknowledgement = ack-nhfb, } @Book{Rogers:1968:UPR, author = "Claude Ambrose Rogers", title = "{Ukladki i pokrytije}. ({Russian}) [Laying and covering]", publisher = "????", address = "????", pages = "134", year = "1968", LCCN = "QA611 .R6317", bibdate = "Wed Jan 9 09:48:56 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib; z3950.loc.gov:7090/Voyager", acknowledgement = ack-nhfb, author-dates = "1920--", language = "Russian", subject = "Generalized spaces", xxnote = "Find original Russian title and check likely-garbled transliteration.", } @Article{Leech:1969:SSD, author = "John Leech", title = "Six and seven dimensional non-lattice sphere packings", journal = j-CAN-MATH-BULL, volume = "12", number = "??", pages = "151--156", month = "????", year = "1969", CODEN = "CMBUA3", DOI = "https://doi.org/10.4153/CMB-1969-014-1", ISSN = "0008-4395 (print), 1496-4287 (electronic)", ISSN-L = "0008-4395", bibdate = "Thu Sep 8 10:03:37 MDT 2011", bibsource = "http://cms.math.ca/cmb/v12/; http://www.math.utah.edu/pub/tex/bib/canmathbull.bib; http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Canadian mathematical bulletin = Bulletin canadien de math{\'e}matiques", journal-URL = "http://cms.math.ca/cmb/", } @Article{Blaisdell:1970:RSP, author = "B. E. Blaisdell and H. Solomon", title = "On random sequential packing in the plane and a conjecture of {Pal{\'a}sti}", journal = j-J-APPL-PROBAB, volume = "7", number = "3", pages = "667--689", month = dec, year = "1970", CODEN = "JPRBAM", ISSN = "0021-9002 (print), 1475-6072 (electronic)", ISSN-L = "0021-9002", bibdate = "Wed Feb 29 21:17:55 2012", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib", URL = "http://www.jstor.org/stable/3211946", acknowledgement = ack-nhfb, fjournal = "Journal of Applied Probability", journal-URL = "http://www.jstor.org/journals/00219002.html", } @Article{Hoylman:1970:DLP, author = "D. J. Hoylman", title = "The densest lattice packing of tetrahedra", journal = j-BULL-AMS, volume = "76", number = "??", pages = "135--137", month = "????", year = "1970", CODEN = "BAMOAD", ISSN = "0002-9904 (print), 1936-881X (electronic)", ISSN-L = "0002-9904", bibdate = "Fri Aug 14 17:28:05 2009", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Bulletin of the American Mathematical Society", journal-URL = "http://www.ams.org/journals/bull/all_issues.html", } @Article{Leech:1971:SPE, author = "John Leech and N. J. A. Sloane", title = "Sphere packings and error-correcting codes", journal = j-CAN-J-MATH, volume = "23", number = "??", pages = "718--745", month = "????", year = "1971", CODEN = "CJMAAB", DOI = "https://doi.org/10.4153/CJM-1971-081-3", ISSN = "0008-414X (print), 1496-4279 (electronic)", ISSN-L = "0008-414X", bibdate = "Sat Sep 10 15:38:42 MDT 2011", bibsource = "http://cms.math.ca/cjm/v23/; http://www.math.utah.edu/pub/tex/bib/canjmath1970.bib; http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Canadian Journal of Mathematics = Journal canadien de math{\'e}matiques", journal-URL = "http://cms.math.ca/cjm/", } @Book{Toth:1972:LEK, author = "L. Fejes T{\'o}th", title = "{Lagerungen in der Ebene, auf der Kugeln und im Raum}. ({German}) [{Ball} deposits in the plane and in space]", volume = "65", publisher = pub-SV, address = pub-SV:adr, edition = "Second", pages = "xi + 238", year = "1972", ISBN = "3-540-05477-4", ISBN-13 = "978-3-540-05477-1", LCCN = "", MRclass = "52.0X", MRnumber = "0057566 (15,248b)", MRreviewer = "W. O. Moser", bibdate = "Wed Jan 09 09:16:25 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", note = "See also first edition \cite{Toth:1953:LEK}.", series = "{Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Ber{\"u}cksichtigung der Anwendungsgebiete}", acknowledgement = ack-nhfb, language = "German", } @Article{Akeda:1975:NTP, author = "Y. Akeda and M. Hori", title = "Numerical test of {Pal{\'a}sti}'s conjecture on two-dimensional random packing density", journal = j-NATURE, volume = "254", number = "5498", pages = "318--319", day = "27", month = mar, year = "1975", CODEN = "NATUAS", DOI = "https://doi.org/10.1038/254318a0", ISSN = "0028-0836 (print), 1476-4687 (electronic)", ISSN-L = "0028-0836", bibdate = "Wed Feb 29 21:19:24 2012", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib", URL = "http://www.nature.com/nature/journal/v254/n5498/pdf/254318a0.pdf", abstract = "Random packing and random space-filling problems have received frequent attention. In one dimension, this topic is usually called the car-parking problem. Pal{\'a}sti generalised the parking problem to two dimensions and conjectured that the random packing density in the two-dimensional case is equal to the square of that in the one-dimensional case. Here we provide a thorough check on her conjecture and confirm its validity.", acknowledgement = ack-nhfb, fjournal = "Nature", journal-URL = "http://www.nature.com/nature/archive/", } @Article{Gilbert:1979:PPT, author = "E. N. Gilbert", title = "The Packing Problem for Twisted Pairs", journal = j-BELL-SYST-TECH-J, volume = "58", number = "10", pages = "2143--2162", month = dec, year = "1979", CODEN = "BSTJAN", ISSN = "0005-8580", bibdate = "Tue Nov 9 11:15:56 MST 2010", bibsource = "http://bstj.bell-labs.com/oldfiles/year.1979/BSTJ.1979.5810.html; http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://bstj.bell-labs.com/BSTJ/images/Vol58/bstj58-10-2143.pdf", abstract = "When wires are packed together in a bundle, as in a cable or on a shelf of a main distribution frame, the packing fraction $f$ is the fraction of cross-sectional area of the bundle occupied by wire. With wires all the same radius, packing fractions as high as $ 0.90690$ can be achieved. However, when the wires are pairs that have been twisted to avoid crosstalk, the packing fraction is much smaller. The largest obtainable packing fraction depends on other properties of the packing. For example, with pairs twisted by machine, all pairs twist at the same rate, and that influences the packing fraction. Several packing problems are considered, but most attention is given to a particularly regular kind of packing in which pairs twist about straight parallel axes located in a lattice arrangement. The densest lattice packing has packing fraction $ 0.56767$. The densest lattice is a complicated one in which each wire touches $ 10$ wires belonging to $6$ other pairs. The numbers $ 10$ and $6$ cannot be increased even with nonlattice packings of pairs with straight parallel axes. These other packings are also conjectured to have packing fractions less than $ 0.56767$, although only $ f < 0.62240$ is proved.", acknowledgement = ack-nhfb, fjournal = "The Bell System Technical Journal", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1538-7305/issues/", } @Book{Schrijver:1979:PCC, author = "A. Schrijver", title = "Packing and covering in combinatorics", volume = "106", publisher = "Mathematisch Centrum", address = "Amsterdam, The Netherlands", pages = "313", year = "1979", ISBN = "90-6196-180-7", ISBN-13 = "978-90-6196-180-2", LCCN = "QA166.7 .P32", bibdate = "Wed Jan 9 09:48:56 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib; z3950.loc.gov:7090/Voyager", series = "Mathematical centre tracts", acknowledgement = ack-nhfb, subject = "Combinatorial packing and covering", } @Article{Tanemura:1979:RCP, author = "M. Tanemura", title = "On random complete packing by discs", journal = j-ANN-INST-STAT-MATH-TOKYO, volume = "31", number = "Part B", pages = "351--365", month = "????", year = "1979", CODEN = "AISXAD", ISSN = "0020-3157 (print), 1572-9052 (electronic)", ISSN-L = "0020-3157", bibdate = "Wed Feb 29 21:24:06 2012", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib", acknowledgement = ack-nhfb, fjournal = "Annals of the Institute of Statistical Mathematics (Tokyo)", } @PhdThesis{Thompson:1979:ECC, author = "Thomas Miller Thompson", title = "From error-correcting codes through sphere packings to simple groups", type = "Thesis ({Ph.D.})", school = "Department of Mathematics, University of California, Davis", address = "Davis, CA, USA", pages = "250", year = "1979", LCCN = "LD781.D5j 1979 T566", bibdate = "Thu Aug 13 18:08:02 2009", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, keywords = "dissertations, academic --- University of California, Davis --- mathematics", } @Article{Senechal:1981:WTF, author = "Marjorie Senechal", title = "Which tetrahedra fill space?", journal = j-MATH-MAG, volume = "54", number = "5", pages = "227--243", month = "????", year = "1981", CODEN = "MAMGA8", DOI = "https://doi.org/10.2307/2689983", ISSN = "0025-570X", MRclass = "52A45 (01A05 05B45 52-03 52A25)", MRnumber = "644075 (83h:52020)", MRreviewer = "A. Florian", bibdate = "Wed Jan 09 09:54:58 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Mathematics Magazine", journal-URL = "http://www.maa.org/pubs/mathmag.html", } @Article{Sloane:1981:TSP, author = "Neil J. A. Sloane", title = "Tables of sphere packings and spherical codes", journal = j-IEEE-TRANS-INF-THEORY, volume = "27", number = "3", pages = "327--338", month = mar, year = "1981", CODEN = "IETTAW", DOI = "https://doi.org/10.1109/TIT.1981.1056351", ISSN = "0018-9448 (print), 1557-9654 (electronic)", ISSN-L = "0018-9448", bibdate = "Thu Aug 13 17:46:36 2009", bibsource = "DBLP; http://dblp.uni-trier.de/db/journals/tit/tit27.html#Sloane81; http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Information Theory", } @Book{Weyl:1982:S, author = "Hermann Weyl", title = "Symmetry", publisher = pub-PRINCETON, address = pub-PRINCETON:adr, pages = "168", year = "1982", ISBN = "0-691-02374-3", ISBN-13 = "978-0-691-02374-8", LCCN = "", bibdate = "Wed Jan 09 15:33:17 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, author-dates = "1885--1955", remark = "Reprint of \cite{Weyl:1952:S}.", } @Article{Bagchi:1983:TAA, author = "Bagchi and Cerjan and Stuart A. Rice", title = "Theoretical analysis of the achievement of random close packing of hard spheres and a conjecture on spinodal decomposition", journal = j-PHYS-REV-B-CMMP, volume = "28", pages = "6411", year = "1983", CODEN = "PRBMDO", ISSN = "1098-0121", ISSN-L = "1098-0121", bibdate = "Thu Aug 13 18:08:02 2009", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Physical Review B: Condensed Matter and Materials Physics", journal-URL = "http://prb.aps.org/browse", keywords = "spin, theoretical", } @Book{Thompson:1983:ECC, author = "Thomas M. Thompson", title = "From error-correcting codes through sphere packings to simple groups", volume = "21", publisher = pub-MATH-ASSOC-AMER, address = pub-MATH-ASSOC-AMER:adr, pages = "xiv + 228", year = "1983", ISBN = "0-88385-023-0", ISBN-13 = "978-0-88385-023-7", LCCN = "QA268 .T56 1983", bibdate = "Thu Aug 13 18:08:02 2009", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", series = "Carus mathematical monographs", acknowledgement = ack-nhfb, keywords = "error-correcting codes (information theory); finite simple groups", } @Article{Sloane:1984:PS, author = "N. J. A. Sloane", title = "The packing of spheres", journal = j-SCI-AMER, volume = "250", number = "1", pages = "92--101", month = jan, year = "1984", CODEN = "SCAMAC", ISSN = "0036-8733 (print), 1946-7087 (electronic)", ISSN-L = "0036-8733", bibdate = "Thu Aug 13 18:08:02 2009", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, classification = "A0550 (Lattice theory and statistics; Ising problems)", fjournal = "Scientific American", journal-URL = "http://www.nature.com/scientificamerican", keywords = "digital signalling; lattice theory and statistics; sphere packing", treatment = "T Theoretical or Mathematical", } @Article{Jodrey:1985:CSC, author = "W. S. Jodrey and E. M. Tory", title = "Computer simulation of close random packing of equal spheres", journal = j-PHYS-REV-A, volume = "32", number = "4", pages = "2347--2351", month = oct, year = "1985", CODEN = "PLRAAN", DOI = "https://doi.org/10.1103/PhysRevA.32.2347", ISSN = "1050-2947 (print), 1094-1622, 1538-4446, 1538-4519", ISSN-L = "1050-2947", bibdate = "Thu Aug 13 17:43:15 2009", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib", URL = "http://link.aps.org/doi/10.1103/PhysRevA.32.2347", acknowledgement = ack-nhfb, fjournal = "Physical Review A (Atomic, Molecular, and Optical Physics)", journal-URL = "http://pra.aps.org/browse", } @InProceedings{Marsaglia:1985:CVR, author = "George Marsaglia", title = "A Current View of Random Number Generators", crossref = "Billard:1985:CSS", pages = "3--10", year = "1985", bibdate = "Thu Dec 18 13:39:28 2003", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/kepler.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib", URL = "http://stat.fsu.edu/pub/diehard/; http://www.evensen.org/marsaglia/keynote.ps", acknowledgement = ack-nhfb, remark = "This paper introduces the Parking Lot test used in the Diehard Battery test suite.", } @Book{Conway:1988:SPL, author = "John Horton Conway and N. J. A. (Neil James Alexander) Sloane and others", title = "Sphere Packings, Lattices and Groups", volume = "290", publisher = pub-SV, address = pub-SV:adr, pages = "xxvii + 663", year = "1988", ISBN = "0-387-96617-X", ISBN-13 = "978-0-387-96617-5", LCCN = "QA166.7 .C66 1988; QA3 .G75; QA3 .G7; QA166.7 .C66 1988", bibdate = "Thu Aug 13 18:08:02 2009", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib; melvyl.cdlib.org:210/CDL90", price = "US\$45.00", series = "{Grundlehren der mathematischen Wissenschaften}", acknowledgement = ack-nhfb, keywords = "book; c1988; c198x; c19xx; codes; constant; constants; G1; G2; group; John Conway; laminated; lattice; packing; polytope; polytopes; quantization; text; zz06099", subject = "Combinatorial packing and covering; Sphere; Lattice theory; Finite groups", } @Article{Toth:1989:FSP, author = "G{\'a}bor Fejes T{\'o}th and Peter Gritzmann and J. M. Wills", title = "Finite Sphere Packing and Sphere Covering", journal = j-DISCRETE-COMPUT-GEOM, volume = "4", number = "1", pages = "19--40", month = dec, year = "1989", CODEN = "DCGEER", ISSN = "0179-5376 (print), 1432-0444 (electronic)", ISSN-L = "0179-5376", bibdate = "Thu Aug 13 17:47:55 2009", bibsource = "DBLP; http://dblp.uni-trier.de/db/journals/dcg/dcg4.html#TothGW89; http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Discrete and Computational Geometry", journal-URL = "http://link.springer.com/journal/454", } @Book{Weyl:1989:S, author = "Hermann Weyl", title = "Symmetry", publisher = pub-PRINCETON, address = pub-PRINCETON:adr, pages = "168", year = "1989", ISBN = "0-691-02374-3", ISBN-13 = "978-0-691-02374-8", LCCN = "", bibdate = "Wed Jan 09 15:33:17 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, author-dates = "1885--1955", remark = "Reprint of \cite{Weyl:1952:S}.", } @Article{Kuperberg:1990:DLP, author = "G. Kuperberg and W. Kuperberg", title = "Double-lattice packings of convex bodies in the plane", journal = j-DISCRETE-COMPUT-GEOM, volume = "5", number = "1", pages = "389--397", month = dec, year = "1990", CODEN = "DCGEER", DOI = "https://doi.org/10.1007/BF02187800", ISSN = "0179-5376 (print), 1432-0444 (electronic)", ISSN-L = "0179-5376", MRclass = "52A45", MRnumber = "1043721 (90k:52027)", MRreviewer = "R. Blind", bibdate = "Wed Jan 09 09:42:32 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Discrete and Computational Geometry", journal-URL = "http://link.springer.com/journal/454", } @Article{Bezdek:1991:MDS, author = "Andr{\'a}s Bezdek and Wlodzimierz Kuperberg and E. Makai", title = "Maximum Density Space Packing with Parallel Strings of Spheres", journal = j-DISCRETE-COMPUT-GEOM, volume = "6", number = "1", pages = "277--283", month = dec, year = "1991", CODEN = "DCGEER", ISSN = "0179-5376 (print), 1432-0444 (electronic)", ISSN-L = "0179-5376", bibdate = "Thu Aug 13 17:49:09 2009", bibsource = "DBLP; http://dblp.uni-trier.de/db/journals/dcg/dcg6.html#BezdekKM91; http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Discrete and Computational Geometry", journal-URL = "http://link.springer.com/journal/454", } @Article{Clare:1991:OPC, author = "B. W. Clare and D. L. Kepert", title = "The optimal packing of circles on a sphere", journal = j-J-MATH-CHEM, volume = "6", number = "1", pages = "325--349", month = dec, year = "1991", CODEN = "JMCHEG", DOI = "https://doi.org/10.1007/BF01192589", ISSN = "0259-9791 (print), 1572-8897 (electronic)", ISSN-L = "0259-9791", bibdate = "Fri Apr 10 06:38:59 MDT 2015", bibsource = "http://link.springer.com/journal/10910/6/1; http://www.math.utah.edu/pub/tex/bib/jmathchem.bib; http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://link.springer.com/article/10.1007/BF01192589", acknowledgement = ack-nhfb, fjournal = "Journal of Mathematical Chemistry", journal-URL = "http://link.springer.com/journal/10910", journalabr = "J. Math. Chem.", } @TechReport{Haussler:1991:SPN, author = "David Haussler", title = "Sphere Packing numbers for subsets of the {Boolean} {$N$}-cube with bounded {Vapnik--Chervonenkis} dimension", number = "UCSC-CRL-91-41", institution = "University of California, Santa Cruz", address = "Santa Cruz, CA, USA", pages = "14", month = oct, year = "1991", bibdate = "Thu Aug 13 18:07:58 2009", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "ftp://ftp.cse.ucsc.edu/pub/tr/ucsc-crl-91-41.ps.Z; mailto:rnalib@ftp.cs.ucsc.edu", abstract = "Let V contained in $ \lbrace $0,1$ \rbrace ${\^{}}n have Vapnk-Chervonenkis dimension d. Let M(k/n,V) denote the cardinality of the largest W contained in V such that any two distinct vectors in W differ on at least k indices. We show that M(k/n,V) $<$ = (cn/(k+d)){\^{}}d for some constant c. This improves on the previous best result of ((cn/k)log(n/k)){\^{}}d. This new result has applications in the theory of empirical processes.", acknowledgement = ack-nhfb, } @InProceedings{Max:1991:CAS, author = "Nelson L. Max", booktitle = "IEEE Visualization", title = "Computer Assisted Sphere Packing in Higher Dimensions", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "102--108", year = "1991", bibdate = "Thu Aug 13 17:48:35 2009", bibsource = "DBLP; http://dblp.uni-trier.de/db/conf/visualization/visualization1991.html#Max91; http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, } @InCollection{Coffman:1992:PAP, author = "E. G. {Coffman, Jr.} and D. S. Johnson and P. W. Shor", title = "Probabilistic Analysis of Packing and Related Partitioning Problems", crossref = "Steele:1992:PA", chapter = "7", pages = "87--108", year = "1992", bibdate = "Wed Jan 09 18:20:11 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, } @Article{Hales:1992:SPP, author = "Thomas C. Hales", title = "The sphere packing problem", journal = j-J-COMPUT-APPL-MATH, volume = "44", number = "1", pages = "41--76", day = "9", month = dec, year = "1992", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 12:20:56 MST 2017", bibsource = "http://www.math.utah.edu/pub/tex/bib/jcomputapplmath1990.bib; http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://www.sciencedirect.com/science/article/pii/037704279290052Y", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @InCollection{Berger:1993:PC, author = "Marcel Berger", booktitle = "Differential geometry and topology ({Alghero}, 1992)", title = "Les paquets de cercles. ({French}) [{Circle} packing]", publisher = pub-WORLD-SCI, address = pub-WORLD-SCI:adr, pages = "23--64", year = "1993", MRclass = "52C15 (52C17)", MRnumber = "1339286 (96e:52043)", MRreviewer = "Igor Rivin", bibdate = "Thu May 22 16:37:35 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, keywords = "Kepler sphere-packing conjecture", language = "French", } @Book{Conway:1993:SPL, author = "John Horton Conway and N. J. A. (Neil James Alexander) Sloane and others", title = "Sphere packings, lattices, and groups", volume = "290", publisher = pub-SV, address = pub-SV:adr, edition = "Second", pages = "xliii + 679", year = "1993", ISBN = "0-387-97912-3, 3-540-97912-3", ISBN-13 = "978-0-387-97912-0, 978-3-540-97912-8", LCCN = "QA166.7 .C66 1993", bibdate = "Thu Aug 13 17:40:44 MDT 2009", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib; z3950.loc.gov:7090/Voyager", series = "{Grundlehren der mathematischen Wissenschaften}", acknowledgement = ack-nhfb, subject = "combinatorial packing and covering; sphere packings; lattice theory; finite groups", } @Article{Hales:1993:RDS, author = "Thomas C. Hales", title = "Remarks on the density of sphere packings in three dimensions", journal = j-COMBINATORICA, volume = "13", number = "2", pages = "181--197", year = "1993", CODEN = "COMBDI", ISSN = "0209-9683 (print), 1439-6912 (electronic)", ISSN-L = "0209-9683", bibdate = "Thu Aug 13 17:49:58 2009", bibsource = "DBLP; http://dblp.uni-trier.de/db/journals/combinatorica/combinatorica13.html#Hales93; http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Combinatorica", journal-URL = "http://link.springer.com/journal/volumesAndIssues/493", } @InCollection{Hsiang:1993:GS, author = "Wu-Yi Hsiang", booktitle = "Differential geometry ({Shanghai}, 1991)", title = "The geometry of spheres", publisher = pub-WORLD-SCI, address = pub-WORLD-SCI:adr, pages = "92--107", year = "1993", MRclass = "52C17", MRnumber = "1341602 (97f:52028)", MRreviewer = "J. B{\"o}hm", bibdate = "Thu May 22 16:37:35 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, keywords = "Kepler sphere-packing conjecture", } @Article{Hsiang:1993:SPPa, author = "Wu-Yi Hsiang", title = "On the sphere packing problem and the proof of {Kepler's Conjecture}", journal = j-INT-J-MATH, volume = "4", number = "5", pages = "739--831", year = "1993", CODEN = "????", DOI = "https://doi.org/10.1142/S0129167X93000364", ISSN = "0129-167X", MRclass = "52C17", MRnumber = "1245351 (95g:52032)", MRreviewer = "G{\'a}bor Fejes T{\'o}th", bibdate = "Thu May 22 16:37:35 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "International Journal of Mathematics", keywords = "Kepler sphere-packing conjecture", } @InCollection{Hsiang:1993:SPPb, author = "Wu-Yi Hsiang", booktitle = "Differential geometry and topology ({Alghero, 1992})", title = "On the sphere packing problem and the proof of {Kepler's Conjecture}", publisher = pub-WORLD-SCI, address = pub-WORLD-SCI:adr, pages = "117--127", year = "1993", MRclass = "52C17", MRnumber = "1339290 (96f:52028)", MRreviewer = "Uwe Schnell", bibdate = "Thu May 22 16:37:35 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, keywords = "Kepler sphere-packing conjecture", } @Article{Kunyavskii:1992:SPC, author = "Boris {\`E}. Kunyavskii", title = "Sphere packings centered at {$S$}-units of algebraic tori", journal = j-LECT-NOTES-MATH, volume = "1518", pages = "108--121", year = "1992", CODEN = "LNMAA2", DOI = "https://doi.org/10.1007/BFb0087996", ISBN = "3-540-55651-6 (print), 3-540-47267-3 (e-book)", ISBN-13 = "978-3-540-55651-0 (print), 978-3-540-47267-4 (e-book)", ISSN = "0075-8434 (print), 1617-9692 (electronic)", ISSN-L = "0075-8434", MRclass = "11H31 (11T71 52C07)", MRnumber = "1186419 (93i:11077)", MRreviewer = "Simon N. Litsyn", bibdate = "Fri May 9 19:07:24 MDT 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib; http://www.math.utah.edu/pub/tex/bib/lnm1990.bib", URL = "http://link.springer.com/chapter/10.1007/BFb0087996/", acknowledgement = ack-nhfb, book-DOI = "https://doi.org/10.1007/BFb0087986", book-URL = "http://www.springerlink.com/content/978-3-540-47267-4", fjournal = "Lecture Notes in Mathematics", journal-URL = "http://link.springer.com/bookseries/304", } @Article{Mathe:1993:KG, author = "Katalin Bogn{\'a}r M{\'a}th{\'e}", title = "{{\"U}ber Kugelsysteme unter Ger{\"a}umigkeitsbedingungen}. ({German}) [{On} sphere systems under spaciousness conditions]", journal = "Studia Sci. Math. Hungar.", volume = "28", number = "3-4", pages = "431--445", year = "1993", CODEN = "SSMHAX", ISSN = "0081-6906", MRclass = "52C17", MRnumber = "1266827 (95b:52032)", MRreviewer = "Uwe Schnell", bibdate = "Thu May 22 16:37:35 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Studia Scientiarum Mathematicarum Hungarica. A Quarterly of the Hungarian Academy of Sciences", keywords = "Kepler sphere-packing conjecture", language = "German", } @Article{Muder:1993:NBL, author = "Douglas J. Muder", title = "A New Bound on the Local Density of Sphere Packings", journal = j-DISCRETE-COMPUT-GEOM, volume = "10", number = "1", pages = "351--375", month = dec, year = "1993", CODEN = "DCGEER", DOI = "https://doi.org/10.1007/BF02573984", ISSN = "0179-5376 (print), 1432-0444 (electronic)", ISSN-L = "0179-5376", MRclass = "52C17", MRnumber = "1243334 (94h:52041)", MRreviewer = "N. J. A. Sloane", bibdate = "Thu May 22 16:37:35 2014", bibsource = "DBLP; http://dblp.uni-trier.de/db/journals/dcg/dcg10.html#Muder93; http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Discrete and Computational Geometry", journal-URL = "http://link.springer.com/journal/454", keywords = "Kepler sphere-packing conjecture", } @Article{Szirmai:1993:OSP, author = "Jen{\H{o}} Szirmai", title = "Optimal sphere packings under some space groups", journal = "Alkalmaz. Mat. Lapok", volume = "17", number = "1--2", pages = "87--99 (1994)", year = "1993", ISSN = "0133-3399", MRclass = "52C17", MRnumber = "1336186 (96g:52038)", MRreviewer = "B{\'e}la Uhrin", bibdate = "Thu Apr 12 18:15:04 2012", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Alkalmazott Matematikai Lapok", } @InCollection{Toth:1993:PCC, author = "G. Fejes T{\'o}th and W. Kuperberg", title = "Packing and covering with convex sets", crossref = "Gruber:1993:HCG", volume = "B", pages = "799--860", year = "1993", bibdate = "Wed Jan 09 09:09:31 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, } @Article{Hales:1994:SKC, author = "Thomas C. Hales", title = "The status of the {Kepler Conjecture}", journal = j-MATH-INTEL, volume = "16", number = "3", pages = "47--58", year = "1994", CODEN = "MAINDC", DOI = "https://doi.org/10.1007/BF03024356", ISSN = "0343-6993 (print), 1866-7414 (electronic)", ISSN-L = "0343-6993", MRclass = "52C17", MRnumber = "1281754 (95g:52033)", MRreviewer = "G{\'a}bor Fejes T{\'o}th", bibdate = "Thu May 22 16:37:35 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "The Mathematical Intelligencer", keywords = "Kepler sphere-packing conjecture", } @Article{Hsiang:1994:GS, author = "Wu-Yi Hsiang", title = "The geometry of spheres", journal = j-NORMAT, volume = "42", number = "1", pages = "1--12", year = "1994", ISSN = "0801-3500", MRclass = "52C17 (51M25)", MRnumber = "1278230 (95f:52028)", MRreviewer = "W. J. Firey", bibdate = "Thu May 22 16:37:35 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Normat. Nordisk Matematisk Tidskrift", keywords = "Kepler sphere-packing conjecture", } @Article{Zinchenko:1994:ARC, author = "Alexander Z. Zinchenko", title = "Algorithm for Random Close Packing of Spheres with Periodic Boundary Conditions", journal = j-J-COMPUT-PHYS, volume = "114", number = "2", pages = "298--307", month = oct, year = "1994", CODEN = "JCTPAH", DOI = "https://doi.org/10.1006/jcph.1994.1168", ISSN = "0021-9991 (print), 1090-2716 (electronic)", ISSN-L = "0021-9991", bibdate = "Mon Jan 2 07:54:57 MST 2012", bibsource = "http://www.math.utah.edu/pub/tex/bib/jcomputphys1990.bib; http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0021999184711685", acknowledgement = ack-nhfb, fjournal = "Journal of Computational Physics", journal-URL = "http://www.sciencedirect.com/science/journal/00219991/", } @Book{Brackenridge:1995:KND, author = "J. Bruce Brackenridge", title = "The key to {Newton}'s dynamics: the {Kepler} problem and the {Principia}: containing an {English} translation of sections 1, 2, and 3 of book one from the first (1687) edition of {Newton}'s {Mathematical Principles of Natural Philosophy}", publisher = pub-U-CAL, address = pub-U-CAL:adr, pages = "xiii + 299", year = "1995", ISBN = "0-520-20065-9 (hardcover), 0-520-20217-1 (paperback)", ISBN-13 = "978-0-520-20065-4 (hardcover), 978-0-520-20217-7 (paperback)", LCCN = "QB355 .B694 1995", bibdate = "Thu Aug 13 16:26:27 MDT 2009", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib; z3950.loc.gov:7090/Voyager", URL = "http://www.loc.gov/catdir/bios/ucal051/95032978.html; http://www.loc.gov/catdir/description/ucal041/95032978.html", acknowledgement = ack-nhfb, subject = "Kepler's laws; celestial mechanics", } @Article{Conway:1995:WAB, author = "John H. Conway and Neil J. A. Sloane", title = "What are All the Best Sphere Packings in Low Dimensions?", journal = j-DISCRETE-COMPUT-GEOM, volume = "13", number = "1", pages = "383--403", month = dec, year = "1995", CODEN = "DCGEER", ISSN = "0179-5376 (print), 1432-0444 (electronic)", ISSN-L = "0179-5376", bibdate = "Thu Aug 13 17:51:55 2009", bibsource = "DBLP; http://dblp.uni-trier.de/db/journals/dcg/dcg13.html#ConwayS95; http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Discrete and Computational Geometry", journal-URL = "http://link.springer.com/journal/454", } @Article{Haussler:1995:SPN, author = "David Haussler", title = "Sphere Packing Numbers for Subsets of the {Boolean} $n$-Cube with Bounded {Vapnik--Chervonenkis} Dimension", journal = j-J-COMB-THEORY-A, volume = "69", number = "2", pages = "217--232", year = "1995", CODEN = "JCBTA7", ISSN = "0097-3165 (print), 1096-0899 (electronic)", ISSN-L = "0097-3165", bibdate = "Thu Aug 13 17:52:38 2009", bibsource = "DBLP; http://dblp.uni-trier.de/db/journals/jct/jcta69.html#Haussler95; http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Combinatorial Theory (Series A)", } @Article{Hsiang:1995:RCH, author = "Wu-Yi Hsiang", title = "A rejoinder to {T. C. Hales}'s article: {``The status of the Kepler Conjecture'' [Math.\ Intelligencer {\bf 16} (1994), no. 3, 47--58; MR1281754 (95g:52033)]}", journal = j-MATH-INTEL, volume = "17", number = "1", pages = "35--42", year = "1995", CODEN = "MAINDC", DOI = "https://doi.org/10.1007/BF03024716", ISSN = "0343-6993 (print), 1866-7414 (electronic)", ISSN-L = "0343-6993", MRclass = "52C17", MRnumber = "1319992 (97f:52029)", bibdate = "Thu May 22 16:37:35 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "The Mathematical Intelligencer", keywords = "Kepler sphere-packing conjecture", } @Article{Zong:1996:KNT, author = "Chuanming Zong", title = "The kissing numbers of tetrahedra", journal = j-DISCRETE-COMPUT-GEOM, volume = "15", number = "3", pages = "251--264", month = mar, year = "1996", CODEN = "DCGEER", DOI = "https://doi.org/10.1007/BF02711493", ISSN = "0179-5376 (print), 1432-0444 (electronic)", ISSN-L = "0179-5376", MRclass = "11H31 (52C07 52C17)", MRnumber = "1380392 (97c:11070)", MRreviewer = "Peter M. Gruber", bibdate = "Wed Jan 09 10:08:13 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Discrete and Computational Geometry", journal-URL = "http://link.springer.com/journal/454", } @Book{Zong:1996:SPC, editor = "Chuanming Zong and James Joseph Dudziak", title = "Strange phenomena in convex and discrete geometry", publisher = pub-SV, address = pub-SV:adr, pages = "x + 158", year = "1996", DOI = "https://doi.org/10.1007/978-1-4613-8481-6", ISBN = "0-387-94734-5 (softcover)", ISBN-13 = "978-0-387-94734-1 (softcover)", LCCN = "QA639.5 .Z66 1996", MRclass = "52-01 (52-02)", MRnumber = "1416567 (97m:52001)", MRreviewer = "Peter M. Gruber", bibdate = "Wed Jan 9 09:28:14 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib; z3950.loc.gov:7090/Voyager", series = "Universitext", acknowledgement = ack-nhfb, subject = "Convex geometry; Discrete geometry", } @Article{Anonymous:1997:EDF, author = "Anonymous", title = "Erratum: {``The dimensional family approach in (hyper)sphere packing: a typological study for new patterns, structures, and interdimensional functions''}", journal = j-APPL-MATH-COMP, volume = "88", number = "1", pages = "??--??", day = "15", month = dec, year = "1997", CODEN = "AMHCBQ", DOI = "https://doi.org/10.1016/S0096-3003(97)00402-5", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", bibdate = "Tue Nov 20 21:02:58 MST 2012", bibsource = "http://www.math.utah.edu/pub/tex/bib/applmathcomput1995.bib; http://www.math.utah.edu/pub/tex/bib/kepler.bib; http://www.sciencedirect.com/science/journal/00963003", note = "See \cite{Muses:1997:DFA,Muses:1998:EDF}.", URL = "http://www.sciencedirect.com/science/article/pii/S0096300397004025", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003", } @Article{Bezdek:1997:IID, author = "K{\'a}roly Bezdek", title = "Isoperimetric inequalities and the dodecahedral conjecture", journal = j-INT-J-MATH, volume = "8", number = "6", pages = "759--780", year = "1997", DOI = "https://doi.org/10.1142/S0129167X9700038X", ISSN = "0129-167X", MRclass = "52C17 (05B40 52B60)", MRnumber = "1470853 (98k:52043)", MRreviewer = "Salvatore Vassallo", bibdate = "Thu May 22 16:37:35 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "International Journal of Mathematics", keywords = "Kepler sphere-packing conjecture", } @Article{Hales:1997:SPa, author = "Thomas C. Hales", title = "Sphere Packings, {I}", journal = j-DISCRETE-COMPUT-GEOM, volume = "17", number = "1", pages = "1--51", month = jan, year = "1997", CODEN = "DCGEER", DOI = "https://doi.org/10.1007/BF02770863", ISSN = "0179-5376 (print), 1432-0444 (electronic)", ISSN-L = "0179-5376", MRclass = "52C17", MRnumber = "1418278 (97k:52025)", MRreviewer = "Uwe Schnell", bibdate = "Thu May 22 16:37:35 2014", bibsource = "DBLP; http://dblp.uni-trier.de/db/journals/dcg/dcg17.html#Hales97a; http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://link.springer.de/link/service/journals/00454/bibs/17n1p1.html", acknowledgement = ack-nhfb, fjournal = "Discrete and Computational Geometry", journal-URL = "http://link.springer.com/journal/454", keywords = "Kepler sphere-packing conjecture", } @Article{Hales:1997:SPb, author = "Thomas C. Hales", title = "Sphere Packings, {II}", journal = j-DISCRETE-COMPUT-GEOM, volume = "18", number = "2", pages = "135--149", month = sep, year = "1997", CODEN = "DCGEER", DOI = "https://doi.org/10.1007/PL00009312", ISSN = "0179-5376 (print), 1432-0444 (electronic)", ISSN-L = "0179-5376", MRclass = "52C17", MRnumber = "1455511 (98h:52033)", MRreviewer = "Uwe Schnell", bibdate = "Thu Aug 13 17:55:00 2009", bibsource = "DBLP; http://dblp.uni-trier.de/db/journals/dcg/dcg18.html#Hales97; http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://link.springer.de/link/service/journals/00454/bibs/18n2p125.html", acknowledgement = ack-nhfb, fjournal = "Discrete and Computational Geometry", journal-URL = "http://link.springer.com/journal/454", } @Article{Hales:1997:SPI, author = "T. C. Hales", title = "Sphere packings. {II}", journal = j-DISCRETE-COMPUT-GEOM, volume = "18", number = "2", pages = "135--149", year = "1997", CODEN = "DCGEER", DOI = "https://doi.org/10.1007/PL00009312", ISSN = "0179-5376 (print), 1432-0444 (electronic)", ISSN-L = "0179-5376", MRclass = "52C17", MRnumber = "1455511 (98h:52033)", MRreviewer = "Uwe Schnell", bibdate = "Thu May 22 16:37:35 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Discrete and Computational Geometry", journal-URL = "http://link.springer.com/journal/454", keywords = "Kepler sphere-packing conjecture", } @InCollection{Leppmeier:1997:KWT, author = "Max Leppmeier", booktitle = "{{\"U}berblicke Mathematik 1996/97}. ({German}) [{Views} of Mathematics 1996/97]", title = "{Kugelpackungen und Wurstkatastrophen oder Zur Theorie der finiten und infiniten Packungen}. ({German}) [{Sphere packings and sausage} disasters or {On} the theory of finite and infinite packings]", publisher = pub-VIEWEG, address = pub-VIEWEG:adr, pages = "96--110", year = "1997", MRclass = "52C07 (52C17)", MRnumber = "1644722 (99i:52018)", MRreviewer = "Konrad J. Swanepoel", bibdate = "Thu May 22 16:37:35 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", series = "{\"U}berbl. Math.", acknowledgement = ack-nhfb, keywords = "Kepler sphere-packing conjecture", language = "German", } @Article{Muses:1997:DFA, author = "C. Mus{\`e}s", title = "The dimensional family approach in (hyper)sphere packing: a typological study of new patterns, structures, and interdimensional functions", journal = j-APPL-MATH-COMP, volume = "88", number = "1", pages = "1--26", day = "15", month = dec, year = "1997", CODEN = "AMHCBQ", DOI = "https://doi.org/10.1016/S0096-3003(97)00004-0", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", bibdate = "Tue Nov 20 21:02:58 MST 2012", bibsource = "http://www.math.utah.edu/pub/tex/bib/applmathcomput1995.bib; http://www.math.utah.edu/pub/tex/bib/kepler.bib; http://www.sciencedirect.com/science/journal/00963003", note = "See errata \cite{Anonymous:1997:EDF,Muses:1998:EDF}.", URL = "http://www.sciencedirect.com/science/article/pii/S0096300397000040", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003", } @Article{Betke:1998:FPS, author = "Ulrich Betke and Martin Henk", title = "Finite Packings of Spheres", journal = j-DISCRETE-COMPUT-GEOM, volume = "19", number = "2", pages = "197--227", month = feb, year = "1998", CODEN = "DCGEER", ISSN = "0179-5376 (print), 1432-0444 (electronic)", ISSN-L = "0179-5376", bibdate = "Thu Aug 13 18:08:01 2009", bibsource = "DBLP; http://dblp.uni-trier.de/db/journals/dcg/dcg19.html#BetkeH98; http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://link.springer.de/link/service/journals/00454/bibs/19n2p197.html", acknowledgement = ack-nhfb, fjournal = "Discrete and Computational Geometry", journal-URL = "http://link.springer.com/journal/454", } @Article{Bezdek:1998:FUS, author = "Andr{\'a}s Bezdek and K{\'a}roly Bezdek and Robert Connelly", title = "Finite and Uniform Stability of Sphere Packings", journal = j-DISCRETE-COMPUT-GEOM, volume = "20", number = "1", pages = "111--130", month = jul, year = "1998", CODEN = "DCGEER", ISSN = "0179-5376 (print), 1432-0444 (electronic)", ISSN-L = "0179-5376", bibdate = "Thu Aug 13 18:08:01 2009", bibsource = "DBLP; http://dblp.uni-trier.de/db/journals/dcg/dcg20.html#BezdekBC98; http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://link.springer.de/link/service/journals/00454/bibs/20n1p111.html", acknowledgement = ack-nhfb, fjournal = "Discrete and Computational Geometry", journal-URL = "http://link.springer.com/journal/454", } @Unpublished{Hales:1998:KC, author = "T. Hales", title = "The {Kepler Conjecture}", year = "1998", bibdate = "Thu Aug 13 16:43:12 2009", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", note = "arXiv:math.MG/9811078", acknowledgement = ack-nhfb, } @Article{Henk:1998:KIC, author = "Martin Henk", title = "{Kugeln im Computer: ein Beweis der Keplerschen Vermutung?}. ({German}) [{Spheres} in the computer: a proof of {Kepler}'s Conjecture?]", journal = j-MDMV, volume = "??", number = "4", pages = "58--60", year = "1998", CODEN = "????", ISSN = "0947-4471", MRclass = "52C17", MRnumber = "1666722", bibdate = "Thu May 22 16:37:35 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Mitteilungen der Deutschen Mathematiker-Vereinigung", keywords = "Kepler sphere-packing conjecture", language = "German", } @Article{Muses:1998:EDF, author = "C. Mus{\`e}s", title = "Erratum: {``The dimensional family approach in (hyper)sphere packing: a typological study for new patterns, structures, and interdimensional functions'': Appl. Math. Comput. {\bf 88} (1997)}", journal = j-APPL-MATH-COMP, volume = "97", number = "2--3", pages = "295--295", day = "15", month = dec, year = "1998", CODEN = "AMHCBQ", DOI = "https://doi.org/10.1016/S0096-3003(98)00030-7", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", bibdate = "Tue Nov 20 21:03:17 MST 2012", bibsource = "http://www.math.utah.edu/pub/tex/bib/applmathcomput1995.bib; http://www.math.utah.edu/pub/tex/bib/kepler.bib; http://www.sciencedirect.com/science/journal/00963003", note = "See \cite{Muses:1997:DFA,Anonymous:1997:EDF}.", URL = "http://www.sciencedirect.com/science/article/pii/S0096300398000307", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003", } @Article{Rintoul:1998:HSS, author = "M. D. Rintoul and S. S. Torquato", title = "Hard-sphere statistics along the metastable amorphous branch", journal = j-PHYS-REV-E, volume = "58", number = "??", pages = "532--537", month = "????", year = "1998", CODEN = "PLEEE8", ISSN = "1539-3755 (print), 1550-2376 (electronic)", ISSN-L = "1539-3755", bibdate = "Fri Aug 14 21:36:55 2009", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Physical Review E (Statistical physics, plasmas, fluids, and related interdisciplinary topics)", journal-URL = "http://pre.aps.org/browse", } @Article{Anonymous:1999:ICH, author = "Anonymous", title = "Interval Computations Help in Proving the {Kepler} Conjecture", journal = j-RELIABLE-COMPUTING, volume = "5", number = "2", pages = "197--199", month = may, year = "1999", CODEN = "RCOMF8", DOI = "https://doi.org/10.1023/A:1017205323815", ISSN = "1385-3139 (print), 1573-1340 (electronic)", ISSN-L = "1385-3139", bibdate = "Sat Jan 5 16:00:57 MST 2013", bibsource = "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1385-3139&volume=5&issue=2; http://www.math.utah.edu/pub/tex/bib/kepler.bib; http://www.math.utah.edu/pub/tex/bib/reliablecomputing.bib", URL = "http://link.springer.com/article/10.1023/A%3A1017205323815/; http://www.springerlink.com/openurl.asp?genre=article&issn=1385-3139&volume=5&issue=2&spage=197-199", acknowledgement = ack-nhfb, fjournal = "Reliable Computing = Nadezhnye vychisleniia", journal-URL = "http://link.springer.com/journal/11155", } @Article{Blinovsky:1999:MPE, author = "Volodia Blinovsky", title = "Multiple Packing of the {Euclidean} Sphere", journal = j-IEEE-TRANS-INF-THEORY, volume = "45", number = "4", pages = "1334--1337", month = apr, year = "1999", CODEN = "IETTAW", DOI = "https://doi.org/10.1109/18.761296", ISSN = "0018-9448 (print), 1557-9654 (electronic)", ISSN-L = "0018-9448", bibdate = "Thu Aug 13 18:08:01 2009", bibsource = "DBLP; http://dblp.uni-trier.de/db/journals/tit/tit45.html#Blinovsky99; http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Information Theory", } @Article{Stephenson:1999:SSD, author = "Paul Stephenson", title = "Solitary Solarians and Densely Packed Spheres: From {Stephen Hales} to {Thomas Hales}", journal = j-MATH-GAZ, volume = "83", number = "498", pages = "426--432", month = nov, year = "1999", CODEN = "MAGAAS", ISSN = "0025-5572", bibdate = "Thu Aug 13 16:36:51 2009", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://www.jstor.org/stable/3620952", acknowledgement = ack-nhfb, fjournal = "Mathematical Gazette", journal-URL = "http://www.m-a.org.uk/jsp/index.jsp?lnk=620", } @Article{Talata:1999:TKN, author = "Istv{\'a}n Talata", title = "The translative kissing number of tetrahedra is $ 18 $", journal = j-DISCRETE-COMPUT-GEOM, volume = "22", number = "2", pages = "231--248", month = sep, year = "1999", CODEN = "DCGEER", DOI = "https://doi.org/10.1007/PL00009457", ISSN = "0179-5376 (print), 1432-0444 (electronic)", ISSN-L = "0179-5376", MRclass = "52C17", MRnumber = "1698544 (2000e:52021)", bibdate = "Wed Jan 09 09:59:02 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Discrete and Computational Geometry", journal-URL = "http://link.springer.com/journal/454", } @Article{Toth:1999:MCP, author = "L. Fejes T{\'o}th", title = "{Minkowski} Circle Packings on the Sphere", journal = j-DISCRETE-COMPUT-GEOM, volume = "22", number = "2", pages = "161--166", month = sep, year = "1999", CODEN = "DCGEER", DOI = "https://doi.org/10.1007/PL00009451", ISSN = "0179-5376 (print), 1432-0444 (electronic)", ISSN-L = "0179-5376", MRclass = "52C15", MRnumber = "1698538 (2000e:52018)", bibdate = "Thu Aug 13 18:08:01 2009", bibsource = "DBLP; http://dblp.uni-trier.de/db/journals/dcg/dcg22.html#Toth99; http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://link.springer.de/link/service/journals/00454/bibs/22n2p161.html", acknowledgement = ack-nhfb, fjournal = "Discrete and Computational Geometry", journal-URL = "http://link.springer.com/journal/454", } @Article{Wang:1999:PUS, author = "Jie Wang", title = "Packing of Unequal Spheres and Automated Radiosurgical Treatment Planning", journal = j-J-COMB-OPTIM, volume = "3", number = "4", pages = "453--463", year = "1999", CODEN = "JCOPFV", ISSN = "1382-6905", ISSN-L = "1382-6905", bibdate = "Thu Aug 13 18:08:01 2009", bibsource = "DBLP; http://dblp.uni-trier.de/db/journals/jco/jco3.html#Wang99; http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Combinatorial Optimization", } @Article{Weaire:1999:SHP, author = "D. Weaire", title = "A short history of packing problems", journal = j-FORMA, volume = "14", number = "4", pages = "279--285", year = "1999", ISSN = "0911-6036", MRclass = "01A05 (52-03 52C17)", MRnumber = "1770214 (2001e:01005)", bibdate = "Thu May 22 16:37:35 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", note = "Proceedings of the 2nd International Katachi U Symmetry Symposium, Part 1 (Tsukuba, 1999).", acknowledgement = ack-nhfb, fjournal = "Forma", keywords = "Kepler sphere-packing conjecture", } @Article{Zhang:1999:OPP, author = "J. Zhang and R. Blaak and E. Trizac and J. A. Cuesta and D. Frenkel", title = "Optimal packing of polydisperse hard-sphere fluids", journal = j-J-CHEM-PHYS, volume = "110", number = "11", pages = "5318--5324", year = "1999", CODEN = "JCPSA6", ISSN = "0021-9606 (print), 1089-7690 (electronic)", ISSN-L = "0021-9606", bibdate = "Fri Aug 14 17:30:31 2009", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Chemical Physics", journal-URL = "http://scitation.aip.org/content/aip/journal/jcp/browse", } @Book{Zong:1999:SP, author = "Chuanming Zong and J. (John) Talbot", title = "Sphere packings", publisher = pub-SV, address = pub-SV:adr, pages = "xiii + 241", year = "1999", ISBN = "0-387-98794-0", ISBN-13 = "978-0-387-98794-1", LCCN = "QA166.7 .Z66 1999", MRclass = "52C22 (11H31)", MRnumber = "1707318 (2000g:52020)", MRreviewer = "Peter M. Gruber", bibdate = "Thu Aug 13 16:26:48 MDT 2009", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib; z3950.loc.gov:7090/Voyager", series = "Universitext", URL = "http://www.loc.gov/catdir/enhancements/fy0816/99015367-d.html; http://www.loc.gov/catdir/enhancements/fy0816/99015367-t.html", acknowledgement = ack-nhfb, subject = "sphere packings", } @Book{Aste:2000:PPP, author = "Tomaso Aste and Denis Weaire", title = "The pursuit of perfect packing", publisher = pub-IOP, address = pub-IOP:adr, pages = "xi + 136", year = "2000", DOI = "https://doi.org/10.1887/0750306483", ISBN = "0-7503-0648-3", ISBN-13 = "978-0-7503-0648-5", LCCN = "QA166.7 .A78 2000", MRclass = "52-01 (52A40 52C15 52C17 52C23)", MRnumber = "1786410 (2001g:52001)", MRreviewer = "Johann Linhart", bibdate = "Tue Mar 13 11:03:37 2001", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, annote = "This is an interesting little monograph on packing problems, a surprising number of which continue to elude proofs and solutions. Packing problems are considered in two-, three-, and higher dimensions, for bubbles, casks of Guinness, concrete, spheres, foam, soap film, \ldots{}.", keywords = "Hilbert's 18th Problem (How can one arrange most densely in space an infinite number of equal solids of given form, so that the ratio of the filled to the unfilled space may be as great as possible?); Kepler sphere-packing conjecture; Malfatti problem; Mallet's model; Thomas Hales' packing-proof programme; Thomson problem; Tammes problem; Vorono{\"\i} construction", libnote = "Not in my library.", } @TechReport{Barg:2000:BPS, author = "A. Barg and D. Nogin", title = "Bounds for Packings of Spheres in the {Grassmann} Manifolds", number = "2000-19", institution = "DIMACS", month = jul # " 27", year = "2000", bibdate = "Fri Aug 14 17:30:34 2009", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", note = "Sat, 5 Aug 2000 13:00:00 GMT", URL = "ftp://dimacs.rutgers.edu/pub/dimacs/TechnicalReports/TechReports/2000/2000-19.ps.gz", abstract = "We derive the Varshamov--Gilbert and Hamming bounds for packings of spheres (codes) in the Grassmann manifolds over $ \mathbb R $ and $ \mathbb C $. The distance between two $k$-planes is defined as $ \rho (p, q) = (\sin^2 \theta_1 + \dots + \sin^2 \theta_k)^{1 / 2}$, where $ \theta_i, 1 \le i \le k$, are the principal angles between $p$ and $q$.", acknowledgement = ack-nhfb, } @Article{Betke:2000:DLP, author = "U. Betke and M. Henk", title = "Densest lattice packings of 3-polytopes", journal = j-COMP-GEOM, volume = "16", number = "3", pages = "157--186", month = jul, year = "2000", DOI = "https://doi.org/10.1016/S0925-7721(00)00007-9", ISSN = "0925-7721", ISSN-L = "0925-7721", bibdate = "Fri Aug 14 17:12:07 2009", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", abstract = "Based on Minkowski's work on critical lattices of 3-dimensional convex bodies we present an efficient algorithm for computing the density of a densest lattice packing of an arbitrary 3-polytope. As an application we calculate densest lattice packings of all regular and Archimedean polytopes.", acknowledgement = ack-nhfb, fjournal = "Computational Geometry. Theory and Applications", } @Article{Cicalese:2000:OCO, author = "Ferdinando Cicalese and Daniele Mundici", title = "Optimal Coding with One Asymmetric Error: Below the Sphere Packing Bound", journal = j-LECT-NOTES-COMP-SCI, volume = "1858", pages = "159--??", year = "2000", CODEN = "LNCSD9", ISSN = "0302-9743 (print), 1611-3349 (electronic)", ISSN-L = "0302-9743", bibdate = "Fri Feb 1 09:16:48 MST 2002", bibsource = "http://link.springer-ny.com/link/service/series/0558/tocs/t1858.htm; http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://link.springer-ny.com/link/service/series/0558/bibs/1858/18580159.htm; http://link.springer-ny.com/link/service/series/0558/papers/1858/18580159.pdf", acknowledgement = ack-nhfb, fjournal = "Lecture Notes in Computer Science", journal-URL = "http://link.springer.com/bookseries/558", } @Article{Hales:2000:ARM, author = "T. C. Hales and P. Sarnak and M. C. Pugh", title = "Advances in Random Matrix Theory, Zeta Functions, and Sphere Packing", journal = j-PROC-NATL-ACAD-SCI-USA, volume = "97", number = "24", pages = "12963--12964", day = "21", month = nov, year = "2000", CODEN = "PNASA6", DOI = "https://doi.org/10.1073/pnas.220396097", ISSN = "0027-8424 (print), 1091-6490 (electronic)", ISSN-L = "0027-8424", MRclass = "11H31 (11M26 52C17)", MRnumber = "1795618 (2001k:11121)", bibdate = "Thu May 22 16:37:35 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://www.jstor.org/stable/123637", abstract = "Over four hundred years ago, Sir Walter Raleigh asked his mathematical assistant to find formulas for the number of cannonballs in regularly stacked piles. These investigations aroused the curiosity of the astronomer Johannes Kepler and led to a problem that has gone centuries without a solution: why is the familiar cannonball stack the most efficient arrangement possible? Here we discuss the solution that Hales found in 1998. Almost every part of the 282-page proof relies on long computer verifications. Random matrix theory was developed by physicists to describe the spectra of complex nuclei. In particular, the statistical fluctuations of the eigenvalues (``the energy levels'') follow certain universal laws based on symmetry types. We describe these and then discuss the remarkable appearance of these laws for zeros of the Riemann zeta function (which is the generating function for prime numbers and is the last special function from the last century that is not understood today.) Explaining this phenomenon is a central problem. These topics are distinct, so we present them separately with their own introductory remarks.", acknowledgement = ack-nhfb, fjournal = "Proceedings of the National Academy of Sciences of the United States of America", journal-URL = "http://www.pnas.org/search", keywords = "Kepler sphere-packing conjecture", } @Article{Hales:2000:CH, author = "Thomas C. Hales", title = "Cannonballs and Honeycombs", journal = j-NAMS, volume = "47", number = "4", pages = "440--449", month = apr, year = "2000", CODEN = "AMNOAN", ISSN = "0002-9920 (print), 1088-9477 (electronic)", ISSN-L = "0002-9920", MRclass = "52C17 (52C20 52C22)", MRnumber = "1745624 (2000m:52027)", MRreviewer = "R. Blind", bibdate = "Thu May 22 16:37:35 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", note = "In January 2003, the author was awarded the Chauvenet Prize of the Mathematical Association of America for this article.", URL = "http://www.ams.org/notices/200004/fea-hales.pdf", acknowledgement = ack-nhfb, fjournal = "Notices of the American Mathematical Society", journal-URL = "http://www.ams.org/notices/", keywords = "Kepler sphere-packing conjecture", } @InCollection{Henk:2000:KIC, author = "Martin Henk and G{\"u}nter M. Ziegler", booktitle = "Alles {Mathematik}. ({German}) [{All} mathematics]", title = "{Kugeln im Computer---die Kepler-Vermutung}. ({German}) [{Spheres} in the computer --- the {Kepler} conjecture]", publisher = pub-VIEWEG, address = pub-VIEWEG:adr, pages = "121--143", year = "2000", MRclass = "52C17", MRnumber = "1801830", bibdate = "Thu May 22 16:37:35 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, keywords = "Kepler sphere-packing conjecture", language = "German", } @Article{Hilbert:2000:MP, author = "David Hilbert", title = "Mathematical problems", journal = j-BULL-AMS, volume = "37", number = "4", pages = "407--436", month = "????", year = "2000", CODEN = "BAMOAD", DOI = "https://doi.org/10.1090/S0273-0979-00-00881-8", ISSN = "0002-9904 (print), 1936-881X (electronic)", ISSN-L = "0002-9904", MRclass = "01A75", MRnumber = "1779412", bibdate = "Wed Jan 09 17:19:41 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", note = "Lecture delivered before the International Congress of Mathematicians at Paris in 1900. Reprint of \cite{Hilbert:1902:MP}. English translation of \cite{Hilbert:1900:MPV, Hilbert:1901:MPV}.", URL = "http://www.ams.org/journals/bull/2000-37-04/S0273-0979-00-00881-8/S0273-0979-00-00881-8.pdf", acknowledgement = ack-nhfb, fjournal = "Bulletin of the American Mathematical Society", journal-URL = "http://www.ams.org/journals/bull/all_issues.html", keywords = "Hilbert's 23 Problems", } @Article{Klarreich:2000:FH, author = "Erica G. Klarreich", title = "Foams and honeycombs", journal = j-AM-SCI, volume = "88", number = "2", pages = "150--161", month = mar # "\slash " # apr, year = "2000", CODEN = "AMSCAC", DOI = "https://doi.org/10.1511/2000.2.152", ISSN = "0003-0996 (print), 1545-2786 (electronic)", ISSN-L = "0003-0996", bibdate = "Wed Jan 09 15:14:16 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://www.americanscientist.org/issues/feature/2000/2/foams-and-honeycombs", acknowledgement = ack-nhfb, fjournal = "American Scientist", journal-URL = "http://www.americanscientist.org/issues/past.aspx", remark = "Morgan writes \cite{Morgan:2005:BRB}, ``So despite similarities, Kepler s Conjecture on sphere packing in $ \mathbb {R}^3 $ was orders of magnitude more difficult than the Hexagonal Honeycomb Conjecture in $ \mathbb {R}^2 $, mainly because $ \mathbb {R}^3 $ provides so many more geometric possibilities than $ \mathbb {R}^2 $. But it was easier in one aspect: it is a packing problem, whereas the Honeycomb Conjecture is a partitioning problem. The optimal two-dimensional packing, with six circles fitting perfectly around every circle, is relatively easy and was proved in 1890 by Thue. (See Klarreich [this paper] for a beautiful account.)", } @Article{Li:2000:BAF, author = "X.-Y. Li and S.-H. Teng and A. {\"U}ng{\"o}r", title = "Biting: advancing front meets sphere packing", journal = j-INT-J-NUMER-METHODS-ENG, volume = "49", number = "1-2", pages = "61--81", year = "2000", CODEN = "IJNMBH", ISSN = "0029-5981", ISSN-L = "0029-5981", bibdate = "Thu Aug 13 17:45:47 2009", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "International Journal for Numerical Methods in Engineering", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1097-0207", } @Article{Muses:2000:VKC, author = "C. Mus{\`e}s", title = "Validating {Kepler's Conjecture}: a new approach", journal = j-APPL-MATH-COMP, volume = "110", number = "1", pages = "99--104", year = "2000", CODEN = "AMHCBQ", DOI = "https://doi.org/10.1016/S0096-3003(99)00058-2", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", MRclass = "52C17", MRnumber = "1743359", bibdate = "Thu May 22 16:37:35 2014", bibsource = "http://www.elsevier.com/locate/issn/00963003; http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://www.elsevier.nl/gej-ng/29/17/20/83/21/24/abstract.html; http://www.elsevier.nl/gej-ng/29/17/20/83/21/24/article.pdf", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003", keywords = "Kepler sphere-packing conjecture", } @Article{Oesterle:2000:DME, author = "Joseph Oesterl{\'e}", title = "Densit{\'e} maximale des empilements de sph{\`e}res en dimension 3 (d'apr{\`e}s {Thomas C. Hales} et {Samuel P. Ferguson}). ({French}) [Maximal 3-dimensional sphere-packing density (according to {Thomas C. Hales} et {Samuel P. Ferguson})]", journal = "Ast{\'e}risque", volume = "266", number = "863(5)", pages = "405--413", year = "2000", ISSN = "0303-1179", MRclass = "52C17 (52A40)", MRnumber = "1772681 (2001k:52030)", MRreviewer = "A. Florian", bibdate = "Thu May 22 16:37:35 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", note = "S{\'e}minaire Bourbaki, Vol. 1998/99. Exp. no. 863, 5.", acknowledgement = ack-nhfb, fjournal = "Ast\'erisque", keywords = "Kepler sphere-packing conjecture", language = "French", } @Article{Schnell:2000:FVH, author = "Uwe Schnell", title = "{FCC} versus {HCP} via parametric density", journal = j-DISCRETE-MATH, volume = "211", number = "1-3", pages = "269--274", year = "2000", CODEN = "DSMHA4", DOI = "https://doi.org/10.1016/S0012-365X(99)00158-2", ISSN = "0012-365X", ISSN-L = "0012-365X", MRclass = "52C17 (52C23)", MRnumber = "1735333 (2001f:52040)", bibdate = "Thu May 22 16:37:35 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Discrete Mathematics", keywords = "Kepler sphere-packing conjecture", } @Article{Wang:2000:MAO, author = "Jie Wang", title = "Medial Axis and Optimal Locations for Min-Max Sphere Packing", journal = j-J-COMB-OPTIM, volume = "4", number = "4", pages = "487--503", year = "2000", CODEN = "JCOPFV", ISSN = "1382-6905", ISSN-L = "1382-6905", bibdate = "Thu Aug 13 18:08:01 2009", bibsource = "DBLP; http://dblp.uni-trier.de/db/journals/jco/jco4.html#Wang00; http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Combinatorial Optimization", } @InProceedings{Backofen:2001:OCF, author = "Rolf Backofen and Sebastian Will", title = "Optimally Compact Finite Sphere Packings --- Hydrophobic Cores in the {FCC}", crossref = "Amir:2001:CPM", volume = "2089", pages = "257--272", year = "2001", bibdate = "Thu Aug 13 18:08:01 2009", bibsource = "DBLP; http://dblp.uni-trier.de/db/conf/cpm/cpm2001.html#BackofenW01; http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://link.springer-ny.com/link/service/series/0558/bibs/2089/20890257.htm; http://link.springer-ny.com/link/service/series/0558/papers/2089/20890257.pdf; http://link.springer-ny.com/link/service/series/0558/tocs/t2089.htm", acknowledgement = ack-nhfb, } @Book{Casti:2001:MMF, author = "J. L. Casti", title = "Mathematical mountaintops: the five most famous problems of all time", publisher = pub-OXFORD, address = pub-OXFORD:adr, pages = "177 + 3", year = "2001", ISBN = "0-19-514171-7", ISBN-13 = "978-0-19-514171-9", LCCN = "QA93 .C386 2001", bibdate = "Wed Jan 9 08:09:14 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib; z3950.loc.gov:7090/Voyager", acknowledgement = ack-nhfb, subject = "Mathematics; Popular works", tableofcontents = "The art of the problem \\ Hilbert's tenth problem \\ The four-color problem \\ The continuum hypothesis \\ The Kepler conjecture \\ Fermat's last theorem \\ The magnificent seven", } @InProceedings{Chen:2001:ACS, author = "Danny Z. Chen and Xiaobo Hu and Yingping Huang and Yifan Li and Jinhui Xu", title = "Algorithms for congruent sphere packing and applications", crossref = "ACM:2001:PSA", pages = "212--221", year = "2001", DOI = "https://doi.org/10.1145/378583.378672", bibdate = "Thu Aug 13 18:08:01 2009", bibsource = "DBLP; http://dblp.uni-trier.de/db/conf/compgeom/compgeom2001.html#ChenHHLX01; http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, } @Book{Cornuejols:2001:COP, author = "G{\'e}rard Cornu{\'e}jols", title = "Combinatorial Optimization: Packing and Covering", volume = "74", publisher = pub-SIAM, address = pub-SIAM:adr, pages = "xi + 132", year = "2001", DOI = "https://doi.org/10.1137/1.9780898717105", ISBN = "0-89871-481-8 (paperback)", ISBN-13 = "978-0-89871-481-4 (paperback)", LCCN = "QA166.7 .C67 2001", MRclass = "90-02 (05-02 52C17 52C45 90C27 90C35)", MRnumber = "1828452 (2002e:90004)", MRreviewer = "Rainer Burkard", bibdate = "Wed Jan 9 09:48:56 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib; z3950.loc.gov:7090/Voyager", series = "CBMS-NSF regional conference series in applied mathematics", URL = "http://www.loc.gov/catdir/enhancements/fy0708/00066073-d.html; http://www.loc.gov/catdir/enhancements/fy0708/00066073-t.html", acknowledgement = ack-nhfb, subject = "Combinatorial packing and covering", } @InCollection{Gardner:2001:PS, author = "Martin Gardner", title = "Packing spheres", crossref = "Gardner:2001:CBM", chapter = "10", pages = "128--136", year = "2001", bibdate = "Wed Jan 09 17:54:03 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", note = "Page 135 mentions a conjecture by Stan Ulam in 1972 that ``the maximal density ($ \approx 0.74048$) for packing congruent spheres is smaller than that for any other convex body'' \cite[page 1547]{Lagarias:2012:MPR}.", acknowledgement = ack-nhfb, } @Article{Hales:2001:HC, author = "Thomas C. Hales", title = "The honeycomb conjecture", journal = j-DISCRETE-COMPUT-GEOM, volume = "25", number = "1", pages = "1--22", month = jan, year = "2001", CODEN = "DCGEER", DOI = "https://doi.org/10.1007/s004540010071", ISSN = "0179-5376 (print), 1432-0444 (electronic)", ISSN-L = "0179-5376", MRclass = "52C20 (51M16)", MRnumber = "1797293 (2002a:52020)", MRreviewer = "W. Kuperberg", bibdate = "Wed Jan 09 15:18:38 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "", acknowledgement = ack-nhfb, fjournal = "Discrete and Computational Geometry", journal-URL = "http://link.springer.com/journal/454", } @InProceedings{Hales:2001:SPG, author = "Thomas C. Hales", title = "Sphere packings and generative [{XXX}?]", crossref = "ACM:2001:PSA", pages = "69--69", year = "2001", DOI = "https://doi.org/10.1145/378583.378624", bibdate = "Thu Aug 13 18:08:01 2009", bibsource = "DBLP; http://dblp.uni-trier.de/db/conf/compgeom/compgeom2001.html#Hales01; http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, } @Book{Hsiang:2001:LAP, author = "Wu Yi Hsiang", title = "Least action principle of crystal formation of dense packing type and {Kepler}'s conjecture", volume = "3", publisher = pub-WORLD-SCI, address = pub-WORLD-SCI:adr, pages = "xxii + 402", year = "2001", DOI = "https://doi.org/10.1142/9789812384911", ISBN = "981-02-4670-6", ISBN-13 = "978-981-02-4670-9", LCCN = "QA166.7 .H85 2001", MRclass = "52C17", MRnumber = "1962807 (2004g:52033)", MRreviewer = "Udo Hertrich-Jeromin", bibdate = "Thu May 22 16:37:35 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib; z3950.loc.gov:7090/Voyager", note = "With a foreword by S. S. Chern", series = "Nankai Tracts in Mathematics", acknowledgement = ack-nhfb, keywords = "Kepler sphere-packing conjecture", subject = "Kepler's conjecture; Sphere packings; Crystallography, Mathematical", xxpages = "xxi + 402", } @Article{Torquato:2001:MGS, author = "S. Torquato and F. H. Stillinger", title = "Multiplicity of generation, selection, and classification procedures for jammed hard-particle packings", journal = j-J-PHYS-CHEM-B, volume = "105", number = "47", pages = "11849--11853", day = "13", month = oct, year = "2001", CODEN = "JPCBFK", DOI = "https://doi.org/10.1021/jp011960q", ISSN = "1089-5647 (print), 1520-6106 (electronic)", ISSN-L = "1520-5207", bibdate = "Fri Aug 14 21:41:30 2009", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Physical Chemistry. B. Condensed matter, materials, surfaces, interfaces \& biophysical", journal-URL = "http://pubs.acs.org/loi/jpcbfk", } @InCollection{Verger-Gaugry:2001:RSP, author = "Jean-Louis Verger-Gaugry", booktitle = "{S{\'e}minaire de Th{\'e}orie Spectrale et G{\'e}om{\'e}trie, Vol. 19, Ann{\'e}e 2000--2001}", title = "Remarks on sphere packings, clusters and {Hales} {Ferguson} theorem", volume = "19", publisher = "Universit{\'e} Grenoble I", address = "Saint-Martin-d'H{\`e}res, France", pages = "165--177", year = "2001", MRclass = "52C17 (52C23)", MRnumber = "1909081 (2003d:52021)", MRreviewer = "Uwe Schnell", bibdate = "Thu May 22 16:37:35 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", series = "S{\'e}min. Th{\'e}or. Spectr. G{\'e}om.", acknowledgement = ack-nhfb, keywords = "Kepler sphere-packing conjecture", } @Article{Barg:2002:BPS, author = "Alexander Barg and Dmitry Yu. Nogin", title = "Bounds on packings of spheres in the {Grassmann} manifold", journal = j-IEEE-TRANS-INF-THEORY, volume = "48", number = "9", pages = "2450--2454", month = sep, year = "2002", CODEN = "IETTAW", DOI = "https://doi.org/10.1109/TIT.2002.801469", ISSN = "0018-9448 (print), 1557-9654 (electronic)", ISSN-L = "0018-9448", bibdate = "Thu Aug 13 18:08:01 2009", bibsource = "DBLP; http://dblp.uni-trier.de/db/journals/tit/tit48.html#BargN02; http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Information Theory", } @InProceedings{Cornforth:2002:EOB, author = "David Cornforth", title = "Evolution in the Orange Box --- {A} New Approach to the Sphere-Packing Problem in {CMAC}-Based Neural Networks", crossref = "McKay:2002:AAA", pages = "333--343", year = "2002", bibdate = "Thu Aug 13 18:08:01 2009", bibsource = "DBLP; http://dblp.uni-trier.de/db/conf/ausai/ausai2002.html#Cornforth02; http://link.springer-ny.com/link/service/series/0558/tocs/t2557.htm; http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://link.springer.de/link/service/series/0558/bibs/2557/25570333.htm; http://link.springer.de/link/service/series/0558/papers/2557/25570333.pdf", acknowledgement = ack-nhfb, } @InProceedings{Gervois:2002:VRT, author = "A. Gervois and Luc Oger and Patrick Richard and Jean Paul Troadec", title = "{Voronoi} and Radical Tessellations of Packings of Spheres", crossref = "Sloot:2002:CSIc", volume = "2331", pages = "95--104", year = "2002", bibdate = "Thu Aug 13 18:08:01 2009", bibsource = "DBLP; http://dblp.uni-trier.de/db/conf/iccS/iccS2002-3.html#GervoisORT02; http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://link.springer.de/link/service/series/0558/bibs/2331/23310095.htm", acknowledgement = ack-nhfb, } @InProceedings{Hales:2002:CVK, author = "Thomas C. Hales", booktitle = "{Proceedings of the International Congress of Mathematicians, Vol. III (Beijing, 2002)}", title = "A computer verification of the {Kepler Conjecture}", publisher = "Higher Education Press", address = "Beijing, People's Republic of China", pages = "795--804", year = "2002", MRclass = "52C17", MRnumber = "1957580 (2004a:52036)", bibdate = "Thu May 22 16:37:35 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, keywords = "Kepler sphere-packing conjecture", } @Article{Lagarias:2002:BLD, author = "J. C. Lagarias", title = "Bounds for local density of sphere packings and the {Kepler Conjecture}", journal = j-DISCRETE-COMPUT-GEOM, volume = "27", number = "2", pages = "165--193", month = jan, year = "2002", CODEN = "DCGEER", DOI = "https://doi.org/10.1007/978-1-4614-1129-1_2", ISSN = "0179-5376 (print), 1432-0444 (electronic)", ISSN-L = "0179-5376", MRclass = "52C17", MRnumber = "1880936 (2002k:52027)", MRreviewer = "Uwe Schnell", bibdate = "Thu May 22 16:37:35 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://link.springer.de/link/service/journals/00454/contents/01/0060/", acknowledgement = ack-nhfb, fjournal = "Discrete and Computational Geometry", journal-URL = "http://link.springer.com/journal/454", keywords = "Kepler sphere-packing conjecture", } @InProceedings{March:2002:EVF, author = "N. H. March and M. P. Tosi", title = "Excluded Volume, Free Volume and Hard Sphere Packing", crossref = "March:2002:ILS", pages = "29--50", year = "2002", DOI = "https://doi.org/10.1142/9789812778482_0002", bibdate = "Fri May 31 08:35:00 MDT 2013", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/march-norman-h.bib; http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://www.worldscientific.com/doi/abs/10.1142/9789812778482_0002", acknowledgement = ack-nhfb, book-URL = "http://www.worldscientific.com/worldscibooks/10.1142/4717", } @Article{Molnar:2002:BTO, author = "E. Moln{\'a}r and I. Prok and J. Szirmai", title = "{Bestimmung der transitiven optimalen Kugelpackungen f{\"u}r die 29 Raumgruppen, die Coxetersche Spiegelungsuntergruppen enthalten}. ({German}) [{Characterization} of transitive optimal sphere packings for the 29 space groups containing {Coxeter} reflection subgroups]", journal = "Studia Sci. Math. Hungar.", volume = "39", number = "3-4", pages = "443--483", year = "2002", CODEN = "SSMHAX", DOI = "https://doi.org/10.1556/SScMath.39.2002.3-4.13", ISSN = "0081-6906", MRclass = "52C17 (20H15 52A40 52B10)", MRnumber = "1956950 (2004a:52038)", bibdate = "Thu Apr 12 18:15:18 2012", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Studia Scientiarum Mathematicarum Hungarica. A Quarterly of the Hungarian Academy of Sciences", language = "German", } @Article{Bowen:2003:DPE, author = "Lewis Bowen and Charles Radin", title = "Densest Packing of Equal Spheres in Hyperbolic Space", journal = j-DISCRETE-COMPUT-GEOM, volume = "29", number = "1", pages = "23--39", month = dec, year = "2003", CODEN = "DCGEER", ISSN = "0179-5376 (print), 1432-0444 (electronic)", ISSN-L = "0179-5376", bibdate = "Thu Aug 13 18:08:01 2009", bibsource = "DBLP; http://dblp.uni-trier.de/db/journals/dcg/dcg29.html#BowenR03; http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://link.springer.de/link/service/journals/00454/contents/02/2791/index.html", acknowledgement = ack-nhfb, fjournal = "Discrete and Computational Geometry", journal-URL = "http://link.springer.com/journal/454", } @Article{Cohn:2003:NUB, author = "Henry Cohn and Noam Elkies", title = "New Upper Bounds on Sphere Packings {I}", journal = j-ANN-MATH-2, volume = "157", number = "2", pages = "698--714", month = mar, year = "2003", CODEN = "ANMAAH", ISSN = "0003-486X (print), 1939-8980 (electronic)", ISSN-L = "0003-486X", bibdate = "Thu Aug 13 16:38:01 2009", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://www.jstor.org/stable/3597215", abstract = "We develop an analogue for sphere packing of the linear programming bounds for error-correcting codes, and use it to prove upper bounds for the density of sphere packings, which are the best bounds known at least for dimensions 4 through 36. We conjecture that our approach can be used to solve the sphere packing problem in dimensions 8 and 24.", acknowledgement = ack-nhfb, fjournal = "Annals of Mathematics (2)", journal-URL = "http://www.jstor.org/action/showPublication?journalCode=annamath", } @Article{Cox:2003:MPI, author = "S. J. Cox and F. Graner and M. Fatima Vaz and C. Monnereaupittet and N. Pittet", title = "Minimal perimeter for {$N$} identical bubbles in two dimensions: calculations and simulations", journal = j-PHIL-MAGAZINE, volume = "83", number = "??", pages = "1393--1406", month = "????", year = "2003", CODEN = "PHMAA4", DOI = "", ISSN = "0031-8086", bibdate = "Wed Jan 09 15:22:36 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "", acknowledgement = ack-nhfb, fjournal = "Philosophical Magazine", journal-URL = "http://www.tandfonline.com/loi/tphm19", } @InCollection{Hales:2003:SAA, author = "Thomas C. Hales", booktitle = "Discrete and computational geometry", title = "Some algorithms arising in the proof of the {Kepler Conjecture}", volume = "25", publisher = pub-SV, address = pub-SV:adr, pages = "489--507", year = "2003", DOI = "https://doi.org/10.1007/978-3-642-55566-4_22", MRclass = "52C17 (90C27)", MRnumber = "2038488 (2005b:52047)", MRreviewer = "Achill Sch{\"u}rmann", bibdate = "Thu May 22 16:37:35 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", series = "Algorithms Combin.", acknowledgement = ack-nhfb, keywords = "Kepler sphere-packing conjecture", } @Article{Hauser:2003:GPM, author = "Kai Hauser and Reinhard Lang", title = "On the geometrical and physical meaning of {Newton}'s solution to {Kepler}'s problem", journal = j-MATH-INTEL, volume = "25", number = "4", pages = "35--44", year = "2003", CODEN = "MAINDC", DOI = "https://doi.org/10.1007/BF02984860", ISSN = "0343-6993", ISSN-L = "0343-6993", MRclass = "70F05 (01A45 52C17 70-03)", MRnumber = "2012561 (2005a:70009)", MRreviewer = "Florin N. Diacu", bibdate = "Thu May 22 16:37:35 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "The Mathematical Intelligencer", keywords = "Kepler sphere-packing conjecture", } @Book{Szpiro:2003:KC, author = "George G. Szpiro", title = "{Kepler}'s conjecture", publisher = pub-WILEY, address = pub-WILEY:adr, pages = "viii + 296", year = "2003", ISBN = "0-471-08601-0", ISBN-13 = "978-0-471-08601-7", MRclass = "52C17 (01A05 05B40 52-03)", MRnumber = "2133723 (2007c:52023)", MRreviewer = "P. D. F. Ion", bibdate = "Thu May 22 16:37:35 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", note = "How some of the greatest minds in history helped solve one of the oldest math problems in the world", acknowledgement = ack-nhfb, keywords = "Kepler sphere-packing conjecture", } @Book{Szpiro:2003:KCH, editor = "George Szpiro", title = "{Kepler's Conjecture}: how some of the greatest minds in history helped solve one of the oldest math problems in the world", publisher = pub-WILEY, address = pub-WILEY:adr, pages = "viii + 296", year = "2003", ISBN = "0-471-08601-0", ISBN-13 = "978-0-471-08601-7", LCCN = "QA93 .S97 2003", MRclass = "52C17 (01A05 05B40 52-03)", MRnumber = "2133723 (2007c:52023)", MRreviewer = "P. D. F. Ion", bibdate = "Thu Aug 13 16:25:44 MDT 2009", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib; z3950.loc.gov:7090/Voyager", URL = "http://www.loc.gov/catdir/bios/wiley046/2002014422.html; http://www.loc.gov/catdir/description/wiley039/2002014422.html; http://www.loc.gov/catdir/toc/wiley031/2002014422.html", acknowledgement = ack-nhfb, subject = "Mathematics; Popular works", tableofcontents = "Cannonballs and tomatoes \\ The puzzle of the dozen spheres \\ Fire hydrants and soccer players \\ Thue's two attempts and Fejes-T\'oth's achievement \\ Twelve's company, thirteen's a crowd \\ Nets and knots \\ Twisted boxes \\ No dancing at this congress \\ The race for the upper bound \\ Right angles for round spaces \\ Wobbly balls and hybrid stars \\ Simplex, Cplex, and symbolic mathematics \\ But is it really a proof? \\ Beehives again \ldots{} This is not an epilogue", } @Article{Szpiro:2003:MDP, author = "George Szpiro", title = "Mathematics: Does the proof stack up?", journal = j-NATURE, volume = "424", number = "6944", pages = "12--13", day = "3", month = jul, year = "2003", CODEN = "NATUAS", DOI = "https://doi.org/10.1038/424012a", ISSN = "0028-0836 (print), 1476-4687 (electronic)", ISSN-L = "0028-0836", bibdate = "Wed Jan 09 15:21:19 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://www.nature.com/nature/journal/v424/n6944/full/424012a.html", acknowledgement = ack-nhfb, fjournal = "Nature", journal-URL = "http://www.nature.com/nature/archive/", remark = "From the introduction: ``Think peer review takes too long? One mathematician has waited four years to have his paper refereed, only to hear that the exhausted reviewers can't be certain whether his proof is correct. George Szpiro investigates.''", } @Book{Boroczky:2004:FPC, author = "K{\'a}roly {B{\"o}r{\"o}czky, Jr.}", title = "Finite Packing and Covering", volume = "154", publisher = pub-CAMBRIDGE, address = pub-CAMBRIDGE:adr, pages = "xvii + 380", year = "2004", DOI = "https://doi.org/10.1017/CBO9780511546587", ISBN = "0-521-80157-5", ISBN-13 = "978-0-521-80157-7", LCCN = "QA166.7 .B67 2004", MRclass = "52C15 (05B40 52C17 52C20 52C22)", MRnumber = "2078625 (2005g:52045)", MRreviewer = "J. M. Wills", bibdate = "Wed Jan 9 08:44:32 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib; z3950.loc.gov:7090/Voyager", series = "Cambridge tracts in mathematics", URL = "http://www.loc.gov/catdir/description/cam041/2003065450.html; http://www.loc.gov/catdir/samples/cam051/2003065450.html; http://www.loc.gov/catdir/toc/cam041/2003065450.html", acknowledgement = ack-nhfb, subject = "Combinatorial packing and covering", } @Article{Donev:2004:LPA, author = "Aleksandar Donev and Salvatore Torquato and Frank H. Stillinger and Robert Connelly", title = "A linear programming algorithm to test for jamming in hard-sphere packings", journal = j-J-COMPUT-PHYS, volume = "197", number = "1", pages = "139--166", day = "10", month = jun, year = "2004", CODEN = "JCTPAH", DOI = "https://doi.org/10.1016/j.jcp.2003.11.022", ISSN = "0021-9991 (print), 1090-2716 (electronic)", ISSN-L = "0021-9991", bibdate = "Mon Jan 2 22:12:40 MST 2012", bibsource = "http://www.math.utah.edu/pub/tex/bib/jcomputphys2000.bib; http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0021999103006235", acknowledgement = ack-nhfb, fjournal = "Journal of Computational Physics", journal-URL = "http://www.sciencedirect.com/science/journal/00219991", } @Article{Donev:2004:UDC, author = "A. Donev and F. H. Stillinger and P. M. Chaikin and S. Torquato", title = "Unusually dense crystal ellipsoid packings", journal = j-PHYS-REV-LET, volume = "92", number = "25", pages = "255506", day = "25", month = jun, year = "2004", CODEN = "PRLTAO", ISSN = "0031-9007 (print), 1079-7114 (electronic), 1092-0145", ISSN-L = "0031-9007", bibdate = "Fri Aug 14 21:29:02 2009", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Physical Review Letters", journal-URL = "http://prl.aps.org/browse", pagecount = "4", } @Article{Gensane:2004:DPE, author = "Thierry Gensane", title = "Dense Packings of Equal Spheres in a Cube", journal = j-ELECT-J-COMB, volume = "11", number = "1", year = "2004", ISSN = "1077-8926 (print), 1097-1440 (electronic)", ISSN-L = "1077-8926", bibdate = "Thu Aug 13 18:08:00 2009", bibsource = "DBLP; http://dblp.uni-trier.de/db/journals/combinatorics/combinatorics11.html#Gensane04; http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://www.combinatorics.org/Volume_11/Abstracts/v11i1r33.html", acknowledgement = ack-nhfb, fjournal = "Electronic Journal of Combinatorics", } @InProceedings{Hales:2004:FPK, author = "Thomas C. Hales", title = "Formalizing the Proof of the {Kepler Conjecture}", crossref = "Slind:2004:TPH", pages = "117--117", year = "2004", bibdate = "Thu Aug 13 17:33:59 2009", bibsource = "DBLP; http://dblp.uni-trier.de/db/conf/tphol/tphol2004.html#Hales04; http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://springerlink.metapress.com/openurl.asp?genre=article&issn=0302-9743&spage=117;volume=3223", acknowledgement = ack-nhfb, } @Article{Radin:2004:OOS, author = "Charles Radin", title = "Orbits of Orbs: Sphere Packing Meets {Penrose} Tilings", journal = j-AMER-MATH-MONTHLY, volume = "111", number = "2", pages = "137--149", month = feb, year = "2004", CODEN = "AMMYAE", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", bibdate = "Thu Aug 13 16:41:03 2009", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://www.jstor.org/stable/4145214", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/page/journal/amermathmont/about.html", } @Article{Barg:2005:CBP, author = "Alexander Barg and Dmitry Yu. Nogin", title = "Correction to ``Bounds on Packings of Spheres in the {Grassmann} Manifold''", journal = j-IEEE-TRANS-INF-THEORY, volume = "51", number = "7", pages = "2732--2732", month = jul, year = "2005", CODEN = "IETTAW", DOI = "https://doi.org/10.1109/TIT.2005.850231", ISSN = "0018-9448 (print), 1557-9654 (electronic)", ISSN-L = "0018-9448", bibdate = "Thu Aug 13 18:08:01 2009", bibsource = "DBLP; http://dblp.uni-trier.de/db/journals/tit/tit51.html#BargN05; http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Information Theory", } @Article{Busa:2005:AFP, author = "J{\'a}n Busa and Jozef Dzurina and Edik Hayryan and Shura Hayryan and Chin-Kun Hu and J{\'a}n Pl{\'a}vka and Imrich Pokorn{\'y} and Jaroslav Skriv{\'a}nek and Ming-Chya Wu", title = "{ARVO}: {A} {Fortran} package for computing the solvent accessible surface area and the excluded volume of overlapping spheres via analytic equations", journal = j-COMP-PHYS-COMM, volume = "165", number = "1", pages = "59--96", year = "2005", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/j.cpc.2004.08.002", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Thu Aug 13 18:08:01 2009", bibsource = "DBLP; http://dblp.uni-trier.de/db/journals/cphysics/cphysics165.html#BusaDHHHPPSW05; http://www.math.utah.edu/pub/tex/bib/compphyscomm2000.bib; http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Article{Davies:2005:WM, author = "Brian Davies", title = "Whither Mathematics?", journal = j-NAMS, volume = "52", number = "11", pages = "1350--1356", month = dec, year = "2005", CODEN = "AMNOAN", ISSN = "0002-9920", bibdate = "Tue Nov 29 05:30:24 2005", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://www.ams.org/notices/200511/comm-davies.pdf", acknowledgement = ack-nhfb, fjournal = "Notices of the American Mathematical Society", journal-URL = "http://www.ams.org/notices/", keywords = "classification of finite simple groups; computer-assisted proofs; consistency of Peano arithmetic; four-color theorem; interval arithmetic; Kepler sphere-packing conjecture; Monster group", } @Article{Donev:2005:NLCa, author = "Aleksandar Donev and Salvatore Torquato and Frank H. Stillinger", title = "Neighbor list collision-driven molecular dynamics simulation for nonspherical hard particles. {I}. {Algorithmic} details", journal = j-J-COMPUT-PHYS, volume = "202", number = "2", pages = "737--764", day = "20", month = jan, year = "2005", CODEN = "JCTPAH", DOI = "https://doi.org/10.1016/j.jcp.2004.08.014", ISSN = "0021-9991 (print), 1090-2716 (electronic)", ISSN-L = "0021-9991", bibdate = "Mon Jan 2 22:12:47 MST 2012", bibsource = "http://www.math.utah.edu/pub/tex/bib/jcomputphys2000.bib; http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0021999104003146", acknowledgement = ack-nhfb, fjournal = "Journal of Computational Physics", journal-URL = "http://www.sciencedirect.com/science/journal/00219991/", } @Article{Donev:2005:NLCb, author = "Aleksandar Donev and Salvatore Torquato and Frank H. Stillinger", title = "Neighbor list collision-driven molecular dynamics simulation for nonspherical hard particles. {II}. {Applications} to ellipses and ellipsoids", journal = j-J-COMPUT-PHYS, volume = "202", number = "2", pages = "765--793", day = "20", month = jan, year = "2005", CODEN = "JCTPAH", DOI = "https://doi.org/10.1016/j.jcp.2004.08.025", ISSN = "0021-9991 (print), 1090-2716 (electronic)", ISSN-L = "0021-9991", bibdate = "Mon Jan 2 22:12:47 MST 2012", bibsource = "http://www.math.utah.edu/pub/tex/bib/jcomputphys2000.bib; http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0021999104003948", acknowledgement = ack-nhfb, fjournal = "Journal of Computational Physics", journal-URL = "http://www.sciencedirect.com/science/journal/00219991/", } @Article{Hales:2005:PKC, author = "Thomas C. Hales", title = "A proof of the {Kepler Conjecture}", journal = j-ANN-MATH-2, volume = "162", number = "3", pages = "1065--1185", year = "2005", CODEN = "ANMAAH", DOI = "https://doi.org/10.4007/annals.2005.162.1065", ISSN = "0003-486X (print), 1939-8980 (electronic)", ISSN-L = "0003-486X", MRclass = "52C17", MRnumber = "2179728 (2006g:52029)", MRreviewer = "G{\'e}za T{\'o}th", bibdate = "Thu May 22 16:37:35 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", note = "This proof is so complex that twelve journal referees were unable to verify it between its submission in 1998 and publication seven years later, so the editors decided to publish the article with a disclaimer about the lack of proof verification; see \cite{Morgan:2005:BRB} for more on this paper, and the associated book \cite{Szpiro:2003:KCH}.", acknowledgement = ack-nhfb, fjournal = "Annals of Mathematics. Second Series", journal-URL = "http://www.jstor.org/action/showPublication?journalCode=annamath", keywords = "Kepler sphere-packing conjecture", } @Article{Henkel:2005:SPB, author = "O. Henkel", title = "Sphere-packing bounds in the {Grassmann} and {Stiefel} manifolds", journal = j-IEEE-TRANS-INF-THEORY, volume = "51", number = "10", pages = "3445--3456", month = oct, year = "2005", CODEN = "IETTAW", DOI = "https://doi.org/10.1109/TIT.2005.855594", ISSN = "0018-9448 (print), 1557-9654 (electronic)", ISSN-L = "0018-9448", bibdate = "Thu Aug 13 18:08:01 2009", bibsource = "DBLP; http://dblp.uni-trier.de/db/journals/tit/tit51.html#Henkel05; http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Information Theory", } @Article{Morgan:2005:BRB, author = "Frank Morgan", title = "Book Review: {{\booktitle{Kepler's Conjecture}} and Hales' Proof}", journal = j-NAMS, volume = "52", number = "1", pages = "44--47", month = jan, year = "2005", CODEN = "AMNOAN", ISSN = "0002-9920 (print), 1088-9477 (electronic)", ISSN-L = "0002-9920", bibdate = "Wed Jan 09 07:58:15 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", note = "Review of \cite{Szpiro:2003:KCH}.", URL = "http://www.ams.org/notices/200501/rev-morgan.pdf", acknowledgement = ack-nhfb, fjournal = "Notices of the American Mathematical Society", journal-URL = "http://www.ams.org/notices/", remark = "\booktitle{Kepler's Conjecture: How some of the greatest minds in history helped solve one of the oldest math problems in the world}, George G. Szpiro, John Wiley \& Sons, Inc., 2003, \$24.95, ISBN 0-471-08601-0.", } @Article{Muraz:2005:LBD, author = "G. Muraz and J.-L. Verger-Gaugry", title = "On Lower Bounds of the Density of {Delone} Sets and Holes in Sequences of Sphere Packings", journal = j-EXP-MATH, volume = "14", number = "1", pages = "47--57", month = "????", year = "2005", CODEN = "????", DOI = "https://doi.org/10.1080/10586458.2005.10128908", ISSN = "1058-6458 (print), 1944-950X (electronic)", ISSN-L = "1058-6458", bibdate = "Mon Mar 5 15:33:58 MST 2012", bibsource = "http://projecteuclid.org/euclid.em; http://www.math.utah.edu/pub/tex/bib/expmath.bib; http://www.math.utah.edu/pub/tex/bib/kepler.bib; http://www.tandfonline.com/toc/uexm20/14/1", URL = "http://projecteuclid.org/euclid.em/1120145569", acknowledgement = ack-nhfb, fjournal = "Experimental Mathematics", journal-URL = "http://www.tandfonline.com/loi/uexm20", onlinedate = "30 Jan 2011", } @InProceedings{Obua:2005:PBR, author = "Steven {Obua}", title = "Proving bounds for real linear programs in {Isabelle\slash HOL}", crossref = "Hurd:2005:TPH", pages = "227--244", year = "2005", DOI = "https://doi.org/10.1007/11541868", MRclass = "68T15 90C05", bibdate = "Thu May 22 16:22:40 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", ZMnumber = "1152.68531", acknowledgement = ack-nhfb, keywords = "Project Flyspeck", } @Article{Ambrus:2006:NLB, author = "G. Ambrus and F. Fodor", title = "A new lower bound on the surface area of a {Voronoi} polyhedron", journal = j-PERIOD-MATH-HUNGAR, volume = "53", number = "1-2", pages = "45--58", year = "2006", CODEN = "PMHGAW", DOI = "https://doi.org/10.1007/s10998-006-0020-5", ISSN = "0031-5303", ISSN-L = "0031-5303", MRclass = "52C17 (52A40)", MRnumber = "2286459 (2007m:52023)", MRreviewer = "Konrad J. Swanepoel", bibdate = "Thu May 22 16:37:35 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Periodica Mathematica Hungarica. Journal of the J\'anos Bolyai Mathematical Society", keywords = "Kepler sphere-packing conjecture", } @Article{Bezdek:2006:SPR, author = "K{\'a}roly Bezdek", title = "Sphere packings revisited", journal = j-EUR-J-COMB, volume = "27", number = "6", pages = "864--883", year = "2006", CODEN = "EJOCDI", DOI = "https://doi.org/10.1016/j.ejc.2005.05.001", ISSN = "0195-6698", ISSN-L = "0195-6698", MRclass = "52C17 (05B40)", MRnumber = "2226423 (2007a:52021)", MRreviewer = "Gy{\"o}rgy D{\'o}sa", bibdate = "Thu May 22 16:37:35 2014", bibsource = "DBLP; http://dblp.uni-trier.de/db/journals/ejc/ejc27.html#Bezdek06; http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "European Journal of Combinatorics", keywords = "Kepler sphere-packing conjecture", } @Article{Bundy:2006:VMD, author = "Alan Bundy", title = "A Very Mathematical Dilemma", journal = j-COMP-J, volume = "49", number = "4", pages = "480--486", month = jul, year = "2006", CODEN = "CMPJA6", DOI = "https://doi.org/10.1093/comjnl/bxl021", ISSN = "0010-4620 (print), 1460-2067 (electronic)", ISSN-L = "0010-4620", bibdate = "Fri Jun 23 11:56:45 MDT 2006", bibsource = "http://comjnl.oxfordjournals.org/content/vol49/issue4/index.dtl; http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://comjnl.oxfordjournals.org/cgi/content/abstract/49/4/480; http://comjnl.oxfordjournals.org/cgi/content/full/49/4/480; http://comjnl.oxfordjournals.org/cgi/reprint/49/4/480", abstract = "Mathematics is facing a dilemma at its heart: the nature of mathematical proof. We have known since Church and Turing independently showed that mathematical provability was undecidable that there are theorems whose shortest proofs are enormous. Within the last half-century we have discovered practical examples of such theorems: the classification of all finite simple groups, the Four Colour Theorem and Kepler's Conjecture. These theorems were only proved with the aid of a computer. But computer proof is very controversial, with many mathematicians refusing to accept a proof that has not been thoroughly checked and understood by a human. The choice seems to be between either abandoning the investigation of theorems whose only proofs are enormous or changing traditional mathematical practice to include computer-aided proofs. Or is there a way to make large computer proofs more accessible to human mathematicians?", acknowledgement = ack-nhfb, fjournal = "The Computer Journal", journal-URL = "http://comjnl.oxfordjournals.org/", } @Article{Conway:2006:PTC, author = "J. H. Conway and S. Torquato", title = "Packing, Tiling, and Covering with Tetrahedra", journal = j-PROC-NATL-ACAD-SCI-USA, volume = "103", number = "28", pages = "10612--10617", day = "11", month = jul, year = "2006", CODEN = "PNASA6", ISSN = "0027-8424 (print), 1091-6490 (electronic)", ISSN-L = "0027-8424", bibdate = "Thu Aug 13 16:39:01 2009", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://www.jstor.org/stable/30049550", abstract = "It is well known that three-dimensional Euclidean space cannot be tiled by regular tetrahedra. But how well can we do? In this work, we give several constructions that may answer the various senses of this question. In so doing, we provide some solutions to packing, tiling, and covering problems of tetrahedra. Our results suggest that the regular tetrahedron may not be able to pack as densely as the sphere, which would contradict a conjecture of Ulam. The regular tetrahedron might even be the convex body having the smallest possible packing density.", acknowledgement = ack-nhfb, fjournal = "Proceedings of the National Academy of Sciences of the United States of America", journal-URL = "http://www.pnas.org/search", } @Article{Ferguson:2006:SPV, author = "Samuel P. Ferguson", title = "Sphere packings. {V}. {Pentahedral} prisms", journal = j-DISCRETE-COMPUT-GEOM, volume = "36", number = "1", pages = "167--204", month = jul, year = "2006", CODEN = "DCGEER", DOI = "https://doi.org/10.1007/s00454-005-1214-y", ISSN = "0179-5376 (print), 1432-0444 (electronic)", ISSN-L = "0179-5376", MRclass = "52C17 (05B40)", MRnumber = "2229661 (2007d:52025)", MRreviewer = "Uwe Schnell", bibdate = "Thu May 22 16:37:35 2014", bibsource = "DBLP; http://dblp.uni-trier.de/db/journals/dcg/dcg36.html#Ferguson06; http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Discrete and Computational Geometry", journal-URL = "http://link.springer.com/journal/454", keywords = "Kepler sphere-packing conjecture", } @Article{Hales:2006:FKC, author = "Thomas C. Hales and Samuel P. Ferguson", title = "A formulation of the {Kepler Conjecture}", journal = j-DISCRETE-COMPUT-GEOM, volume = "36", number = "1", pages = "21--69", month = jul, year = "2006", CODEN = "DCGEER", DOI = "https://doi.org/10.1007/s00454-005-1211-1", ISSN = "0179-5376 (print), 1432-0444 (electronic)", ISSN-L = "0179-5376", MRclass = "52C17 (05B40)", MRnumber = "2229658 (2007d:52022)", MRreviewer = "Uwe Schnell", bibdate = "Thu May 22 16:37:35 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Discrete and Computational Geometry", journal-URL = "http://link.springer.com/journal/454", keywords = "Kepler sphere-packing conjecture", } @Article{Hales:2006:HOK, author = "Thomas C. Hales", title = "Historical overview of the {Kepler Conjecture}", journal = j-DISCRETE-COMPUT-GEOM, volume = "36", number = "1", pages = "5--20", month = jul, year = "2006", CODEN = "DCGEER", DOI = "https://doi.org/10.1007/s00454-005-1210-2", ISSN = "0179-5376 (print), 1432-0444 (electronic)", ISSN-L = "0179-5376", MRclass = "52C17 (05B40)", MRnumber = "2229657 (2007d:52021)", MRreviewer = "Uwe Schnell", bibdate = "Thu May 22 16:37:35 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Discrete and Computational Geometry", journal-URL = "http://link.springer.com/journal/454", keywords = "Kepler sphere-packing conjecture", } @Article{Hales:2006:SPIa, author = "Thomas C. Hales", title = "Sphere packings. {III}. {Extremal} cases", journal = j-DISCRETE-COMPUT-GEOM, volume = "36", number = "1", pages = "71--110", month = jul, year = "2006", CODEN = "DCGEER", DOI = "https://doi.org/10.1007/s00454-005-1212-0", ISSN = "0179-5376 (print), 1432-0444 (electronic)", ISSN-L = "0179-5376", MRclass = "52C17 (05B40)", MRnumber = "2229659 (2007d:52023)", MRreviewer = "Uwe Schnell", bibdate = "Thu May 22 16:37:35 2014", bibsource = "DBLP; http://dblp.uni-trier.de/db/journals/dcg/dcg36.html#Hales06a; http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Discrete and Computational Geometry", journal-URL = "http://link.springer.com/journal/454", keywords = "Kepler sphere-packing conjecture", } @Article{Hales:2006:SPIb, author = "Thomas C. Hales", title = "Sphere Packings. {IV}. {Detailed} Bounds", journal = j-DISCRETE-COMPUT-GEOM, volume = "36", number = "1", pages = "111--166", month = jul, year = "2006", CODEN = "DCGEER", DOI = "https://doi.org/10.1007/s00454-005-1213-z", ISSN = "0179-5376 (print), 1432-0444 (electronic)", ISSN-L = "0179-5376", MRclass = "52C17 (05B40)", MRnumber = "2229660 (2007d:52024)", MRreviewer = "Uwe Schnell", bibdate = "Thu May 22 16:37:35 2014", bibsource = "DBLP; http://dblp.uni-trier.de/db/journals/dcg/dcg36.html#Hales06b; http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Discrete and Computational Geometry", journal-URL = "http://link.springer.com/journal/454", keywords = "Kepler sphere-packing conjecture", } @Article{Hales:2006:SPV, author = "Thomas C. Hales", title = "Sphere packings. {VI}. {Tame} graphs and linear programs", journal = j-DISCRETE-COMPUT-GEOM, volume = "36", number = "1", pages = "205--265", month = jul, year = "2006", CODEN = "DCGEER", DOI = "https://doi.org/10.1007/s00454-005-1215-x", ISSN = "0179-5376 (print), 1432-0444 (electronic)", ISSN-L = "0179-5376", MRclass = "52C17 (05B40 90C08)", MRnumber = "2229662 (2007d:52026)", MRreviewer = "Uwe Schnell", bibdate = "Thu May 22 16:37:35 2014", bibsource = "DBLP; http://dblp.uni-trier.de/db/journals/dcg/dcg36.html#Hales06c; http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Discrete and Computational Geometry", journal-URL = "http://link.springer.com/journal/454", keywords = "Kepler sphere-packing conjecture", } @InCollection{Nipkow:2006:FTG, author = "Tobias Nipkow and Gertrud Bauer and Paula Schultz", booktitle = "Automated reasoning", title = "Flyspeck. {I}. {Tame} graphs", volume = "4130", publisher = pub-SV, address = pub-SV:adr, pages = "21--35", year = "2006", DOI = "https://doi.org/10.1007/11814771_4", MRclass = "Database Expansion Item", MRnumber = "2354670", bibdate = "Thu May 22 17:30:35 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", series = "Lecture Notes in Comput. Sci.", acknowledgement = ack-nhfb, } @Article{Parisi:2006:APH, author = "G. Parisi and F. Zamponi", title = "Amorphous packings of hard spheres in large space dimension", journal = "Computing Research Repository (CoRR)", volume = "abs/cond-mat/0601573", year = "2006", bibdate = "Thu Aug 13 18:08:01 2009", bibsource = "DBLP; http://dblp.uni-trier.de/db/journals/corr/corr0601.html#abs-cond-mat-0601573; http://www.math.utah.edu/pub/tex/bib/kepler.bib", note = "informal publication", URL = "http://arxiv.org/abs/cond-mat/0601573", acknowledgement = ack-nhfb, } @Article{Sahai:2006:HBS, author = "Anant Sahai", title = "How to beat the sphere-packing bound with feedback", journal = "Computing Research Repository (CoRR)", volume = "abs/cs/0610139", year = "2006", bibdate = "Thu Aug 13 18:08:01 2009", bibsource = "DBLP; http://dblp.uni-trier.de/db/journals/corr/corr0610.html#abs-cs-0610139; http://www.math.utah.edu/pub/tex/bib/kepler.bib", note = "informal publication", URL = "http://arxiv.org/abs/cs/0610139", acknowledgement = ack-nhfb, } @Article{Schurmann:2006:PSC, author = "Achill Sch{\"u}rmann", title = "On packing spheres into containers", journal = j-DOC-MATH, volume = "11", pages = "393--406 (electronic)", year = "2006", ISSN = "1431-0635", ISSN-L = "1431-0635", MRclass = "52C17 (01A45 05B40)", MRnumber = "2288074 (2007j:52020)", bibdate = "Thu May 22 16:37:35 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Documenta Mathematica", keywords = "Kepler sphere-packing conjecture", } @Article{Skoge:2006:PHH, author = "Monica Skoge and Aleksandar Donev and Frank H. Stillinger and Salvatore Torquato", title = "Packing hyperspheres in high-dimensional {Euclidean} spaces", journal = j-PHYS-REV-E, volume = "74", number = "4", pages = "041127", month = oct, year = "2006", CODEN = "PLEEE8", DOI = "https://doi.org/10.1103/PhysRevE.74.041127", ISSN = "1539-3755 (print), 1550-2376 (electronic)", ISSN-L = "1539-3755", bibdate = "Wed Feb 29 20:59:25 2012", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib", URL = "http://link.aps.org/doi/10.1103/PhysRevE.74.041127", acknowledgement = ack-nhfb, fjournal = "Physical Review E (Statistical physics, plasmas, fluids, and related interdisciplinary topics)", journal-URL = "http://pre.aps.org/browse", pagecount = "11", } @Article{Torquato:2006:NCL, author = "S. Torquato and F. H. Stillinger", title = "New Conjectural Lower Bounds on the Optimal Density of Sphere Packings", journal = j-EXP-MATH, volume = "15", number = "3", pages = "307--331", month = "????", year = "2006", CODEN = "????", DOI = "https://doi.org/10.1080/10586458.2006.10128964", ISSN = "1058-6458 (print), 1944-950X (electronic)", ISSN-L = "1058-6458", bibdate = "Mon Mar 5 15:43:50 MST 2012", bibsource = "http://projecteuclid.org/euclid.em; http://www.math.utah.edu/pub/tex/bib/expmath.bib; http://www.math.utah.edu/pub/tex/bib/kepler.bib; http://www.tandfonline.com/toc/uexm20/15/3", URL = "http://projecteuclid.org/euclid.em/1175789761", acknowledgement = ack-nhfb, fjournal = "Experimental Mathematics", journal-URL = "http://www.tandfonline.com/loi/uexm20", onlinedate = "30 Jan 2011", xxauthor = "F. H. Stillinger and S. Torquato", xxpages = "307--332", } @Article{Torquato:2006:RSA, author = "S. Torquato and O. U. Uche and F. H. Stillinger", title = "Random sequential addition of hard spheres in high {Euclidean} dimensions", journal = j-PHYS-REV-E, volume = "74", number = "6 (part 1)", pages = "061308", month = dec, year = "2006", CODEN = "PLEEE8", DOI = "https://doi.org/10.1103/PhysRevE.74.061308", ISSN = "1539-3755", ISSN-L = "1539-3755", bibdate = "Fri Jun 7 07:03:20 MDT 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Physical Review E (Statistical physics, plasmas, fluids, and related interdisciplinary topics)", journal-URL = "http://pre.aps.org/browse", } @Article{Toth:2006:GEF, author = "G{\'a}bor Fejes Toth and Jeffrey C. Lagarias", title = "Guest editors' foreword {[The Kepler Conjecture by Thomas C. Hales, with Samuel P.\ Ferguson]}", journal = j-DISCRETE-COMPUT-GEOM, volume = "36", number = "1", pages = "1--3", month = jul, year = "2006", CODEN = "DCGEER", DOI = "https://doi.org/10.1007/s00454-005-1209-8", ISSN = "0179-5376 (print), 1432-0444 (electronic)", ISSN-L = "0179-5376", MRclass = "52-06 (52C17)", MRnumber = "2229656", bibdate = "Thu May 22 16:37:35 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Discrete and Computational Geometry", journal-URL = "http://link.springer.com/journal/454", keywords = "Kepler sphere-packing conjecture", } @Book{Zong:2006:CWC, author = "Chuanming Zong", title = "The Cube: a Window to Convex and Discrete Geometry", volume = "168", publisher = pub-CAMBRIDGE, address = pub-CAMBRIDGE:adr, pages = "x + 174", year = "2006", DOI = "https://doi.org/10.1017/CBO9780511543173", ISBN = "0-521-85535-7 (hardcover)", ISBN-13 = "978-0-521-85535-8 (hardcover)", LCCN = "QA639.5 .Z64 2006", MRclass = "52B12 (05C99 52B05 52B70 94A08)", MRnumber = "2221660 (2007a:52016)", bibdate = "Wed Jan 9 09:28:14 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib; z3950.loc.gov:7090/Voyager", series = "Cambridge tracts in mathematics", URL = "http://www.loc.gov/catdir/enhancements/fy0642/2006296040-d.html; http://www.loc.gov/catdir/enhancements/fy0642/2006296040-t.html; http://www.loc.gov/catdir/enhancements/fy0733/2006296040-b.html", acknowledgement = ack-nhfb, subject = "Convex geometry; Discrete geometry", } @InProceedings{Anikeenko:2007:RQR, author = "Alexey V. Anikeenko and Nikolai N. Medvedev", title = "The Role of Quasi-Regular Tetrahedra in Dense Disordered Packings of Hard Spheres", crossref = "IEEE:2007:ISV", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "130--134", year = "2007", DOI = "https://doi.org/10.1109/ISVD.2007.36", bibdate = "Thu Aug 13 18:08:01 2009", bibsource = "DBLP; http://dblp.uni-trier.de/db/conf/isvd/isvd2007.html#AnikeenkoM07; http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, } @Article{Donev:2007:UJP, author = "A. Donev and R. Connelly and F. H. Stillinger and S. Torquato", title = "Underconstrained jammed packings of nonspherical hard particles: ellipses and ellipsoids", journal = j-PHYS-REV-E, volume = "75", number = "5", pages = "051304", month = "????", year = "2007", CODEN = "PLEEE8", DOI = "https://doi.org/10.1103/PhysRevE.75.051304", ISSN = "1539-3755 (print), 1550-2376 (electronic)", ISSN-L = "1539-3755", bibdate = "Fri Aug 14 21:43:34 2009", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://link.aps.org/doi/10.1103/PhysRevE.75.051304", abstract = "Continuing on recent computational and experimental work on jammed packings of hard ellipsoids [Donev et al., Science 303, 990 (2004)] we consider jamming in packings of smooth strictly convex nonspherical hard particles. We explain why an isocounting conjecture, which states that for large disordered jammed packings the average contact number per particle is twice the number of degrees of freedom per particle ($ \bar {Z} = 2 d_f$), does not apply to nonspherical particles. We develop first- and second-order conditions for jamming and demonstrate that packings of nonspherical particles can be jammed even though they are underconstrained (hypoconstrained, $ \bar {Z} < 2 d_f$). We apply an algorithm using these conditions to computer-generated hypoconstrained ellipsoid and ellipse packings and demonstrate that our algorithm does produce jammed packings, even close to the sphere point. We also consider packings that are nearly jammed and draw connections to packings of deformable (but stiff) particles. Finally, we consider the jamming conditions for nearly spherical particles and explain quantitatively the behavior we observe in the vicinity of the sphere point.", acknowledgement = ack-nhfb, fjournal = "Physical Review E (Statistical physics, plasmas, fluids, and related interdisciplinary topics)", journal-URL = "http://pre.aps.org/browse", pagecount = "32", } @Book{Gruber:2007:CDG, author = "Peter M. Gruber", title = "Convex and discrete geometry", volume = "336", publisher = pub-SV, address = pub-SV:adr, pages = "xiii + 578", year = "2007", ISBN = "3-540-71132-5 (hardcover)", ISBN-13 = "978-3-540-71132-2 (hardcover)", LCCN = "QA639.5 .G78 2007", MRclass = "52-02 (11-02 46B20 49-02 49J53 90C25)", MRnumber = "2335496 (2008f:52001)", MRreviewer = "Aleksandr Koldobsky", bibdate = "Wed Jan 9 09:28:14 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib; z3950.loc.gov:7090/Voyager", series = "Grundlehren der mathematischen Wissenschaften", URL = "http://www.loc.gov/catdir/enhancements/fy0825/2007922936-b.html; http://www.loc.gov/catdir/enhancements/fy0825/2007922936-d.html; http://www.loc.gov/catdir/enhancements/fy0825/2007922936-t.html", acknowledgement = ack-nhfb, subject = "Convex geometry; Discrete geometry", } @InCollection{Hales:2007:EQR, author = "Thomas C. {Hales}", title = "Equidecomposable quadratic regions", crossref = "Botana:2007:ADG", pages = "24--38", year = "2007", DOI = "https://doi.org/10.1007/978-3-540-77356-6_2", MRclass = "68U05 51M20 51M25 68T15", bibdate = "Thu May 22 16:19:28 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", ZMnumber = "1195.68104", acknowledgement = ack-nhfb, keywords = "Project Flyspeck", } @InProceedings{Imamichi:2007:MSS, author = "Takashi Imamichi and Hiroshi Nagamochi", title = "A Multi-sphere Scheme for {$2$D} and {$3$D} Packing Problems", crossref = "Stutzle:2007:ESL", pages = "207--211", year = "2007", DOI = "https://doi.org/10.1007/978-3-540-74446-7_19", bibdate = "Thu Aug 13 18:08:01 2009", bibsource = "DBLP; http://dblp.uni-trier.de/db/conf/sls/sls2007.html#ImamichiN07; http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, } @Article{Kuperberg:2007:OAP, author = "Wlodzimierz Kuperberg", title = "Optimal Arrangements in Packing Congruent Balls in a Spherical Container", journal = j-DISCRETE-COMPUT-GEOM, volume = "37", number = "2", pages = "205--212", month = feb, year = "2007", CODEN = "DCGEER", DOI = "https://doi.org/10.1007/s00454-006-1303-6", ISSN = "0179-5376 (print), 1432-0444 (electronic)", ISSN-L = "0179-5376", bibdate = "Thu Aug 13 18:08:01 2009", bibsource = "DBLP; http://dblp.uni-trier.de/db/journals/dcg/dcg37.html#Kuperberg07; http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Discrete and Computational Geometry", journal-URL = "http://link.springer.com/journal/454", } @InProceedings{Nipkow:2007:FTG, author = "Tobias Nipkow and Gertrud Bauer and Paula Schultz", title = "{Flyspeck I}. {Tame} graphs", crossref = "Furbach:2007:ART", pages = "21--35", year = "2007", DOI = "https://doi.org/10.1007/11814771_4", ISSN = "0302-9743 (print), 1611-3349 (electronic)", ISSN-L = "0302-9743", MRclass = "52C17 (Packing and covering in n dimensions (discrete geometry)); 68T15 (Theorem proving (deduction, resolution, etc.))", MRnumber = "2354670", bibdate = "Thu May 22 15:38:33 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", series = "Lecture Notes in Computer Science", ZMnumber = "1222.52018", acknowledgement = ack-nhfb, keywords = "Project Flyspeck", } @Article{Trovato:2007:SSO, author = "Antonio Trovato and Trinh Xuan Hoang and Jayanth R. Banavar and Amos Maritan", title = "Symmetry, Shape, and Order", journal = j-PROC-NATL-ACAD-SCI-USA, volume = "104", number = "49", pages = "19187--19192", month = dec, year = "2007", CODEN = "PNASA6", DOI = "https://doi.org/10.1073/pnas.0707523104", ISSN = "0027-8424", ISSN-L = "0027-8424", bibdate = "Fri Jun 7 05:51:55 MDT 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Proceedings of the National Academy of Sciences of the United States of America", journal-URL = "http://www.pnas.org/search", } @Article{Zammataro:2007:ECM, author = "Luca Zammataro and Guido Serini and Todd Rowland and Federico Bussolino", title = "Embryonic cleavage modeling as a computational approach to sphere packing problem", journal = j-J-THEOR-BIOL, volume = "245", number = "1", pages = "77--82", year = "2007", CODEN = "JTBIAP", DOI = "https://doi.org/10.1016/j.jtbi.2006.09.032", ISSN = "0022-5193", ISSN-L = "0022-5193", MRclass = "92C37 (52C17)", MRnumber = "2306459 (2007m:92023)", bibdate = "Thu May 22 16:37:35 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Theoretical Biology", journal-URL = "http://www.sciencedirect.com/science/journal/00225193", keywords = "Kepler sphere-packing conjecture", } @InCollection{Anikeenko:2008:SDS, author = "Alexey V. Anikeenko and Marina L. Gavrilova and Nikolai N. Medvedev", title = "Shapes of {Delaunay} Simplexes and Structural Analysis of Hard Sphere Packings", crossref = "Gavrilova:2008:GVD", volume = "158", publisher = pub-SV, address = pub-SV:adr, pages = "13--45", year = "2008", DOI = "https://doi.org/10.1007/978-3-540-85126-4_2", ISBN = "3-540-85125-9", ISBN-13 = "978-3-540-85125-7", bibdate = "Thu Aug 13 18:08:01 2009", bibsource = "DBLP; http://dblp.uni-trier.de/db/series/sci/sci158.html#AnikeenkoGM08; http://www.math.utah.edu/pub/tex/bib/kepler.bib", series = "Studies in Computational Intelligence", acknowledgement = ack-nhfb, } @Article{Birgin:2008:MOD, author = "Ernesto G. Birgin and F. N. C. Sobral", title = "Minimizing the object dimensions in circle and sphere packing problems", journal = j-COMP-OP-RES, volume = "35", number = "7", pages = "2357--2375", year = "2008", CODEN = "CMORAP", DOI = "https://doi.org/10.1016/j.cor.2006.11.002", ISSN = "0305-0548 (print), 1873-765X (electronic)", ISSN-L = "0305-0548", bibdate = "Thu Aug 13 18:08:01 2009", bibsource = "DBLP; http://dblp.uni-trier.de/db/journals/cor/cor35.html#BirginS08; http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Computers and Operations Research", } @Article{Chen:2008:DPR, author = "Elizabeth R. Chen", title = "A Dense Packing of Regular Tetrahedra", journal = j-DISCRETE-COMPUT-GEOM, volume = "40", number = "2", pages = "214--240", month = sep, year = "2008", CODEN = "DCGEER", DOI = "https://doi.org/10.1007/s00454-008-9101-y", ISSN = "0179-5376 (print), 1432-0444 (electronic)", ISSN-L = "0179-5376", MRclass = "52C17 (52B10 52B55)", MRnumber = "2438925 (2009g:52035)", MRreviewer = "Marek Lassak", bibdate = "Fri Aug 14 13:59:06 2009", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://www.springerlink.com/content/f06j66h83424q440/fulltext.pdf", acknowledgement = ack-nhfb, fjournal = "Discrete and Computational Geometry", journal-URL = "http://link.springer.com/journal/454", } @InCollection{Chen:2008:SPP, author = "Danny Z. Chen", title = "Sphere Packing Problem", crossref = "Kao:2008:EA", year = "2008", DOI = "https://doi.org/10.1007/978-0-387-30162-4_391", bibdate = "Thu Aug 13 18:08:01 2009", bibsource = "DBLP; http://dblp.uni-trier.de/db/reference/algo/algo2008.html#Chen08; http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, } @Article{Hales:2008:FP, author = "Thomas C. Hales", title = "Formal Proof", journal = j-NAMS, volume = "55", number = "11", pages = "1370--1380", month = dec, year = "2008", CODEN = "AMNOAN", ISSN = "0002-9920 (print), 1088-9477 (electronic)", ISSN-L = "0002-9920", MRclass = "03B35 (68T15)", MRnumber = "2463990 (2010h:03012)", MRreviewer = "Jean-Fran{\c{c}}ois Dufourd", bibdate = "Wed Jan 09 07:49:58 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://www.ams.org/notices/200811/index.html; http://www.ams.org/notices/200811/tx081101370p.pdf", acknowledgement = ack-nhfb, fjournal = "Notices of the American Mathematical Society", journal-URL = "http://www.ams.org/notices/", keywords = "FPK (Formal Proof of the Kepler Conjecture); Project Flyspeck", remark = "There are three other related papers in this issue on Formal Proof.", } @Article{Harrison:2008:FPT, author = "John Harrison", title = "Formal proof---theory and practice", journal = j-NAMS, volume = "55", number = "11", pages = "1395--1406", year = "2008", CODEN = "AMNOAN", ISSN = "0002-9920 (print), 1088-9477 (electronic)", ISSN-L = "0002-9920", MRclass = "03B35 (68T15)", MRnumber = "2463992 (2010h:03013)", MRreviewer = "Jean-Fran{\c{c}}ois Dufourd", bibdate = "Thu May 22 15:23:56 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Notices of the American Mathematical Society", journal-URL = "http://www.ams.org/notices/", keywords = "Project Flyspeck", } @Article{Scardicchio:2008:EOD, author = "A. Scardicchio and F. H. Stillinger and S. Torquato", title = "Estimates of the optimal density of sphere packings in high dimensions", journal = j-J-MATH-PHYS, volume = "49", number = "4", pages = "043301", month = apr, year = "2008", CODEN = "JMAPAQ", DOI = "https://doi.org/10.1063/1.2897027", ISSN = "0022-2488 (print), 1089-7658 (electronic), 1527-2427", ISSN-L = "0022-2488", bibdate = "Wed Oct 26 09:05:55 MDT 2011", bibsource = "http://www.aip.org/ojs/jmp.html; http://www.math.utah.edu/pub/tex/bib/jmathphys2005.bib; http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://jmp.aip.org/resource/1/jmapaq/v49/i4/p043301_s1", acknowledgement = ack-nhfb, fjournal = "Journal of Mathematical Physics", journal-URL = "http://jmp.aip.org/", onlinedate = "3 April 2008", pagecount = "15", } @InCollection{Toth:2008:CCB, author = "G{\'a}bor Fejes T{\'o}th", booktitle = "Number theory \& discrete geometry", title = "Covering with convex bodies", volume = "6", publisher = "Ramanujan Mathematical Society", address = "Mysore, India", pages = "181--189", year = "2008", MRclass = "52C17", MRnumber = "2454307 (2010e:52046)", bibdate = "Thu May 22 16:37:35 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", series = "Ramanujan Math. Soc. Lect. Notes Ser.", acknowledgement = ack-nhfb, keywords = "Kepler sphere-packing conjecture", } @Article{Cohn:2009:MPT, author = "Henry Cohn", title = "Mathematical physics: {A} tight squeeze", journal = j-NATURE, volume = "460", number = "7257", pages = "801--802", day = "13", month = aug, year = "2009", CODEN = "NATUAS", DOI = "https://doi.org/10.1038/460801a", ISSN = "0028-0836 (print), 1476-4687 (electronic)", ISSN-L = "0028-0836", bibdate = "Thu Aug 13 16:53:32 2009", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://www.nature.com/nature/journal/v460/n7257/; http://www.princeton.edu/main/news/archive/S25/00/22A50/; http://www.nature.com/nature/journal/v460/n7257/pdf/460801a.pdf", abstract = "How can identical particles be crammed together as densely as possible? A combination of theory and computer simulations shows how the answer to this intricate problem depends on the shape of the particles.", acknowledgement = ack-nhfb, fjournal = "Nature", journal-URL = "http://www.nature.com/nature/archive/", remark = "Advances in packing of tetrahedra and more complex solids.", } @InCollection{Entezari:2009:SPT, author = "Alireza Entezari and Ramsay Dyer and Torsten M{\"o}ller", booktitle = "Mathematical foundations of scientific visualization, computer graphics, and massive data exploration", title = "From sphere packing to the theory of optimal lattice sampling", publisher = pub-SV, address = pub-SV:adr, pages = "227--255", year = "2009", DOI = "https://doi.org/10.1007/b106657_12", MRclass = "94A20 (52C17 65D18)", MRnumber = "2560516 (2012i:94103)", MRreviewer = "Jordi Marzo", bibdate = "Thu May 22 16:37:35 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", series = "Math. Vis.", acknowledgement = ack-nhfb, keywords = "Kepler sphere-packing conjecture", } @Article{Haji-Akbari:2009:DQC, author = "Amir Haji-Akbari and Michael Engel and Aaron S. Keys and Xiaoyu Zheng and Rolfe G. Petschek and Peter Palffy-Muhoray and Sharon C. Glotzer", title = "Disordered, quasicrystalline and crystalline phases of densely packed tetrahedra", journal = j-NATURE, volume = "462", number = "7274", pages = "773--777", day = "10", month = dec, year = "2009", CODEN = "NATUAS", DOI = "https://doi.org/10.1038/nature08641", ISSN = "0028-0836 (print), 1476-4687 (electronic)", ISSN-L = "0028-0836", bibdate = "Wed Jan 09 09:34:06 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://www.nature.com/nature/journal/v462/n7274/full/nature08641.html", acknowledgement = ack-nhfb, fjournal = "Nature", journal-URL = "http://www.nature.com/nature/archive/", } @Article{Jiao:2009:OPS, author = "Y. Jiao and F. H. Stillinger and S. Torquato", title = "Optimal packings of superballs", journal = j-PHYS-REV-E, volume = "79", number = "4", pages = "041309", month = "????", year = "2009", CODEN = "PLEEE8", DOI = "https://doi.org/10.1103/PhysRevE.79.041309", ISSN = "1539-3755 (print), 1550-2376 (electronic)", ISSN-L = "1539-3755", bibdate = "Fri Aug 14 17:20:44 2009", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://link.aps.org/doi/10.1103/PhysRevE.79.041309", abstract = "Dense hard-particle packings are intimately related to the structure of low-temperature phases of matter and are useful models of heterogeneous materials and granular media. Most studies of the densest packings in three dimensions have considered spherical shapes, and it is only more recently that nonspherical shapes (e.g., ellipsoids) have been investigated. Superballs (whose shapes are defined by $ |x1 |^{2p} + |x2 |^{2p} + |x3 |^{2p} <= 1$) provide a versatile family of convex particles ($ p \geq 0.5$) with both cubic-like and octahedral-like shapes as well as concave particles ($ 0 < p < 0.5$) with octahedral-like shapes. In this paper, we provide analytical constructions for the densest known superball packings for all convex and concave cases. The candidate maximally dense packings are certain families of Bravais lattice packings (in which each particle has 12 contacting neighbors) possessing the global symmetries that are consistent with certain symmetries of a superball. We also provide strong evidence that our packings for convex superballs ($ p >= 0.5$) are most likely the optimal ones. The maximal packing density as a function of $p$ is nonanalytic at the sphere point ($ p = 1$) and increases dramatically as p moves away from unity. Two more nontrivial nonanalytic behaviors occur at $ p_c^* >= 1.150 9...$ and $ p_o^* = \ln 3 / \ln 4 = 0.792 4...$ for cubic and octahedral superballs, respectively, where different Bravais lattice packings possess the same densities. The packing characteristics determined by the broken rotational symmetry of superballs are similar to but richer than their two-dimensional superdisk counterparts [Y. Jiao et al., Phys. Rev. Lett. 100, 245504 (2008)] and are distinctly different from that of ellipsoid packings. Our candidate optimal superball packings provide a starting point to quantify the equilibrium phase behavior of superball systems, which should deepen our understanding of the statistical thermodynamics of nonspherical-particle systems.", acknowledgement = ack-nhfb, fjournal = "Physical Review E (Statistical physics, plasmas, fluids, and related interdisciplinary topics)", journal-URL = "http://pre.aps.org/browse", pagecount = "12", } @Article{Kallus:2009:DPP, author = "Y. Kallus and V. Elser and S. Gravel", title = "Dense periodic packings of tetrahedra with small repeating units", journal = "{arXiv.org}", pages = "??", day = "27", month = oct, year = "2009", bibdate = "Wed Jan 09 10:04:27 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", note = "Published in \cite{Kallus:2010:DPP}.", URL = "http://arxiv.org/abs/0910.5226", acknowledgement = ack-nhfb, } @Article{Obua:2009:FIB, author = "Steven Obua and Tobias Nipkow", title = "{Flyspeck II}: the basic linear programs", journal = j-ANN-MATH-ARTIF-INTELL, volume = "56", number = "3-4", pages = "245--272", year = "2009", CODEN = "AMAIEC", DOI = "https://doi.org/10.1007/s10472-009-9168-z", ISSN = "1012-2443 (print), 1573-7470 (electronic)", ISSN-L = "1012-2443", MRclass = "68T15 (03B35)", MRnumber = "2595204 (2011j:68135)", bibdate = "Thu May 22 15:53:21 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Annals of Mathematics and Artificial Intelligence", journal-URL = "http://link.springer.com/journal/10472", keywords = "Project Flyspeck", } @Article{Schiftner:2009:PCS, author = "Alexander Schiftner and Mathias H{\"o}binger and Johannes Wallner and Helmut Pottmann", title = "Packing circles and spheres on surfaces", journal = j-TOG, volume = "28", number = "5", pages = "139:1--139:8", month = dec, year = "2009", CODEN = "ATGRDF", DOI = "https://doi.org/10.1145/1661412.1618485", ISSN = "0730-0301 (print), 1557-7368 (electronic)", ISSN-L = "0730-0301", bibdate = "Mon Mar 15 09:01:55 MDT 2010", bibsource = "http://www.acm.org/pubs/contents/journals/tog/; http://www.math.utah.edu/pub/tex/bib/kepler.bib", abstract = "Inspired by freeform designs in architecture which involve circles and spheres, we introduce a new kind of triangle mesh whose faces' incircles form a packing. As it turns out, such meshes have a rich geometry and allow us to cover surfaces with circle patterns, sphere packings, approximate circle packings, hexagonal meshes which carry a torsion-free support structure, hybrid tri-hex meshes, and others. We show how triangle meshes can be optimized so as to have the incircle packing property. We explain their relation to conformal geometry and implications on solvability of optimization. The examples we give confirm that this kind of meshes is a rich source of geometric structures relevant to architectural geometry.", acknowledgement = ack-nhfb, articleno = "139", fjournal = "ACM Transactions on Graphics", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J778", keywords = "architectural geometry; circle packing; computational conformal geometry; computational differential geometry; freeform surface; sphere packing; supporting structures", } @Article{Torquato:2009:ACF, author = "S. Torquato and Y. Jiao", title = "Analytical Constructions of a Family of Dense Tetrahedron Packings and the Role of Symmetry", journal = "{arXiv.org}", pages = "16", day = "21", month = dec, year = "2009", bibdate = "Wed Jan 09 10:02:53 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://arxiv.org/abs/0912.4210", acknowledgement = ack-nhfb, } @Article{Torquato:2009:DPPa, author = "S. Torquato and Y. Jiao", title = "Dense packings of the {Platonic} and {Archimedean} solids", journal = j-NATURE, volume = "460", number = "7257", pages = "876--880", day = "13", month = aug, year = "2009", CODEN = "NATUAS", DOI = "https://doi.org/10.1038/nature08239", ISSN = "0028-0836 (print), 1476-4687 (electronic)", ISSN-L = "0028-0836", bibdate = "Fri Aug 14 13:52:02 2009", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://www.nature.com/nature/journal/v460/n7257/pdf/nature08239.pdf", abstract = "Kepler's conjecture gives the densest possible packing for spherical particles but, until now, there has not been an analogous system for determining dense polyhedral packings. Using a variety of multiparticle initial configurations it has now been possible to determine the densest known packings for the Platonic solids and to conjecture that those of Platonic and Archimedean solids with central symmetry are given by their corresponding densest lattice packings.", acknowledgement = ack-nhfb, fjournal = "Nature", journal-URL = "http://www.nature.com/nature/archive/", xxpages = "876--879", } @Article{Torquato:2009:DPPb, author = "S. Torquato and Y. Jiao", title = "Dense packings of polyhedra: {Platonic} and {Archimedean} solids", journal = j-PHYS-REV-E, volume = "80", number = "4", pages = "041104:1--041104:21", year = "2009", CODEN = "PLEEE8", DOI = "https://doi.org/10.1103/PhysRevE.80.041104", ISSN = "1539-3755 (print), 1550-2376 (electronic)", ISSN-L = "1539-3755", MRclass = "52C17 (82D25)", MRnumber = "2607418 (2010i:52027)", bibdate = "Thu May 22 16:37:35 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", abstract = "Understanding the nature of dense particle packings is a subject of intense research in the physical, mathematical, and biological sciences. The preponderance of previous work has focused on spherical particles and very little is known about dense polyhedral packings. We formulate the problem of generating dense packings of nonoverlapping, nontiling polyhedra within an adaptive fundamental cell subject to periodic boundary conditions as an optimization problem, which we call the adaptive shrinking cell (ASC) scheme. This optimization problem is solved here (using a variety of multiparticle initial configurations) to find the dense packings of each of the Platonic solids in three-dimensional Euclidean space $ \mathbb {R}^3 $, except for the cube, which is the only Platonic solid that tiles space. We find the densest known packings of tetrahedra, icosahedra, dodecahedra, and octahedra with densities 0.823\ldots, 0.836\ldots, 0.904\ldots, and 0.947\ldots, respectively. It is noteworthy that the densest tetrahedral packing possesses no long-range order. Unlike the densest tetrahedral packing, which must not be a Bravais lattice packing, the densest packings of the other nontiling Platonic solids that we obtain are their previously known optimal (Bravais) lattice packings. We also derive a simple upper bound on the maximal density of packings of congruent nonspherical particles and apply it to Platonic solids, Archimedean solids, superballs, and ellipsoids. Provided that what we term the ``asphericity'' (ratio of the circumradius to inradius) is sufficiently small, the upper bounds are relatively tight and thus close to the corresponding densities of the optimal lattice packings of the centrally symmetric Platonic and Archimedean solids. Our simulation results, rigorous upper bounds, and other theoretical arguments lead us to the conjecture that the densest packings of Platonic and Archimedean solids with central symmetry are given by their corresponding densest lattice packings. This can be regarded to be the analog of Kepler's sphere conjecture for these solids. The truncated tetrahedron is the only non-centrally symmetric Archimedean solid, the densest known packing of which is a non-lattice packing with density at least as high as $ 23 / 24 = 0.958 ~333 ~ \ldots {} $. We discuss the validity of our conjecture to packings of superballs, prisms, and antiprisms as well as to high-dimensional analogs of the Platonic solids. In addition, we conjecture that the optimal packing of any convex, congruent polyhedron without central symmetry generally is not a lattice packing. Finally, we discuss the possible applications and generalizations of the ASC scheme in predicting the crystal structures of polyhedral nanoparticles and the study of random packings of hard polyhedra.", acknowledgement = ack-nhfb, fjournal = "Physical Review E (Statistical physics, plasmas, fluids, and related interdisciplinary topics)", journal-URL = "http://pre.aps.org/browse", keywords = "Kepler sphere-packing conjecture", PACS = "05.20.Jj; 45.70.An; 61.50.Ah", } @Article{vanMeel:2009:GFS, author = "J. A. van Meel and D. Frenkel and P. Charbonneau", title = "Geometrical frustration: a study of four-dimensional hard spheres", journal = j-PHYS-REV-E, volume = "79", number = "3", pages = "030201", month = mar, year = "2009", CODEN = "PLEEE8", DOI = "https://doi.org/10.1103/PhysRevE.79.030201", ISSN = "1539-3755 (print), 1550-2376 (electronic)", ISSN-L = "1539-3755", bibdate = "Wed Feb 29 21:06:47 2012", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib", URL = "http://link.aps.org/doi/10.1103/PhysRevE.79.030201", acknowledgement = ack-nhfb, fjournal = "Physical Review E (Statistical physics, plasmas, fluids, and related interdisciplinary topics)", journal-URL = "http://pre.aps.org/browse", numpages = "4", } @Article{Agapie:2010:RPH, author = "Stefan C. Agapie and Paula A. Whitlock", title = "Random packing of hyperspheres and {Marsaglia}'s parking lot test", journal = j-MONTE-CARLO-METHODS-APPL, volume = "16", number = "3--4", pages = "197--209", month = dec, year = "2010", CODEN = "MCMAC6", DOI = "https://doi.org/10.1515/mcma.2010.019", ISSN = "0929-9629 (print), 1569-3961 (electronic)", ISSN-L = "0929-9629", MRclass = "65C05 (65C10)", MRnumber = "2747812", bibdate = "Wed Feb 29 09:27:54 MST 2012", bibsource = "http://www.degruyter.com/view/j/mcma.2010.16.issue-3/issue-files/mcma.2010.16.issue-3.xml; http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/kepler.bib; http://www.math.utah.edu/pub/tex/bib/mcma.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib", URL = "http://www.degruyter.com/view/j/mcma.2010.16.issue-3-4/mcma.2010.019/mcma.2010.019.xml", acknowledgement = ack-nhfb, fjournal = "Monte Carlo Methods and Applications", journal-URL = "http://www.degruyter.com/view/j/mcma", keywords = "CDC 48-bit multiplicative congruential generator {\tt rannyu()}", remark = "The authors investigate the connection between the hypersphere packing problem and Marsaglia's parking lot test \cite{Marsaglia:1985:CVR} for correlations in random number generator output.", } @Article{Baker:2010:MMS, author = "Jessica Baker and Arshad Kudrolli", title = "Maximum and minimum stable random packings of {Platonic} solids", journal = j-PHYS-REV-E, volume = "82", number = "6 (part 1)", pages = "061304", month = dec, year = "2010", CODEN = "PLEEE8", DOI = "https://doi.org/10.1103/PhysRevE.82.061304", ISSN = "1550-2376", ISSN-L = "1539-3755", bibdate = "Fri Jun 7 06:46:20 MDT 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Physical Review E (Statistical physics, plasmas, fluids, and related interdisciplinary topics)", journal-URL = "http://pre.aps.org/browse", } @Article{Chen:2010:DCD, author = "Elizabeth R. Chen and Michael Engel and Sharon C. Glotzer", title = "Dense crystalline dimer packings of regular tetrahedra", journal = j-DISCRETE-COMPUT-GEOM, volume = "44", number = "2", pages = "253--280", month = sep, year = "2010", CODEN = "DCGEER", DOI = "https://doi.org/10.1007/s00454-010-9273-0", ISSN = "0179-5376 (print), 1432-0444 (electronic)", ISSN-L = "0179-5376", MRclass = "52C17", MRnumber = "2671012 (2011k:52028)", MRreviewer = "Peter M. Gruber", bibdate = "Wed Jan 09 08:52:40 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "", acknowledgement = ack-nhfb, fjournal = "Discrete and Computational Geometry", journal-URL = "http://link.springer.com/journal/454", } @PhdThesis{Chen:2010:PTP, author = "Elizabeth R. Chen", title = "A picturebook of tetrahedral packings", type = "{Ph.D.} Thesis", school = "University of Michigan", address = "Ann Arbor, MI. USA", pages = "????", year = "2010", bibdate = "Wed Jan 09 08:47:42 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://hdl.handle.net/2027.42/75860; http://mirlyn.lib.umich.edu/Record/008012603", abstract = "We explore many different packings of regular tetrahedra, with various clusters & lattices & symmetry groups. We construct a dense packing of regular tetrahedra, with packing density $ D > 0.7786157 $.", acknowledgement = ack-nhfb, } @Article{Cox:2010:MPC, author = "S. J. Cox and E. Flikkema", title = "The minimal perimeter for {$N$} confined deformable bubbles of equal area", journal = j-ELECT-J-COMB, volume = "17", number = "1", pages = "23", year = "2010", ISSN = "1077-8926 (print), 1097-1440 (electronic)", ISSN-L = "1077-8926", MRclass = "05B40 (52B05 52C17 74G65)", MRnumber = "2607331 (2011e:05054)", MRreviewer = "Ren Ding", bibdate = "Wed Jan 9 16:40:33 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", note = "Research Paper 45.", URL = "http://www.combinatorics.org/Volume_17/Abstracts/v17i1r45.html", acknowledgement = ack-nhfb, fjournal = "Electronic Journal of Combinatorics", } @Article{Hales:2010:DC, author = "Thomas C. Hales and Sean McLaughlin", title = "The dodecahedral conjecture", journal = j-J-AM-MATH-SOC, volume = "23", number = "2", pages = "299--344", year = "2010", DOI = "https://doi.org/10.1090/S0894-0347-09-00647-X", ISSN = "0894-0347", ISSN-L = "0894-0347", MRclass = "52C17", MRnumber = "2601036 (2011d:52037)", MRreviewer = "Peter M. Gruber", bibdate = "Thu May 22 16:37:35 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the American Mathematical Society", keywords = "Kepler sphere-packing conjecture", } @Article{Hales:2010:LPK, author = "Thomas C. Hales", title = "Linear Programs for the {Kepler} Conjecture", journal = j-LECT-NOTES-COMP-SCI, volume = "6327", pages = "149--151", year = "2010", CODEN = "LNCSD9", DOI = "https://doi.org/10.1007/978-3-642-15582-6_28", ISSN = "0302-9743 (print), 1611-3349 (electronic)", ISSN-L = "0302-9743", MRclass = "68-99 65-99", bibdate = "Sat Aug 9 15:34:18 MDT 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib; http://www.math.utah.edu/pub/tex/bib/lncs2010a.bib", URL = "http://link.springer.com/content/pdf/10.1007/978-3-642-15582-6_28.pdf", ZMnumber = "05785562", acknowledgement = ack-nhfb, book-DOI = "https://doi.org/10.1007/978-3-642-15582-6", book-URL = "http://www.springerlink.com/content/978-3-642-15582-6", fjournal = "Lecture Notes in Computer Science", keywords = "Project Flyspeck", } @Article{Hales:2010:RPK, author = "Thomas C. Hales and John Harrison and Sean McLaughlin and Tobias Nipkow and Steven Obua and Roland Zumkeller", title = "A revision of the proof of the {Kepler Conjecture}", journal = j-DISCRETE-COMPUT-GEOM, volume = "44", number = "1", pages = "1--34", year = "2010", CODEN = "DCGEER", DOI = "https://doi.org/10.1007/s00454-009-9148-4", ISSN = "0179-5376 (print), 1432-0444 (electronic)", ISSN-L = "0179-5376", MRclass = "52C17 (03B35 05B40 90C05)", MRnumber = "2639816 (2012h:52044)", MRreviewer = "Uwe Schnell", bibdate = "Thu May 22 16:37:35 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Discrete and Computational Geometry", journal-URL = "http://link.springer.com/journal/454", keywords = "Kepler sphere-packing conjecture; Project Flyspeck", } @Article{Hopkins:2010:SCM, author = "Adam B. Hopkins and Frank H. Stillinger and Salvatore Torquato", title = "Spherical codes, maximal local packing density, and the golden ratio", journal = j-J-MATH-PHYS, volume = "51", number = "4", pages = "043302", month = apr, year = "2010", CODEN = "JMAPAQ", DOI = "https://doi.org/10.1063/1.3372627", ISSN = "0022-2488 (print), 1089-7658 (electronic), 1527-2427", ISSN-L = "0022-2488", bibdate = "Wed Oct 26 16:59:49 MDT 2011", bibsource = "http://www.aip.org/ojs/jmp.html; http://www.math.utah.edu/pub/tex/bib/jmathphys2010.bib; http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://jmp.aip.org/resource/1/jmapaq/v51/i4/p043302_s1", acknowledgement = ack-nhfb, journal-URL = "http://jmp.aip.org/", onlinedate = "12 April 2010", pagecount = "6", } @Article{Kallus:2010:DPP, author = "Yoav Kallus and Veit Elser and Simon Gravel", title = "Dense periodic packings of tetrahedra with small repeating units", journal = j-DISCRETE-COMPUT-GEOM, volume = "44", number = "2", pages = "245--252", month = sep, year = "2010", CODEN = "DCGEER", DOI = "https://doi.org/10.1007/s00454-010-9254-3", ISSN = "0179-5376 (print), 1432-0444 (electronic)", ISSN-L = "0179-5376", bibdate = "Wed Jan 09 09:41:04 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "", acknowledgement = ack-nhfb, fjournal = "Discrete and Computational Geometry", journal-URL = "http://link.springer.com/journal/454", } @Article{Li:2010:MPD, author = "ShuiXiang Li and Jian Zhao and Peng Lu and Yu Xie", title = "Maximum packing densities of basic {$3$D} objects", journal = "Chinese Science Bulletin", volume = "55", number = "2", pages = "114--119", month = jan, year = "2010", DOI = "https://doi.org/10.1007/s11434-009-0650-0", bibdate = "Fri Jun 7 05:51:55 MDT 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Chinese Science Bulletin", } @Article{Smith:2010:AJS, author = "Kyle C. Smith and Meheboob Alam and Timothy S. Fisher", title = "Athermal jamming of soft frictionless {Platonic} solids", journal = j-PHYS-REV-E, volume = "82", number = "5 (part 1)", pages = "051304", month = nov, year = "2010", CODEN = "PLEEE8", DOI = "https://doi.org/10.1103/PhysRevE.82.051304", ISSN = "1550-2376", ISSN-L = "1539-3755", bibdate = "Fri Jun 7 06:46:20 MDT 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Physical Review E (Statistical physics, plasmas, fluids, and related interdisciplinary topics)", journal-URL = "http://pre.aps.org/browse", } @Article{Torquato:2010:DPP, author = "S. Torquato and Y. Jiao", title = "Dense packings of the {Platonic} and {Archimedean} solids (vol 460, pg 876, 2009)", journal = j-NATURE, volume = "463", number = "7284", pages = "??--??", month = "????", year = "2010", CODEN = "NATUAS", DOI = "https://doi.org/10.1038/nature08847", ISSN = "0028-0836", ISSN-L = "0028-0836", bibdate = "Fri Jun 7 06:46:20 MDT 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Nature", journal-URL = "http://www.nature.com/nature/archive/", onlinedate = "2010-FEB-25", } @Article{Torquato:2010:ECF, author = "S. Torquato and Y. Jiao", title = "Exact constructions of a family of dense periodic packings of tetrahedra", journal = j-PHYS-REV-E, volume = "81", number = "4 (part 1)", pages = "041310", month = apr, year = "2010", CODEN = "PLEEE8", DOI = "https://doi.org/10.1103/PhysRevE.81.041310", ISSN = "1539-3755 (print), 1550-2376 (electronic)", ISSN-L = "1539-3755", MRclass = "82D25 (52C22)", MRnumber = "2736208 (2012j:82052)", bibdate = "Wed Jan 09 10:06:16 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Physical Review E (Statistical physics, plasmas, fluids, and related interdisciplinary topics)", journal-URL = "http://pre.aps.org/browse", } @Article{Torquato:2010:JHP, author = "S. Torquato and F. H. Stillinger", title = "Jammed hard-particle packings: From {Kepler} to {Bernal} and beyond", journal = j-REV-MOD-PHYS, volume = "82", number = "3", pages = "2633--2672", month = jul, year = "2010", CODEN = "RMPHAT", DOI = "https://doi.org/10.1103/RevModPhys.82.2633", ISSN = "0034-6861 (print), 1538-4527 (electronic), 1539-0756", ISSN-L = "0034-6861", bibdate = "Tue May 22 16:37:33 MDT 2012", bibsource = "http://rmp.aps.org/toc/RMP/v82/i3; http://www.math.utah.edu/pub/tex/bib/kepler.bib; http://www.math.utah.edu/pub/tex/bib/revmodphys2010.bib", note = "See note \cite{Torquato:2010:PNJ}.", URL = "http://link.aps.org/doi/10.1103/RevModPhys.82.2633; http://rmp.aps.org/abstract/RMP/v82/i3/p2633_1", acknowledgement = ack-nhfb, fjournal = "Reviews of Modern Physics", journal-URL = "http://rmp.aps.org/browse", onlinedate = "15 September 2010", } @Article{Torquato:2010:PNJ, author = "S. Torquato and F. H. Stillinger", title = "{Publisher}'s Note: {Jammed hard-particle packings: From Kepler to Bernal and beyond [Rev. Mod. Phys. {\bf 82}, 2633 (2010)]}", journal = j-REV-MOD-PHYS, volume = "82", number = "4", pages = "3197--3197", month = oct, year = "2010", CODEN = "RMPHAT", DOI = "https://doi.org/10.1103/RevModPhys.82.3197", ISSN = "0034-6861 (print), 1538-4527 (electronic), 1539-0756", ISSN-L = "0034-6861", bibdate = "Tue May 22 16:37:34 MDT 2012", bibsource = "http://rmp.aps.org/toc/RMP/v82/i4; http://www.math.utah.edu/pub/tex/bib/kepler.bib; http://www.math.utah.edu/pub/tex/bib/revmodphys2010.bib", note = "See \cite{Torquato:2010:JHP}.", URL = "http://link.aps.org/doi/10.1103/RevModPhys.82.3197; http://rmp.aps.org/abstract/RMP/v82/i4/p3197_1", acknowledgement = ack-nhfb, fjournal = "Reviews of Modern Physics", journal-URL = "http://rmp.aps.org/browse", onlinedate = "20 December 2010", } @Article{Arkus:2011:DFS, author = "Natalie Arkus and Vinothan N. Manoharan and Michael P. Brenner", title = "Deriving Finite Sphere Packings", journal = j-SIAM-J-DISCR-MATH, volume = "25", number = "4", pages = "1860--1901", month = "????", year = "2011", CODEN = "SJDMEC", DOI = "https://doi.org/10.1137/100784424", ISSN = "0895-4801 (print), 1095-7146 (electronic)", ISSN-L = "0895-4801", bibdate = "Thu Feb 9 07:01:44 MST 2012", bibsource = "http://epubs.siam.org/sam-bin/dbq/toclist/SIDMA/25/4; http://www.math.utah.edu/pub/tex/bib/kepler.bib; http://www.math.utah.edu/pub/tex/bib/siamjdiscrmath.bib", URL = "http://epubs.siam.org/sidma/resource/1/sjdmec/v25/i4/p1860_s1", acknowledgement = ack-nhfb, fjournal = "SIAM Journal on Discrete Mathematics", journal-URL = "http://epubs.siam.org/sidma", onlinedate = "December 20, 2011", } @PhdThesis{Balou:2011:THT, author = "A. Haji Akbari Balou", title = "Thermodynamics of the hard tetrahedron system", type = "{Ph.D.} thesis", school = "University of Michigan", address = "Ann Arbor, MI, USA", pages = "????", year = "2011", bibdate = "Wed Jan 09 09:31:25 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://hdl.handle.net/2027.42/91590", acknowledgement = ack-nhfb, } @Article{Fuchs:2011:SEI, author = "Elena Fuchs and Katherine Sanden", title = "Some Experiments with Integral {Apollonian} Circle Packings", journal = j-EXP-MATH, volume = "20", number = "4", pages = "380--399", month = "????", year = "2011", CODEN = "????", DOI = "https://doi.org/10.1080/10586458.2011.565255", ISSN = "1058-6458 (print), 1944-950X (electronic)", ISSN-L = "1058-6458", bibdate = "Mon Mar 5 15:46:22 MST 2012", bibsource = "http://projecteuclid.org/euclid.em; http://www.math.utah.edu/pub/tex/bib/expmath.bib; http://www.math.utah.edu/pub/tex/bib/kepler.bib; http://www.tandfonline.com/toc/uexm20/20/4", URL = "http://projecteuclid.org/euclid.em/1323367153", acknowledgement = ack-nhfb, fjournal = "Experimental Mathematics", journal-URL = "http://www.tandfonline.com/loi/uexm20", onlinedate = "28 Nov 2011", } @Article{Gravel:2011:UBP, author = "Simon Gravel and Veit Elser and Yoav Kallus", title = "Upper bound on the packing density of regular tetrahedra and octahedra", journal = j-DISCRETE-COMPUT-GEOM, volume = "46", number = "4", pages = "799--818", month = dec, year = "2011", CODEN = "DCGEER", DOI = "https://doi.org/10.1007/s00454-010-9304-x", ISSN = "0179-5376 (print), 1432-0444 (electronic)", ISSN-L = "0179-5376", MRclass = "52C17", MRnumber = "2846180 (2012g:52035)", MRreviewer = "W. Kuperberg", bibdate = "Wed Jan 09 09:23:37 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Discrete and Computational Geometry", journal-URL = "http://link.springer.com/journal/454", } @Article{Haji-Akbari:2011:PDH, author = "Amir Haji-Akbari and Michael Engel and Sharon C. Glotzer", title = "Phase diagram of hard tetrahedra", journal = j-J-CHEM-PHYS, volume = "135", number = "19", pages = "194101:1--194101:10", day = "15", month = nov, year = "2011", CODEN = "JCPSA6", DOI = "https://doi.org/10.1063/1.3651370", ISSN = "0021-9606 (print), 1089-7690 (electronic)", ISSN-L = "0021-9606", bibdate = "Wed Jan 09 09:32:31 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://jcp.aip.org/resource/1/jcpsa6/v135/i19/p194101_s1", acknowledgement = ack-nhfb, fjournal = "Journal of Chemical Physics", journal-URL = "http://scitation.aip.org/content/aip/journal/jcp/browse", } @Book{Hales:2011:KC, author = "Thomas Hales and Samuel Ferguson", title = "The {Kepler Conjecture}", publisher = pub-SV, address = pub-SV:adr, pages = "xiv + 456", year = "2011", ISBN = "1-4614-1128-9, 1-4614-1129-7", ISBN-13 = "978-1-4614-1128-4; 978-1-4614-1129-1", MRclass = "52-02 (11H31 52C17)", MRnumber = "3075372", bibdate = "Thu May 22 16:37:35 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", note = "The Hales-Ferguson proof, Including papers reprinted from Discrete Comput. Geom. {{\bf{3}}6} (2006), no. 1, Edited by Jeffrey C. Lagarias", acknowledgement = ack-nhfb, keywords = "Kepler sphere-packing conjecture", } @Article{Jiao:2011:MRJ, author = "Yang Jiao and Salvatore Torquato", title = "Maximally random jammed packings of {Platonic} solids: hyperuniform long-range correlations and isostaticity", journal = j-PHYS-REV-E, volume = "84", number = "4 (part 1)", pages = "041309", month = oct, year = "2011", CODEN = "PLEEE8", ISSN = "1550-2376", ISSN-L = "1539-3755", bibdate = "Fri Jun 7 06:46:20 MDT 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Physical Review E (Statistical physics, plasmas, fluids, and related interdisciplinary topics)", journal-URL = "http://pre.aps.org/browse", } @InCollection{Lagarias:2011:KCP, author = "Jeffrey C. Lagarias", title = "The {Kepler Conjecture} and its proof", crossref = "Lagarias:2011:KCH", chapter = "1", publisher = pub-SV, address = pub-SV:adr, pages = "3--26", year = "2011", DOI = "https://doi.org/10.1007/978-1-4614-1129-1_1", MRclass = "52C17 (11H31)", MRnumber = "3050907", bibdate = "Thu May 22 16:37:35 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, keywords = "Kepler sphere-packing conjecture", } @Article{Marchal:2011:SKC, author = "Christian Marchal", title = "Study of the {Kepler}'s conjecture: the problem of the closest packing", journal = j-MATH-Z, volume = "267", number = "3-4", pages = "737--765", year = "2011", CODEN = "MAZEAX", DOI = "https://doi.org/10.1007/s00209-009-0644-2", ISSN = "0025-5874", ISSN-L = "0025-5874", MRclass = "52C17 (05B40)", MRnumber = "2776056 (2012b:52032)", MRreviewer = "M{\'a}rton Nasz{\'o}di", bibdate = "Thu May 22 16:37:35 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Mathematische Zeitschrift", keywords = "Kepler sphere-packing conjecture", } @Article{Andreanov:2012:RPL, author = "A. Andreanov and A. Scardicchio", title = "Random perfect lattices and the sphere packing problem", journal = j-PHYS-REV-E, volume = "86", number = "4 (part 1)", pages = "041117", month = oct, year = "2012", CODEN = "PLEEE8", ISSN = "1550-2376", ISSN-L = "1539-3755", bibdate = "Fri Jun 7 07:03:20 MDT 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Physical Review E (Statistical physics, plasmas, fluids, and related interdisciplinary topics)", journal-URL = "http://pre.aps.org/browse", } @Article{Gabbrielli:2012:FTS, author = "Ruggero Gabbrielli and Yang Jiao and Salvatore Torquato", title = "Families of tessellations of space by elementary polyhedra via retessellations of face-centered-cubic and related tilings", journal = j-PHYS-REV-E, volume = "86", number = "4 (part 1)", pages = "041141", month = oct, year = "2012", CODEN = "PLEEE8", ISSN = "1550-2376", ISSN-L = "1539-3755", bibdate = "Fri Jun 7 06:46:20 MDT 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Physical Review E (Statistical physics, plasmas, fluids, and related interdisciplinary topics)", journal-URL = "http://pre.aps.org/browse", } @Article{Garcke:2012:KKC, author = "Harald Garcke", title = "{Kepler, Kristalle und Computer. Mathematik und numerische Simulationen helfen Kristallwachstum zu verstehen}. ({German}) [{Kepler}, crystals and computer. {Mathematics} and numerical simulations help crystal-growth understanding]", journal = j-MDMV, volume = "20", number = "4", pages = "219--228", year = "2012", ISSN = "0947-4471", MRclass = "52C17 (92E10)", MRnumber = "3060788", bibdate = "Thu May 22 16:37:35 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Mitteilungen der Deutschen Mathematiker-Vereinigung", keywords = "Kepler sphere-packing conjecture", language = "German", } @Book{Hales:2012:DSP, author = "Thomas Callister Hales", title = "Dense sphere packings: a blueprint for formal proofs", volume = "400", publisher = pub-CAMBRIDGE, address = pub-CAMBRIDGE:adr, pages = "xiv + 271", year = "2012", ISBN = "0-521-61770-7 (paperback)", ISBN-13 = "978-0-521-61770-3 (paperback)", LCCN = "QA166.7 .H35 2012", bibdate = "Mon Aug 18 08:26:33 MDT 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib; z3950.loc.gov:7090/Voyager", series = "London Mathematical Society lecture note series", abstract = "The 400-year-old Kepler Conjecture asserts that no packing of congruent balls in three dimensions can have a density exceeding the familiar pyramid-shaped cannonball arrangement. In this book, a new proof of the conjecture is presented that makes it accessible for the first time to a broad mathematical audience. The book also presents solutions to other previously unresolved conjectures in discrete geometry, including the strong dodecahedral conjecture on the smallest surface area of a Voronoi cell in a sphere packing. This book is also currently being used as a blueprint for a large-scale formal proof project, which aims to check every logical inference of the proof of the Kepler conjecture by computer. This is an indispensable resource for those who want to be brought up to date with research on the Kepler Conjecture", acknowledgement = ack-nhfb, subject = "Sphere packings; Kepler's conjecture", tableofcontents = "1. Close packing \\ 2. Trigonometry \\ 3. Volume \\ 4. Hypermap \\ 5. Fan \\ 6. Packing \\ 7. Local fan \\ 8. Tame hypermap", } @Article{Joswig:2012:KHB, author = "Michael Joswig", title = "From {Kepler} to {Hales}, and back to {Hilbert}", journal = j-DOC-MATH, volume = "??", pages = "439--446", year = "2012", ISSN = "1431-0635", ISSN-L = "1431-0635", MRclass = "01A65 (03B35 05B40 52-03 52C17)", MRnumber = "2991505", bibdate = "Thu May 22 16:37:35 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", note = "Extra volume: Optimization stories.", acknowledgement = ack-nhfb, fjournal = "Documenta Mathematica", keywords = "Kepler sphere-packing conjecture", } @Article{Lagarias:2012:MPR, author = "Jeffrey C. Lagarias and Chuanming Zong", title = "Mysteries in Packing Regular Tetrahedra", journal = j-NAMS, volume = "59", number = "11", pages = "1540--1549", month = dec, year = "2012", CODEN = "AMNOAN", ISSN = "0002-9920 (print), 1088-9477 (electronic)", ISSN-L = "0002-9920", bibdate = "Wed Jan 09 07:36:55 2013", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/u/ulam-stanislaw-m.bib; http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://www.ams.org/notices/201211/rtx121101540p.pdf", acknowledgement = ack-nhfb, fjournal = "Notices of the American Mathematical Society", journal-URL = "http://www.ams.org/notices/", remark = "This article gives a nice historical survey of the Kepler sphere-packing problem, and its extensions to packing of Platonic solids, such as tetrahedra, octahedra, cubeoctahedra, and icosahedra. It also shows that several early `proofs' of their packing fractions were wrong, and gives pointers to recent work that is more reliable.", } @Article{Phillips:2012:OFS, author = "Carolyn L. Phillips and Joshua A. Anderson and Greg Huber and Sharon C. Glotzer", title = "Optimal Filling of Shapes", journal = "arxiv.org", volume = "??", number = "??", pages = "??--??", day = "11", month = feb, year = "2012", bibdate = "Wed May 16 15:23:31 2012", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", note = "arXiv:1202.2450v1", URL = "http://arxiv.org/abs/1202.2450; http://www.rdmag.com/News/2012/05/Life-Science-Mathematics-Nanotechnology-Twist-on-ancient-math-problem-could-improve-medicine-microelectronics/", abstract = "We present filling as a type of spatial subdivision problem similar to covering and packing. Filling addresses the optimal placement of overlapping objects lying entirely inside an arbitrary shape so as to cover the most interior volume. In $n$-dimensional space, if the objects are polydisperse $n$-balls, we show that solutions correspond to sets of maximal n-balls. For polygons, we provide a heuristic for finding solutions of maximal discs. We consider the properties of ideal distributions of $N$ discs as $N$ approaches infinity. We note an analogy with energy landscapes.", acknowledgement = ack-nhfb, pagecount = "5", } @Article{Torquato:2012:OPD, author = "Salvatore Torquato and Yang Jiao", title = "Organizing principles for dense packings of nonspherical hard particles: not all shapes are created equal", journal = j-PHYS-REV-E, volume = "86", number = "1 (part 1)", pages = "011102", month = jul, year = "2012", CODEN = "PLEEE8", ISSN = "1550-2376", ISSN-L = "1539-3755", bibdate = "Fri Jun 7 06:46:20 MDT 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Physical Review E (Statistical physics, plasmas, fluids, and related interdisciplinary topics)", journal-URL = "http://pre.aps.org/browse", } @InProceedings{Allamigeon:2013:CBN, author = "Xavier Allamigeon and St{\'e}phane Gaubert and Victor Magron and Benjamin Werner", title = "Certification of bounds of non-linear functions: the templates method", crossref = "Carette:2013:ICM", pages = "51--65", year = "2013", DOI = "https://doi.org/10.1007/978-3-642-39320-4_4", MRclass = "68Txx", bibdate = "Thu May 22 16:08:58 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", ZMnumber = "06197174", acknowledgement = ack-nhfb, keywords = "Project Flyspeck", } @Article{Bezdek:2013:SVK, author = "K{\'a}roly Bezdek", title = "On a strong version of the {Kepler Conjecture}", journal = j-MATHEMATIKA, volume = "59", number = "1", pages = "23--30", year = "2013", CODEN = "MTKAAB", DOI = "https://doi.org/10.1112/S0025579312001039", ISSN = "0025-5793", MRclass = "52C22 (05B40 05B45 52B60 52C17)", MRnumber = "3028169", MRreviewer = "Brigitte Servatius", bibdate = "Thu May 22 16:37:35 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, fjournal = "Mathematika. A Journal of Pure and Applied Mathematics", keywords = "Kepler sphere-packing conjecture", } @Article{Chen:2013:GCL, author = "Hao Chen", title = "On a generalization of {Craig} lattices", journal = "J. Th\'eor. Nombres Bordeaux", volume = "25", number = "1", pages = "59--70", year = "2013", ISSN = "1246-7405", MRclass = "11H31 (11H71 52C07 52C17)", MRnumber = "3063830", MRreviewer = "Ranjeet Kaur Sehmi", bibdate = "Thu May 22 16:37:35 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://jtnb.cedram.org/item?id=JTNB_2013__25_1_59_0", acknowledgement = ack-nhfb, fjournal = "Journal de Th\'eorie des Nombres de Bordeaux", keywords = "Kepler sphere-packing conjecture", } @Article{Harrison:2013:HLT, author = "John Harrison", title = "The {HOL Light} Theory of {Euclidean} Space", journal = j-J-AUTOM-REASON, volume = "50", number = "2", pages = "173--190", month = feb, year = "2013", CODEN = "JAREEW", DOI = "https://doi.org/10.1007/s10817-012-9250-9", ISSN = "0168-7433 (print), 1573-0670 (electronic)", ISSN-L = "0168-7433", MRclass = "03B35 (68T15)", MRnumber = "3016800", bibdate = "Sat Apr 2 11:10:18 MDT 2016", bibsource = "http://www.math.utah.edu/pub/tex/bib/jautomreason.bib; http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://link.springer.com/article/10.1007/s10817-012-9250-9", acknowledgement = ack-nhfb, ajournal = "J. Autom. Reason.", fjournal = "Journal of Automated Reasoning", journal-URL = "http://link.springer.com/journal/10817", keywords = "Project Flyspeck", } @InProceedings{Kaliszyk:2013:ARS, author = "Cezary Kaliszyk and Josef Urban", title = "Automated reasoning service for {HOL Light}", crossref = "Carette:2013:ICM", pages = "120--135", year = "2013", DOI = "https://doi.org/10.1007/978-3-642-39320-4_8", MRclass = "68Txx", bibdate = "Thu May 22 16:06:05 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", ZMnumber = "06197178", acknowledgement = ack-nhfb, keywords = "Project Flyspeck", } @Article{Li:2013:SCA, author = "Yanheng Li and Wei Ji", title = "Stability and convergence analysis of a dynamics-based collective method for random sphere packing", journal = j-J-COMPUT-PHYS, volume = "250", number = "??", pages = "373--387", day = "1", month = oct, year = "2013", CODEN = "JCTPAH", ISSN = "0021-9991 (print), 1090-2716 (electronic)", ISSN-L = "0021-9991", bibdate = "Mon Jul 8 16:59:11 MDT 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/jcomputphys2010.bib; http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://www.sciencedirect.com/science/article/pii/S002199911300363X", acknowledgement = ack-nhfb, fjournal = "Journal of Computational Physics", journal-URL = "http://www.sciencedirect.com/science/journal/00219991", } @InProceedings{Tankink:2013:FMD, author = "Carst Tankink and Cezary Kaliszyk and Josef Urban and Herman Geuvers", title = "Formal mathematics on display: a {Wiki} for {Flyspeck}", crossref = "Carette:2013:ICM", publisher = pub-SV, address = pub-SV:adr, pages = "152--167", year = "2013", DOI = "https://doi.org/10.1007/978-3-642-39320-4_10", ISBN = "3-642-39319-5 (paperback)", ISBN-13 = "978-3-642-39319-8 (paperback)", MRclass = "68Txx", bibdate = "Thu May 22 15:56:15 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", ZMnumber = "06197180", acknowledgement = ack-nhfb, keywords = "Project Flyspeck", } @Article{Xu:2013:PPM, author = "Bai-Xiang Xu and Yang Gao and Min-Zhong Wang", title = "Particle packing and the mean theory", journal = j-PHYS-LET-A, volume = "377", number = "3--4", pages = "145--147", year = "2013", CODEN = "PYLAAG", DOI = "https://doi.org/10.1016/j.physleta.2012.11.022", ISSN = "0375-9601 (print), 1873-2429 (electronic)", ISSN-L = "0375-9601", bibdate = "Tue Mar 14 16:13:28 2017", bibsource = "http://www.math.utah.edu/pub/tex/bib/agm.bib; http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0375960112011772", abstract = "This Letter presents two mean relations between the densities/porosities of random and regular packing modes. The two mean relations work very well for the packing of spherical particles, cubic particles and circular discs. Results confirm the corresponding experimental and computational results.", acknowledgement = ack-nhfb, fjournal = "Physics Letters A", journal-URL = "http://www.sciencedirect.com/science/journal/03759601", keywords = "Arithmetic--geometric mean; Harmonic--geometric mean; Packing density; Particle packing", } @Article{Zong:2013:TPD, author = "Chuanming Zong", title = "On the translative packing densities of tetrahedra and cubooctahedra", journal = "{arXiv.org}", pages = "37", day = "2", month = aug, year = "2013", DOI = "", bibdate = "Wed Jan 09 10:10:16 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://arxiv.org/abs/1208.0420", abstract = "In this paper, upper bounds for the densities of the densest translative tetrahedron packings and the densest translative cubooctahedron packings are obtained.", acknowledgement = ack-nhfb, } @InProceedings{Adams:2014:FF, author = "Mark Adams", title = "Flyspecking {Flyspeck}", crossref = "Hong:2014:MSI", pages = "16--20", year = "2014", DOI = "https://doi.org/10.1007/978-3-662-44199-2_3", bibdate = "Tue Feb 06 10:26:02 2018", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", abstract = "The formalisation of mathematics by use of theorem provers has reached the stage where previously questioned mathematical proofs have been formalised. However, sceptics will argue that lingering doubts remain about the efficacy of these formalisations. In this paper we motivate and describe a capability for addressing such concerns. We concentrate on the nearly-complete Flyspeck Project, which uses the HOL Light system to formalise the Kepler Conjecture proof. We first explain why a sceptic might doubt the formalisation. We go on to explain how the formal proof can be ported to the highly-trustworthy HOL Zero system and then independently audited, thus resolving any doubts.", acknowledgement = ack-nhfb, } @Article{Avigad:2014:FVM, author = "Jeremy Avigad and John Harrison", title = "Formally verified mathematics", journal = j-CACM, volume = "57", number = "4", pages = "66--75", month = apr, year = "2014", CODEN = "CACMA2", DOI = "https://doi.org/10.1145/2591012", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Wed May 21 10:20:09 MDT 2014", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; http://www.math.utah.edu/pub/tex/bib/cacm2010.bib; http://www.math.utah.edu/pub/tex/bib/kepler.bib", abstract = "With the help of computational proof assistants, formal verification could become the new standard for rigor in mathematics.", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J79", keywords = "ACL2 proof system; Boyer--Moore NQTHM theorem prover; Brouwer Fixed-Point Theorem; Coq proof system; de Bruijn factor; EQP theorem prover; Four Color Theorem; G{\"o}del Incompleteness Theorem; HOL proof system; HOL Light proof system; Isabelle proof checker; LCF proof checker; Milawa proof system; Mizar proof checker; Nuprl proof system; Prime Number Theorem; Project Flyspeck (Formal Proof of the Kepler Conjecture); Prototype Verification System (PVS) proof system; SSReflect proof language; Univalent Foundations Project", } @Article{deGoes:2014:WTG, author = "Fernando de Goes and Pooran Memari and Patrick Mullen and Mathieu Desbrun", title = "Weighted Triangulations for Geometry Processing", journal = j-TOG, volume = "33", number = "3", pages = "28:1--28:??", month = may, year = "2014", CODEN = "ATGRDF", DOI = "https://doi.org/10.1145/2602143", ISSN = "0730-0301 (print), 1557-7368 (electronic)", ISSN-L = "0730-0301", bibdate = "Mon Jun 9 12:26:19 MDT 2014", bibsource = "http://www.acm.org/pubs/contents/journals/tog/; http://www.math.utah.edu/pub/tex/bib/kepler.bib; http://www.math.utah.edu/pub/tex/bib/tog.bib", abstract = "In this article we investigate the use of weighted triangulations as discrete, augmented approximations of surfaces for digital geometry processing. By incorporating a scalar weight per mesh vertex, we introduce a new notion of discrete metric that defines an orthogonal dual structure for arbitrary triangle meshes and thus extends weighted Delaunay triangulations to surface meshes. We also present alternative characterizations of this primal-dual structure (through combinations of angles, areas, and lengths) and, in the process, uncover closed-form expressions of mesh energies that were previously known in implicit form only. Finally, we demonstrate how weighted triangulations provide a faster and more robust approach to a series of geometry processing applications, including the generation of well-centered meshes, self-supporting surfaces, and sphere packing.", acknowledgement = ack-nhfb, articleno = "28", fjournal = "ACM Transactions on Graphics", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J778", } @Article{Hu:2014:RKP, author = "Shengda Hu and Manuele Santoprete", title = "Regularization of the {Kepler} Problem on the Three-sphere", journal = j-CAN-J-MATH, volume = "66", number = "4", pages = "760--??", month = aug, year = "2014", CODEN = "CJMAAB", DOI = "https://doi.org/10.4153/CJM-2012-039-9", ISSN = "0008-414X (print), 1496-4279 (electronic)", ISSN-L = "0008-414X", bibdate = "Sat Sep 13 12:48:06 MDT 2014", bibsource = "http://cms.math.ca/cjm/v66/n4; http://www.math.utah.edu/pub/tex/bib/canjmath2010.bib; http://www.math.utah.edu/pub/tex/bib/kepler.bib", abstract = "In this paper we regularize the Kepler problem on $ S^3 $ in several different ways. First, we perform a Moser-type regularization. Then, we adapt the Ligon-Schaaf regularization to our problem. Finally, we show that the Moser regularization and the Ligon-Schaaf map we obtained can be understood as the composition of the corresponding maps for the Kepler problem in Euclidean space and the gnomonic transformation.", acknowledgement = ack-nhfb, journal-URL = "http://cms.math.ca/cjm/", } @Article{NIST:2014:CDM, author = "{NIST}", title = "The clumping density of many things seems to be a standard", journal = j-RES-DEV, volume = "??", number = "??", pages = "??--??", day = "11", month = jun, year = "2014", CODEN = "REDEEA", ISSN = "0746-9179", ISSN-L = "0746-9179", bibdate = "Wed Jun 11 18:42:06 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", note = "See research in \cite{Zangmeister:2014:PDR}", URL = "http://www.rdmag.com/news/2014/06/nist-clumping-density-many-things-seems-be-standard", abstract = "Particles of soot floating through the air and comets hurtling through space have at least one thing in common: 0.36. That, reports a research group at the National Institute of Standards and Technology (NIST), is the measure of how dense they will get under normal conditions, and it s a value that seems to be constant for similar aggregates across an impressively wide size range from nanometers to tens of meters.", acknowledgement = ack-nhfb, fjournal = "Research \& Development", } @Article{Zangmeister:2014:PDR, author = "Christopher D. Zangmeister and James G. Radney and Lance T. Dockery and Jessica T. Young and Xiaofei Ma and Rian You and Michael R. Zachariah", title = "Packing density of rigid aggregates is independent of scale", journal = j-PROC-NATL-ACAD-SCI-USA, volume = "??", number = "??", pages = "??--??", month = "????", year = "2014", CODEN = "PNASA6", DOI = "https://doi.org/10.1073/pnas.1403768111", ISSN = "0027-8424 (print), 1091-6490 (electronic)", ISSN-L = "0027-8424", bibdate = "Sat Apr 8 10:47:39 2017", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; http://www.math.utah.edu/pub/tex/bib/cacm2010.bib; http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://www.pnas.org/content/early/2014/06/04/1403768111.abstract", abstract = "Large planetary seedlings, comets, microscale pharmaceuticals, and nanoscale soot particles are made from rigid, aggregated subunits that are compacted under low compression into larger structures spanning over 10 orders of magnitude in dimensional space. Here, we demonstrate that the packing density ($ \theta_f$) of compacted rigid aggregates is independent of spatial scale for systems under weak compaction. The $ \theta_f$ of rigid aggregated structures across six orders of magnitude were measured using nanoscale spherical soot aerosol composed of aggregates with 17-nm monomeric subunits and aggregates made from uniform monomeric 6-mm spherical subunits at the macroscale. We find $ \theta_f = 0.36 \pm 0.02$ at both dimensions. These values are remarkably similar to $ \theta_f$ observed for comet nuclei and measured values of other rigid aggregated systems across a wide variety of spatial and formative conditions. We present a packing model that incorporates the aggregate morphology and show that $ \theta_f$ is independent of both monomer and aggregate size. These observations suggest that the $ \theta_f$ of rigid aggregates subject to weak compaction forces is independent of spatial dimension across varied formative conditions.", eprint = "http://www.pnas.org/content/early/2014/06/04/1403768111.full.pdf+html", fjournal = "Proceedings of the National Academy of Sciences of the United States of America", journal-URL = "http://www.pnas.org/search", } @Article{Zhang:2014:OAS, author = "Yong Zhang and Francis Y. L. Chin and Hing-Fung Ting and Xin Han and Chung Keung Poon and Yung H. Tsin and Deshi Ye", title = "Online algorithms for $1$-space bounded $2$-dimensional bin packing and square packing", journal = j-THEOR-COMP-SCI, volume = "554", number = "??", pages = "135--149", day = "16", month = oct, year = "2014", CODEN = "TCSCDI", ISSN = "0304-3975 (print), 1879-2294 (electronic)", ISSN-L = "0304-3975", bibdate = "Fri Oct 31 17:22:55 MDT 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib; http://www.math.utah.edu/pub/tex/bib/tcs2010.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0304397514001716", acknowledgement = ack-nhfb, fjournal = "Theoretical Computer Science", journal-URL = "http://www.sciencedirect.com/science/journal/03043975/", } @Article{Hales:2015:FPK, author = "Thomas Hales and Mark Adams and Gertrud Bauer and Dat Tat Dang and John Harrison and Truong Le Hoang and Cezary Kaliszyk and Victor Magron and Sean McLaughlin and Thang Tat Nguyen and Truong Quang Nguyen and Tobias Nipkow and Steven Obua and Joseph Pleso and Jason Rute and Alexey Solovyev and An Hoai Thi Ta and Trung Nam Tran and Diep Thi Trieu and Josef Urban and Ky Khac Vu and Roland Zumkeller", title = "A formal proof of the {Kepler} conjecture", journal = "ArXiv.org", pages = "21", month = jan, year = "2015", bibdate = "Tue Jul 5 07:04:31 2016", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://arxiv.org/abs/1501.02155", abstract = "This article describes a formal proof of the Kepler conjecture on dense sphere packings in a combination of the HOL Light and Isabelle proof assistants. This paper constitutes the official published account of the now completed Flyspeck project.", acknowledgement = ack-nhfb, keywords = "Mathematics --- Metric Geometry, Computer Science --- Logic in Computer Science", primaryclass = "math.MG", } @Article{Khirevich:2015:CFG, author = "Siarhei Khirevich and Irina Ginzburg and Ulrich Tallarek", title = "Coarse- and fine-grid numerical behavior of {MRT\slash TRT} lattice-{Boltzmann} schemes in regular and random sphere packings", journal = j-J-COMPUT-PHYS, volume = "281", number = "??", pages = "708--742", day = "15", month = jan, year = "2015", CODEN = "JCTPAH", ISSN = "0021-9991 (print), 1090-2716 (electronic)", ISSN-L = "0021-9991", bibdate = "Sat Nov 29 05:07:36 MST 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/jcomputphys2010.bib; http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0021999114007207", acknowledgement = ack-nhfb, fjournal = "Journal of Computational Physics", journal-URL = "http://www.sciencedirect.com/science/journal/00219991/", } @Misc{Bailey:2016:SPP, author = "David H. Bailey and Jonathan M. Borwein", title = "Sphere packing problem solved in $8$ and $24$ dimensions", howpublished = "Blog posting", day = "1", month = apr, year = "2016", bibdate = "Sat Apr 02 06:35:18 2016", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", note = "See research papers \cite{Viazovska:2016:SPP,Cohn:2016:SPP}.", URL = "http://experimentalmath.info/blog/2016/04/sphere-packing-problem-solved-in-8-and-24-dimensions/", acknowledgement = ack-nhfb, } @Article{Cohn:2016:SPP, author = "Henry Cohn and Abhinav Kumar and Stephen D. Miller and Danylo Radchenko and Maryna Viazovska", title = "The sphere packing problem in dimension $24$", journal = "arxiv.org", volume = "??", number = "??", pages = "1--12", day = "21", month = mar, year = "2016", bibdate = "Sat Apr 02 06:35:47 2016", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://arxiv.org/abs/1603.06518", acknowledgement = ack-nhfb, } @Article{Edwards:2016:NAP, author = "Chris Edwards", title = "News: Automating proofs", journal = j-CACM, volume = "59", number = "4", pages = "13--15", month = apr, year = "2016", CODEN = "CACMA2", DOI = "https://doi.org/10.1145/2892710", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Apr 15 14:58:10 MDT 2016", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; http://www.math.utah.edu/pub/tex/bib/cacm2010.bib; http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://cacm.acm.org/magazines/2016/4/200157/fulltext", abstract = "Math struggles with the usability of formal proofs.", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J79", keywords = "Archive of Formal Proofs; Coq theorem prover; Four-Color Theorem; Isabelle theorem prover; Kepler Theorem; Project Flyspeck; Thomas Hales", } @Article{Holmes-Cerfon:2016:ERS, author = "Miranda C. Holmes-Cerfon", title = "Enumerating Rigid Sphere Packings", journal = j-SIAM-REVIEW, volume = "58", number = "2", pages = "229--244", month = "????", year = "2016", CODEN = "SIREAD", DOI = "https://doi.org/10.1137/140982337", ISSN = "0036-1445 (print), 1095-7200 (electronic)", ISSN-L = "0036-1445", bibdate = "Fri May 27 07:15:57 MDT 2016", bibsource = "http://epubs.siam.org/toc/siread/58/2; http://www.math.utah.edu/pub/tex/bib/kepler.bib; http://www.math.utah.edu/pub/tex/bib/siamreview.bib", acknowledgement = ack-nhfb, fjournal = "SIAM Review", journal-URL = "http://epubs.siam.org/sirev", onlinedate = "January 2016", } @Article{Lozano:2016:EAG, author = "Elias Lozano and Deane Roehl and Waldemar Celes and Marcelo Gattass", title = "An efficient algorithm to generate random sphere packs in arbitrary domains", journal = j-COMPUT-MATH-APPL, volume = "71", number = "8", pages = "1586--1601", month = apr, year = "2016", CODEN = "CMAPDK", ISSN = "0898-1221 (print), 1873-7668 (electronic)", ISSN-L = "0898-1221", bibdate = "Wed Mar 1 21:51:55 MST 2017", bibsource = "http://www.math.utah.edu/pub/tex/bib/computmathappl2010.bib; http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0898122116300864", acknowledgement = ack-nhfb, fjournal = "Computers and Mathematics with Applications", journal-URL = "http://www.sciencedirect.com/science/journal/08981221", } @Misc{Pinter:2016:GOF, author = "Janos Pinter and Frank J. Kampas and Ignacio Castillo", title = "Globally Optimized Finite Packings of Arbitrary Size Spheres in {$ R^d $}", howpublished = "Web document", day = "31", month = mar, year = "2016", bibdate = "Tue Apr 05 06:49:15 2016", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://www.optimization-online.org/DB_HTML/2016/03/5392.html", abstract = "This work discusses the following general packing problem-class: given a finite collection of $d$-dimensional spheres with arbitrarily chosen radii, find the smallest sphere in R^d that contains the entire collection of these spheres in a non-overlapping arrangement. Generally speaking, analytical solution approaches cannot be expected to apply to this general problem-type, except for very small or certain specially structured sphere configurations. In order to find high-quality numerical (approximate) solutions, we propose a suitable combination of heuristic strategies with constrained global and local nonlinear optimization. We present numerical results for non-trivial model instance-classes of optimized sphere configurations with up to $ n = 50$ spheres in dimensions $ d = 2, 3, 4$. Our numerical results for an intensively studied model-class in $ R^2$ are on average within 1\% of the entire set of best known results, with new optimized (conjectured) packings for previously unexplored generalizations of the same model-class in $ R^d$ with $ d = 3, 4$. The results obtained also support the estimation of the optimized container sphere radii and of the packing fraction as functions of the model instance parameters $n$ and $ 1 / n$, respectively.", acknowledgement = ack-nhfb, } @Article{Viazovska:2016:SPP, author = "Maryna Viazovska", title = "The sphere packing problem in dimension $8$", journal = "arxiv.org", volume = "??", number = "??", pages = "1--22", day = "14", month = mar, year = "2016", bibdate = "Sat Apr 02 06:35:47 2016", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://arxiv.org/abs/1603.04246", acknowledgement = ack-nhfb, } @Article{Zauner:2016:AFF, author = "Thomas Zauner", title = "Application of a force field algorithm for creating strongly correlated multiscale sphere packings", journal = j-J-COMPUT-PHYS, volume = "313", number = "??", pages = "662--673", day = "15", month = may, year = "2016", CODEN = "JCTPAH", ISSN = "0021-9991 (print), 1090-2716 (electronic)", ISSN-L = "0021-9991", bibdate = "Tue Apr 5 16:33:45 MDT 2016", bibsource = "http://www.math.utah.edu/pub/tex/bib/jcomputphys2015.bib; http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0021999116001029", acknowledgement = ack-nhfb, fjournal = "Journal of Computational Physics", journal-URL = "http://www.sciencedirect.com/science/journal/00219991/", } @Article{Cohn:2017:CBS, author = "Henry Cohn", title = "A Conceptual Breakthrough in Sphere Packing", journal = j-NAMS, volume = "64", number = "2", pages = "102--115", month = feb, year = "2017", CODEN = "AMNOAN", ISSN = "0002-9920 (print), 1088-9477 (electronic)", ISSN-L = "0002-9920", bibdate = "Mon Jan 30 16:21:57 2017", bibdate = "Mon Jan 30 16:18:54 2017", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://www.ams.org/publications/journals/notices/201702/rnoti-p102.pdf", acknowledgement = ack-nhfb, fjournal = "Notices of the American Mathematical Society", journal-URL = "http://www.ams.org/notices/", keywords = "8-D sphere-packing problem; Kepler 3-D sphere-packing problem; Maryna Viazovska", } @Article{Degond:2017:DAH, author = "Pierre Degond and Marina A. Ferreira and Sebastien Motsch", title = "{Damped Arrow-Hurwicz} algorithm for sphere packing", journal = j-J-COMPUT-PHYS, volume = "332", number = "??", pages = "47--65", day = "1", month = mar, year = "2017", CODEN = "JCTPAH", ISSN = "0021-9991 (print), 1090-2716 (electronic)", ISSN-L = "0021-9991", bibdate = "Fri Jan 6 14:09:06 MST 2017", bibsource = "http://www.math.utah.edu/pub/tex/bib/jcomputphys2015.bib; http://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0021999116306398", acknowledgement = ack-nhfb, fjournal = "Journal of Computational Physics", journal-URL = "http://www.sciencedirect.com/science/journal/00219991/", }

%%% ==================================================================== %%% Cross-referenced entries must come last:

@Book{Steele:1992:PA, editor = "John Michael Steele and William F. Eddy", title = "Probability and Algorithms", publisher = pub-NAP, address = pub-NAP:adr, pages = "ix + 178", year = "1992", ISBN = "0-309-04776-5", ISBN-13 = "978-0-309-04776-0", LCCN = "QA273.P7953 1992", bibdate = "Wed Jan 9 18:09:54 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib; jenson.stanford.edu:2210/unicorn; library.mit.edu:9909/mit01", URL = "http://site.ebrary.com/lib/stanford/Doc?id=10056784; http://www.nap.edu/books/0309047765/html/", acknowledgement = ack-nhfb, remark = "Panel on Probability and Algorithms, J. Michael Steele, chair.. Committee on Applied and Theoretical Statistics, William F. Eddy, chair.", subject = "Probabilities; Algorithms; Combinatorial optimization", tableofcontents = "1: Introduction / J. Michael Steele and David Aldous \\ 2: Simulated Annealing / Dimitris Bertsimas and John Tsitsiklis \\ 3: Approximate Counting Via Markov Chains / David Aldous \\ 4: Probabilistic Algorithms for Speedup / Joan Feigenbaum and Jeffrey C. Lagarias \\ 5: Probabilistic Algorithms for Defeating Adversaries / Joan Feigenbaum \\ 6: Pseudorandom Numbers / Jeffrey C. Lagarias \\ 7: Probabilistic Analysis of Packing and Related Partitioning Problems / E. G. Coffman, Jr., D. S. Johnson, P. W. Shor and G. S. Lueker \\ 8: Probability and Problems in Euclidean Combinatorial Optimization / J. Michael Steele \\ 9: Probabilistic Analysis in Linear Programming / Ron Shamir \\ 10: Randomization in Parallel Algorithms / Vijaya Ramachandran \\ 11: Randomly Wired Multistage Networks / Bruce M. Maggs \\ 12: Missing Pieces, Derandomization, and Concluding Remarks / J. Michael Steele", } @Proceedings{Billard:1985:CSS, editor = "L. (Lynne) Billard", booktitle = "Computer science and statistics: proceedings of the Sixteenth Symposium on the Interface, Atlanta, Georgia, March 1984", title = "Computer science and statistics: proceedings of the Sixteenth Symposium on the Interface, Atlanta, Georgia, March 1984", publisher = pub-ELS, address = pub-ELS:adr, bookpages = "xi + 296", pages = "xi + 296", year = "1985", ISBN = "0-444-87725-8", ISBN-13 = "978-0-444-87725-3", LCCN = "QA276.4 .S95 1984", bibdate = "Thu Dec 18 13:41:50 2003", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib", acknowledgement = ack-nhfb, } @Book{Gruber:1993:HCG, editor = "Peter M. Gruber and J{\"o}rg M. Wills", booktitle = "Handbook of Convex Geometry", title = "Handbook of Convex Geometry", publisher = pub-NORTH-HOLLAND, address = pub-NORTH-HOLLAND:adr, pages = "xi + 735 (vol. A), lxvi + 739--1438 (vol. B)", year = "1993", ISBN = "0-444-89598-1 (set of vols. A and B), 0-444-89596-5 (vol. A), 0-444-89597-3 (vol. B)", ISBN-13 = "978-0-444-89598-1 (set of vols. A and B), 978-0-444-89596-7 (vol. A), 978-0-444-89597-4 (vol. B)", LCCN = "QA639.5 .H36 1993", bibdate = "Wed Jan 9 08:55:44 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib; z3950.loc.gov:7090/Voyager", acknowledgement = ack-nhfb, subject = "Convex geometry", xxpages = "xi + 1438", xxpages = "xlvi + 735 (vol. A), lxvi + 739--1438 (vol. B)", xxpages = "lxvi + 735 (vol. A), lxvi + 739--1438 (vol. B)", } @Book{Conway:1999:SPL, editor = "John Horton Conway and N. J. A. (Neil James Alexander) Sloane", booktitle = "Sphere packings, lattices, and groups", title = "Sphere packings, lattices, and groups", volume = "290", publisher = pub-SV, address = pub-SV:adr, edition = "Third", pages = "lxxiv + 703", year = "1999", ISBN = "0-387-98585-9 (hardcover)", ISBN-13 = "978-0-387-98585-5 (hardcover)", LCCN = "QA166.7 .C66 1999; QA3 .G75", bibdate = "Fri Aug 14 17:15:48 MDT 2009", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib; melvyl.cdlib.org:210/CDL90", series = "Grundlehren der mathematischen Wissenschaften", acknowledgement = ack-nhfb, remark = "Ch. 1. Sphere Packings and Kissing Numbers / J. H. Conway and N. J. A. Sloane \\ Ch. 2. Coverings, Lattices and Quantizers / J. H. Conway and N. J. A. Sloane \\ Ch. 3. Codes, Designs and Groups / J. H. Conway and N. J. A. Sloane \\ Ch. 4. Certain Important Lattices and Their Properties / J. H. Conway and N. J. A. Sloane \\ Ch. 5. Sphere Packing and Error-Correcting Codes / J. Leech and N. J. A. Sloane \\ Ch. 6. Laminated Lattices / J. H. Conway and N. J. A. Sloane \\ Ch. 7. Further Connections Between Codes and Lattices / N. J. A. Sloane \\ Ch. 8. Algebraic Constructions for Lattices / J. H. Conway and N. J. A. Sloane \\ Ch. 9. Bounds for Codes and Sphere Packings / N. J. A. Sloane \\ Ch. 10. Three Lectures on Exceptional Groups / J. H. Conway \\ Ch. 11. The Golay Codes and the Mathieu Groups / J. H. Conway \\ Ch. 12. A Characterization of the Leech Lattice / J. H. Conway \\ Ch. 13. Bounds on Kissing Numbers / A. M. Odlyzko and N. J. A. Sloane\ldots{} Ch. 14. Uniqueness of Certain Spherical Codes / E. Bannai and N. J. A. Sloane \\ Ch. 15. On the Classification of Integral Quadratic Forms / J. H. Conway and N. J. A. Sloane \\ Ch. 16. Enumeration of Unimodular Lattices / J. H. Conway and N. J. A. Sloane \\ Ch. 17. The 24-Dimensional Odd Unimodular Lattices / R. E. Borcherds \\ Ch. 18. Even Unimodular 24-Dimensional Lattices / B. B. Venkov \\ Ch. 19. Enumeration of Extremal Self-Dual Lattices / J. H. Conway, A. M. Odlyzko and N. J. A. Sloane \\ Ch. 20. Finding the Closest Lattice Point / J. H. Conway and N. J. A. Sloane \\ Ch. 21. Voronoi Cells of Lattices and Quantization Errors / J. H. Conway and N. J. A. Sloane \\ Ch. 22. A Bound for the Covering Radius of the Leech Lattice / S. P. Norton \\ Ch. 23. The Covering Radius of the Leech Lattice / J. H. Conway, R. A. Parker and N. J. A. Sloane \\ Ch. 24. Twenty-Three Constructions for the Leech Lattice / J. H. Conway and N. J. A. Sloane\ldots{} Ch. 25. The Cellular Structure of the Leech Lattice / R. E. Borcherds, J. H. Conway and L. Queen \\ Ch. 26. Lorentzian Forms for the Leech Lattice / J. H. Conway and N. J. A. Sloane \\ Ch. 27. The Automorphism Group of the 26-Dimensional Even Unimodular Lorentzian Lattice / J. H. Conway \\ Ch. 28. Leech Roots and Vinberg Groups / J. H. Conway and N. J. A. Sloane \\ Ch. 29. The Monster Group and its 196884-Dimensional Space / J. H. Conway \\ Ch. 30. A Monster Lie Algebra? / R. E. Borcherds, J. H. Conway and L. Queen [et al.]", subject = "combinatorial packing and covering; sphere; lattice theory; finite groups", xxauthor = "John Horton Conway and N. J. A. (Neil James Alexander) Sloane and Eiichi Bannai and others", } @Proceedings{ACM:2001:PSA, editor = "ACM", booktitle = "{Proceedings of the seventeenth annual Symposium on Computational Geometry (SCG'01): June 3--5, 2001, Medford, Massachusetts, USA}", title = "{Proceedings of the seventeenth annual Symposium on Computational Geometry (SCG'01): June 3--5, 2001, Medford, Massachusetts, USA}", publisher = pub-ACM, address = pub-ACM:adr, pages = "x + 334", year = "2001", ISBN = "1-58113-357-X", ISBN-13 = "978-1-58113-357-8", LCCN = "????", bibdate = "Fri Jan 6 12:10:47 MST 2006", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib; library.ox.ac.uk:210/ADVANCE", acknowledgement = ack-nhfb, remark = "ACM order number 429010.", subject = "Geometry; Data processing; Congresses", } @Proceedings{Amir:2001:CPM, editor = "Amihood Amir and Gad M. Landau", booktitle = "{Combinatorial pattern matching: 12th annual symposium, CPM 2001, Jerusalem, Israel, July 1--4, 2001: Proceedings}", title = "{Combinatorial pattern matching: 12th annual symposium, CPM 2001, Jerusalem, Israel, July 1--4, 2001: Proceedings}", volume = "2089", publisher = pub-SV, address = pub-SV:adr, pages = "viii + 271", year = "2001", CODEN = "LNCSD9", DOI = "????", ISBN = "3-540-42271-4", ISBN-13 = "978-3-540-42271-6", ISSN = "0302-9743 (print), 1611-3349 (electronic)", ISSN-L = "0302-9743", LCCN = "QA76.9.A43 C65 2001", bibdate = "Thu Jan 17 11:49:19 MST 2002", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", series = ser-LNCS # " and " # ser-LNAI, URL = "http://link.springer-ny.com/link/service/series/0558/tocs/t2089.htm; http://www.springerlink.com/openurl.asp?genre=issue&issn=0302-9743&volume=2089", acknowledgement = ack-nhfb, keywords = "combinatorial analysis --- congresses; computer algorithms --- congresses", } @Book{Gardner:2001:CBM, author = "Martin Gardner", booktitle = "The colossal book of mathematics: classic puzzles, paradoxes, and problems: number theory, algebra, geometry, probability, topology, game theory, infinity, and other topics of recreational mathematics", title = "The colossal book of mathematics: classic puzzles, paradoxes, and problems: number theory, algebra, geometry, probability, topology, game theory, infinity, and other topics of recreational mathematics", publisher = pub-NORTON, address = pub-NORTON:adr, pages = "xi + 724", year = "2001", ISBN = "0-393-02023-1", ISBN-13 = "978-0-393-02023-6", LCCN = "QA95 .G245 2001", bibdate = "Wed Jan 9 17:40:28 MST 2013", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/u/ulam-stanislaw-m.bib; http://www.math.utah.edu/pub/tex/bib/kepler.bib; z3950.loc.gov:7090/Voyager", acknowledgement = ack-nhfb, author-dates = "1914--2010", subject = "Mathematical recreations", tableofcontents = "Preface \\ I. Arithmetic and algebra \\ 1. The monkey and the coconuts \\ 2. The calculus of finite differences \\ 3. Palindromes : words and numbers \\ II. Plane geometry \\ 4. Curves of constant width \\ 5. Rep-tiles \\ 6. Piet Hein's superellipse \\ 7. Penrose tiles \\ 8. The wonders of a planiverse \\ III. Solid geometry and higher dimensions \\ 9. The helix \\ 10. Packing spheres \\ 11. Spheres and hyperspheres \\ 12. The Church of the Fourth Dimension \\ 13. Hypercubes \\ 14. Non-Euclidean geometry \\ IV. Symmetry \\ 15. Rotations and reflections \\ 16. The amazing creations of Scott Kim \\ 17. The art of M. C. Escher \\ V. Topology \\ 18. Klein bottles and other surfaces \\ 19. Knots \\ 20. Doughnuts : linked and knotted \\ VI. Probability \\ 21. Probability and ambiguity \\ 22. Nontransitive dice and other paradoxes \\ 23. More nontransitive paradoxes \\ VII. Infinity \\ 24. Infinite regress \\ 25. Aleph-null and aleph-one \\ 26. Supertasks \\ 27. Fractal music \\ 28. Surreal numbers \\ VIII. Combinatorics \\ 29. Hexaflexagons \\ 30. The Soma cube \\ 31. The game of life \\ 32. Paper folding \\ 33. Ramsey theory \\ 34. Bulgarian solitaire and other seemingly endless tasks \\ IX. Games and decision theory \\ 35. A matchbox game-learning machine \\ 36. Sprouts and Brussels sprouts \\ 37. Harary's generalized ticktacktoe \\ 38. The new Eleusis \\ X. Physics \\ 39. Time travel \\ 40. Does time ever stop? \\ 41. Induction and probability \\ 42. Simplicity \\ XI. Logic and philosophy \\ 43. The unexpected hanging \\ 44. Newcomb's paradox \\ 45. Nothing \\ 46. Everything \\ XII. Miscellaneous \\ 47. Melody-making machines \\ 48. Mathematical zoo \\ 49. G{\"o}del, Escher, Bach \\ 50. Six sensational discoveries \\ Selected titles by the author on mathematics \\ Index", } @Book{March:2002:ILS, author = "Norman H. (Norman Henry) March and M. P. Tosi", booktitle = "Introduction to Liquid State Physics", title = "Introduction to Liquid State Physics", publisher = pub-WORLD-SCI, address = pub-WORLD-SCI:adr, pages = "xvii + 431", year = "2002", DOI = "https://doi.org/10.1142/4717", ISBN = "981-02-4652-8 (paperback), 981-02-4639-0, 981-277-848-9 (e-book)", ISBN-13 = "978-981-277-848-2 (e-book), 978-981-02-4652-5 (paperback), 978-981-02-4639-6", LCCN = "QC145.2 .M42 2002eb", bibdate = "Tue May 28 14:24:20 MDT 2013", bibsource = "aubrey.tamu.edu:7090/voyager; catalog.princeton.edu:7090/voyager; clas.caltech.edu:210/INNOPAC; fsz3950.oclc.org:210/WorldCat; http://www.math.utah.edu/pub/bibnet/authors/m/march-norman-h.bib; http://www.math.utah.edu/pub/tex/bib/kepler.bib; jenson.stanford.edu:2210/unicorn; prodorbis.library.yale.edu:7090/voyager", URL = "http://site.ebrary.com/lib/princeton/Doc?id=10201245; http://site.ebrary.com/lib/yale/Doc?id=10201245; http://www.worldscientific.com/worldscibooks/10.1142/4717", acknowledgement = ack-nhfb, subject = "Liquids; Thermodynamics; Statistical mechanics; SCIENCE; Physics; General;Mechanics; Energy", } @Proceedings{McKay:2002:AAA, editor = "Bob McKay and J. K. Slaney", booktitle = "{AI 2002}: advances in artificial intelligence: 15th Australian Joint Conference on Artificial Intelligence, Canberra, Australia, December 2--6, 2002: Proceedings", title = "{AI 2002}: advances in artificial intelligence: 15th Australian Joint Conference on Artificial Intelligence, Canberra, Australia, December 2--6, 2002: Proceedings", volume = "2557", publisher = pub-SV, address = pub-SV:adr, pages = "xv + 730", year = "2002", CODEN = "LNCSD9", DOI = "????", ISBN = "3-540-00197-2 (paperback)", ISBN-13 = "978-3-540-00197-3 (paperback)", ISSN = "0302-9743 (print), 1611-3349 (electronic)", ISSN-L = "0302-9743", LCCN = "Q334 .A97 2002", bibdate = "Thu Nov 11 19:26:30 MST 2004", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", series = ser-LNAI # " and " # ser-LNCS, URL = "http://link.springer-ny.com/link/service/series/0558/tocs/t2557.htm; http://www.springerlink.com/openurl.asp?genre=issue&issn=0302-9743&volume=2557", acknowledgement = ack-nhfb, keywords = "artificial intelligence --- congresses", } @Proceedings{Sloot:2002:CSIc, editor = "P. M. A. Sloot and C. J. Kenneth Tan and J. J. Dongarra and A. G. Hoekstra", booktitle = "Computational Science--ICCS 2002: International Conference, Amsterdam, The Netherlands, April 21--24, 2002. Proceedings, Part III", title = "Computational Science--{ICCS 2002}: International Conference, Amsterdam, The Netherlands, April 21--24, 2002. Proceedings, {Part III}", volume = "2331", publisher = pub-SV, address = pub-SV:adr, pages = "xli + 1227", year = "2002", CODEN = "LNCSD9", ISBN = "3-540-43594-8 (paperback)", ISBN-13 = "978-3-540-43594-5 (paperback)", ISSN = "0302-9743 (print), 1611-3349 (electronic)", ISSN-L = "0302-9743", LCCN = "QA75.5 .I13 2002", bibdate = "Thu Nov 11 19:26:30 MST 2004", bibsource = "ftp://ftp.math.utah.edu/pub/bibnet/authors/d/dongarra-jack-j.bib; http://www.math.utah.edu/pub/tex/bib/kepler.bib", series = ser-LNCS, URL = "http://link.springer-ny.com/link/service/series/0558/tocs/t2331.htm", acknowledgement = ack-nhfb, } @Proceedings{Slind:2004:TPH, editor = "Konrad Slind and Annette Bunker and Ganesh Gopalakrishnan", booktitle = "{Theorem proving in higher order logics: 17th international conference, TPHOLs 2004, Park City, Utah, USA, September 14--17, 2004: proceedings}", title = "{Theorem proving in higher order logics: 17th international conference, TPHOLs 2004, Park City, Utah, USA, September 14--17, 2004: proceedings}", volume = "3223", publisher = pub-SV, address = pub-SV:adr, pages = "viii + 336", year = "2004", DOI = "https://doi.org/10.1007/b100400", ISBN = "3-540-23017-3 (paperback)", ISBN-13 = "978-3-540-23017-5 (paperback)", ISSN = "0302-9743 (print), 1611-3349 (electronic)", ISSN-L = "0302-9743", LCCN = "QA76.9.A96 T655 2004", MRclass = "68-06 (03B15 03B35 68T15)", MRnumber = "2142103 (2005m:68012)", bibdate = "Thu Aug 13 17:35:01 MDT 2009", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib; z3950.loc.gov:7090/Voyager", series = "Lecture notes in computer science", URL = "http://www.loc.gov/catdir/enhancements/fy0823/2004111288-d.html; http://www.loc.gov/catdir/summary/eng0801/2004111288.html; http://www.loc.gov/catdir/toc/fy0716/2004111288.html; http://www.springerlink.com/openurl.asp?genre=issue&issn=0302-9743&volume=3223", acknowledgement = ack-nhfb, meetingname = "TPHOLs 2004 (2004 : Park City, Utah)", subject = "Automatic theorem proving; Congresses", } @Proceedings{Hurd:2005:TPH, editor = "Joe Hurd and T. F. (Tom F.) Melham", booktitle = "{Theorem proving in higher order logics: 18th international conference, TPHOLs 2005, Oxford, UK, August 22--25, 2005: proceedings}", title = "{Theorem proving in higher order logics: 18th international conference, TPHOLs 2005, Oxford, UK, August 22--25, 2005: proceedings}", volume = "3603", publisher = pub-SV, address = pub-SV:adr, pages = "ix + 408", year = "2005", ISBN = "3-540-28372-2 (paperback)", ISBN-13 = "978-3-540-28372-0 (paperback)", ISSN = "0302-9743 (print), 1611-3349 (electronic)", ISSN-L = "0302-9743", LCCN = "QA76.9.A96 T655 2005", bibdate = "Thu May 22 16:23:45 MDT 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib; z3950.loc.gov:7090/Voyager", series = "Lecture notes in computer science", URL = "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=0302-9743&volume=3603; http://www.loc.gov/catdir/enhancements/fy0663/2005930490-d.html; http://www.loc.gov/catdir/toc/fy0612/2005930490.html", acknowledgement = ack-nhfb, meetingname = "TPHOLs 2005 (2005 : Oxford, England)", subject = "Automatic theorem proving; Congresses", } @Proceedings{Botana:2007:ADG, editor = "Francisco Botana and Tomas Recio", booktitle = "{Automated deduction in geometry: 6th international workshop, ADG 2006, Pontevedra, Spain, August 31--September 2, 2006. revised papers}", title = "{Automated deduction in geometry: 6th international workshop, ADG 2006, Pontevedra, Spain, August 31--September 2, 2006. revised papers}", volume = "4869", publisher = pub-SV, address = pub-SV:adr, pages = "x + 211", year = "2007", DOI = "https://doi.org/10.1007/978-3-540-77356-6", ISBN = "3-540-77355-X (softcover)", ISBN-13 = "978-3-540-77355-9 (softcover)", LCCN = "QA448.D38 I577 2006", MRclass = "68-06 (03B35 51-06 68T15 68U05)", MRnumber = "2866844 (2012h:68002)", bibdate = "Thu May 22 15:42:50 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib; http://www.math.utah.edu/pub/tex/bib/lncs2007a.bib", note = "Revised papers from the 6th International Workshop (ADG 2006) held at the University of Vigo, Pontevedra, August 31--September 2, 2006", series = "Lecture Notes in Computer Science", abstract = "This book constitutes the thoroughly refereed post-proceedings of the 6th International Workshop on Automated Deduction in Geometry, ADG 2006, held at Pontevedra, Spain, in August/September 2006 as a satellite event of the International Congress of Mathematicians, ICM 2006. The 13 revised full papers presented were carefully selected from the submissions made due to a call for papers - within the scope of ADG shortly after the meeting. The papers show the lively variety of topics and methods and the current applicability of automated deduction in geometry to different branches of mathematics and to other sciences and technologies.", acknowledgement = ack-nhfb, keywords = "Project Flyspeck", } @Proceedings{Furbach:2007:ART, editor = "Ulrich Furbach and Natarajan Shankar", booktitle = "{Automated Reasoning: Third International Joint Conference, IJCAR 2006, Seattle, WA, USA, August 17--20, 2006: Proceedings}", title = "{Automated Reasoning: Third International Joint Conference, IJCAR 2006, Seattle, WA, USA, August 17--20, 2006: Proceedings}", volume = "4130", publisher = pub-SV, address = pub-SV:adr, pages = "xv + 680", year = "2007", ISBN = "3-540-37187-7 (paperback), 3-540-37188-5 (e-book)", ISBN-13 = "978-3-540-37187-8 (paperback), 978-3-540-37188-5 (e-book)", LCCN = "QA76.9.A96 I33 2006", bibdate = "Thu May 22 15:32:05 MDT 2014", bibsource = "fsz3950.oclc.org:210/WorldCat; http://www.math.utah.edu/pub/tex/bib/kepler.bib; http://www.math.utah.edu/pub/tex/bib/lncs2007a.bib", series = "Lecture Notes in Computer Science Ser.", URL = "http://WX7CF7ZP2H.search.serialssolutions.com/?V=1.0&L=WX7CF7ZP2H&S=JCs&C=TC", abstract = "Here are the proceedings of the Third International Joint Conference on Automated Reasoning, IJCAR 2006, held in Seattle, Washington, USA, August 2006. The book presents 41 revised full research papers and 8 revised system descriptions, with 3 invited papers and a summary of a systems competition. The papers are organized in topical sections on proofs, search, higher-order logic, proof theory, proof checking, combination, decision procedures, CASC-J3, rewriting, and description logic.", acknowledgement = ack-nhfb, keywords = "Project Flyspeck", subject = "Automatic theorem proving; Artificial intelligence", } @Proceedings{IEEE:2007:ISV, editor = "IEEE", booktitle = "{4th International Symposium on Voronoi Diagrams in Science and Engineering (ISVD 2007): School of Computing, University of Glamorgan, Pontypprid, Wales, UK}", title = "{4th International Symposium on Voronoi Diagrams in Science and Engineering (ISVD 2007): School of Computing, University of Glamorgan, Pontypprid, Wales, UK}", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "????", year = "2007", ISBN = "????", ISBN-13 = "????", LCCN = "????", bibdate = "Thu Aug 13 18:49:38 2009", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib", acknowledgement = ack-nhfb, } @Proceedings{Stutzle:2007:ESL, editor = "Thomas St{\"u}tzle and Mauro Birattari and Holger H. Hoos", booktitle = "{Engineering Stochastic Local Search Algorithms. Designing, Implementing and Analyzing Effective Heuristics, International Workshop, SLS 2007, Brussels, Belgium, September 6--8, 2007, Proceedings}", title = "{Engineering Stochastic Local Search Algorithms. Designing, Implementing and Analyzing Effective Heuristics, International Workshop, SLS 2007, Brussels, Belgium, September 6--8, 2007, Proceedings}", volume = "4638", publisher = pub-SV, address = pub-SV:adr, year = "2007", ISBN = "3-540-74445-2", ISBN-13 = "978-3-540-74445-0", bibdate = "Thu Aug 13 18:38:32 2009", bibsource = "DBLP; http://dblp.uni-trier.de/db/conf/sls/sls2007.html; http://www.math.utah.edu/pub/tex/bib/kepler.bib", series = ser-LNCS, acknowledgement = ack-nhfb, } @Book{Gavrilova:2008:GVD, editor = "Marina L. Gavrilova", booktitle = "Generalized {Voronoi} diagram: a geometry-based approach to computational intelligence", title = "Generalized {Voronoi} diagram: a geometry-based approach to computational intelligence", volume = "158", publisher = pub-SV, address = pub-SV:adr, pages = "xxi + 312", year = "2008", DOI = "https://doi.org/10.1007/978-3-540-85126-4", ISBN = "3-540-85126-7, 3-540-85125-9", ISBN-13 = "978-3-540-85126-4, 978-3-540-85125-7", ISSN = "1860-949X", LCCN = "QA278.2 .G46 2008ebeb; QA278.2 .G46 2008eb; QA278.2; QA278.2 .G46 2008", bibdate = "Thu Aug 13 18:45:14 MDT 2009", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib; melvyl.cdlib.org:210/CDL90", series = "Studies in computational intelligence", acknowledgement = ack-nhfb, subject = "Voronoi polygons; Computational intelligence", } @Book{Kao:2008:EA, editor = "Ming-Yang Kao", booktitle = "Encyclopedia of algorithms", title = "Encyclopedia of algorithms", publisher = pub-SV, address = pub-SV:adr, pages = "liii + 1166", year = "2008", DOI = "https://doi.org/10.1007/978-0-387-30162-4", ISBN = "0-387-30162-3, 0-387-30770-2", ISBN-13 = "978-0-387-30162-4, 978-0-387-30770-1", LCCN = "QA9.58 .E52 2007eb; QA9.58 .E52 2007ebeb; QA9.58; QA9.58 .E52 2007", bibdate = "Thu Aug 13 18:40:37 MDT 2009", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib; melvyl.cdlib.org:210/CDL90", series = "Springer reference", acknowledgement = ack-nhfb, subject = "Computer algorithms; Encyclopedias; Algorithms", } @Proceedings{Fukuda:2010:MSI, editor = "Komei Fukuda and Joris {Van der Hoeven} and Michael Joswig and Nobuki Takayama", booktitle = "{Mathematical Software --- ICMS 2010: Third International Congress on Mathematical Software, Kobe, Japan, September 13--17, 2010, Proceedings}", title = "{Mathematical Software --- ICMS 2010: Third International Congress on Mathematical Software, Kobe, Japan, September 13--17, 2010, Proceedings}", publisher = pub-SV, address = pub-SV:adr, pages = "xvi + 368", year = "2010", ISBN = "3-642-15581-2 (paperback)", ISBN-13 = "978-3-642-15581-9 (paperback)", LCCN = "QA76.95 .I5654 2010", bibdate = "Thu May 22 16:13:39 MDT 2014", bibsource = "fsz3950.oclc.org:210/WorldCat; http://www.math.utah.edu/pub/tex/bib/kepler.bib", series = "Lecture Notes in Computer Science / Theoretical Computer Science and General Issues Ser.", URL = "http://link.springer.com/openurl?genre=book&isbn=978-3-642-15581-9", abstract = "This book constitutes the refereed proceedings of the Third International Congress on Mathematical Software, ICMS 2010, held in Kobe, Japan in September 2010. The 49 revised full papers presented were carefully reviewed and selected for presentation. The papers are organized in topical sections on computational group theory, computation of special functions, computer algebra and reliable computing, computer tools for mathematical editing and scientific visualization, exact numeric computation for algebraic and geometric computation, formal proof, geometry and visualization, Groebner bases and applications, number theoretical software as well as software for optimization and polyhedral computation.", acknowledgement = ack-nhfb, keywords = "Project Flyspeck", meetingname = "International Congress of Mathematical Software (3rd : 2010 : K{\=o}be-shi, Japan)", subject = "Mathematics; Data processing; Congresses; Computer software; Computer software.; Data processing.", } @Book{Lagarias:2011:KCH, editor = "Jeffrey C. Lagarias", booktitle = "The {Kepler Conjecture}: the {Hales--Ferguson} proof by {Thomas C. Hales} and {Samuel P. Ferguson}", title = "The {Kepler Conjecture}: the {Hales--Ferguson} proof by {Thomas C. Hales} and {Samuel P. Ferguson}", publisher = pub-SV, address = pub-SV:adr, pages = "xiv + 456", year = "2011", ISBN = "1-4614-1128-9 (paperback), 1-4614-1129-7 (e-book)", ISBN-13 = "978-1-4614-1128-4 (paperback), 978-1-4614-1129-1 (e-book)", LCCN = "QA166.7 .K47 2011", bibdate = "Wed Jan 9 08:09:14 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/kepler.bib; z3950.loc.gov:7090/Voyager", acknowledgement = ack-nhfb, keywords = "Samuel P. Ferguson; Thomas C. Hales", remark = "Includes a special issue of \booktitle{Discrete and Computational Geometry}.", subject = "Kepler's conjecture", tableofcontents = "Part 1. Introduction and survey \\ Chapter 1. The Kepler Conjecture and Its Proof / J. C. Lagarias / 3--26 \\ Chapter 2. Bounds for Local Density of Sphere Packings and the Kepler Conjecture / J. C. Lagarias / 27--61 \\ Part 2. Proof of the Kepler Conjecture \\ Chapter 3. Historical Overview of the Kepler Conjecture / T. C. Hales / 65--82 \\ Chapter 4. A Formulation of the Kepler Conjecture / T. C. Hales and S. P. Ferguson / 83--134 \\ Chapter 5. Sphere Packings III. Extremal Cases / T. C. Hales / 135--173 \\ Chapter 6. Sphere Packings IV. Detailed Bounds / T. C. Hales / 177--234 \\ Chapter 7. Sphere Packings V. Pentahedral Prisms / S. P. Ferguson / 235--274 \\ Chapter 8. Sphere Packings VI. Tame Graphs and Linear Programs / T. C. Hales / 275--340 \\ Part 3. A revision to the proof of the Kepler conjecture \\ Chapter 9. A Revision of the Proof of the Kepler Conjecture / T. C. Hales, J. Harrison, S. McLaughlin, T. Nipkow, S. Obua; R. Zumkeller / 341--378 \\ Part 4. Initial papers of the Hales program \\ Chapter 10. Sphere Packings I / T. C. Hales / 379--432 \\ Chapter 11. Sphere Packings II / T. C. Hales / 433--450 \\ Index of Symbols / 451--452 \\ Index of Subjects / 453--456", } @Proceedings{Carette:2013:ICM, editor = "Jacques Carette and David Aspinall and Christoph Lange and Petr Sojka and Wolfgang Windsteiger", booktitle = "{Intelligent computer mathematics: MKM, Calculemus, DML, and Systems and Projects 2013, held as part of CICM 2013, Bath, UK, July 8--12, 2013. Proceedings}", title = "{Intelligent computer mathematics: MKM, Calculemus, DML, and Systems and Projects 2013, held as part of CICM 2013, Bath, UK, July 8--12, 2013. Proceedings}", volume = "7961", publisher = pub-SV, address = pub-SV:adr, pages = "xv + 384", year = "2013", DOI = "https://doi.org/10.1007/978-3-642-39320-4", ISBN = "3-642-39319-5 (paperback), 3-642-39320-9 (e-book)", ISBN-13 = "978-3-642-39319-8 (paperback), 978-3-642-39320-4 (e-book)", LCCN = "QA76.9.M35 I58 2013", bibdate = "Thu May 22 15:57:56 MDT 2014", bibsource = "fsz3950.oclc.org:210/WorldCat; http://www.math.utah.edu/pub/tex/bib/kepler.bib; http://www.math.utah.edu/pub/tex/bib/lncs2013a.bib", series = "Lecture notes in computer science", acknowledgement = ack-nhfb, keywords = "Project Flyspeck", subject = "Informatique; Math{\'e}matiques; Congr{\`e}s; Intelligence artificielle; Computer science; Algebra; Data processing; Information storage and retrieval systems; Artificial intelligence; Text processing (Computer science)", tableofcontents = "Calculemus: The rooster and the butterflies / Assia Mahboubi \\ Optimising Problem Formulation for Cylindrical Algebraic Decomposition / Russell Bradford, James H. Davenport, Matthew England, and David Wilson \\ The Formalization of Syntax-Based Mathematical Algorithms Using Quotation and Evaluation / William M. Farmer \\ Certification of bounds of non-linear functions: the templates method / Xavier Allamigeon, St{\'e}phane Gaubert, Victor Magron, and Benjamin Werner \\ Verifying a Platform for Digital Imaging: A Multi-tool Strategy / Jonathan Heras, Gadea Mata, Ana Romero, Julio Rubio, and Rub{\'e}n S{\'a}enz \\ A Universal Machine for Biform Theory Graphs / Michael Kohlhase, Felix Manse and Florian Rabe \\ MKM: Mathematical Practice, Crowdsourcing, and Social Machines / Ursula Martin and Alison Pease \\ Automated Reasoning Service for HOL Light / Cezary Kaliszyk and Josef Urban \\ Understanding Branch Cuts of Expressions / Matthew England, Russell Bradford, James H. Davenport, and David Wilson \\ Formal Mathematics on Display: A Wiki for Flyspeck / Carst Tankink, Cezary Kaliszyk, Josef Urban, and Herman Geuvers \\ Determining Points on Handwritten Mathematical Symbols / Rui Hu and Stephen M. Watt \\ Capturing Hiproofs in HOL Light / Steven Obua, Mark Adams, and David Aspinall \\ A Qualitative Comparison of the Suitability of Four Theorem Provers for Basic Auction Theory / Christoph Lange, Marco B. Caminati, Manfred Kerber, Till Mossakowski, Colin Rowat, Makarius Wenzel, and Wolfgang Windsteiger \\ Students' Comparison of Their Trigonometric Answers with the Answers of a Computer Algebra System / Eno Tonisson \\ DML: Mathematics and the World Wide Web / Patrick D.F. Ion \\ Structural Similarity Search for Mathematics Retrieval / Shahab Kamali and Frank Wm. Tompa \\ Towards Machine-Actionable Modules of a Digital Mathematics Library: The Example of DML-CZ / Michal R{\^e}u\?zi\?cka, Petr Sojka, and Vlastimil Krej\?c\c \?r \\ A Hybrid Approach for Semantic Enrichment of MathML Mathematical Expressions / Minh-Quoc Nghiem, Giovanni Yoko Kristianto, Goran Topi{\'c}, and Akiko Aizawa \\ Three Years of DLMF: Web, Math and Search / Bruce R. Miller \\ Escaping the Trap of Too Precise Topic Queries / Paul Libbrecht \\ Using MathML to Represent Units of Measurement for Improved Ontology Alignment / Chau Do and Eric J. Pauwels \\ Systems and projects: A Web Interface for Isabelle: The Next Generation / Christoph L{\"u}th and Martin Ring \\ The ForMaRE Project--Formal Mathematical Reasoning in Economics / Christoph Lange, Colin Rowat, and Manfred Kerber \\ LATExml 2012--A Year of LATExml / Deyan Ginev and Bruce R. Miller \\ The MMT API: A Generic MKM System / Florian Rabe \\ Math-Net.Ru as a Digital Archive of the Russian Mathematical Knowledge from the XIX Century to Today / Dmitry E. Chebukov, Alexander D. Izaak, Olga G. Misyurina, Yuri A. Pupyrev, and Alexey B. Zhizhchenko \\ A Dynamic Symbolic Geometry Environment Based on the Gr{\"o}bnerCover Algorithm for the Computation of Geometric Loci and Envelopes / Miguel A. Ab{\'a}nades and Francisco Botana \\ ML4PG in computer algebra verification / J{\'o}nathan Heras and Ekaterina Komendantskaya \\ Pervasive Parallelism in Highly-Trustable Interactive Theorem Proving Systems / Bruno Barras, Lourdes del Carmen Gonz{\'a}lez Huesca, Hugo Herbelin, Yann R{\'e}gis-Gianas, Enrico Tassi, Makarius Wenzel, and Burkhart Wolff \\ The Web geometry laboratory project / Pedro Quaresma, Vanda Santos, and Seifeddine Bouallegue \\ swMATH--A New Information Service for Mathematical Software / Sebastian B{\"o}bisch, Michael Brickenstein, Hagen Chrapary, Gert-Martin Greuel, and Wolfram Sperber \\ Software for evaluating relevance of steps in algebraic transformations / Rein Prank \\ The DeLiverMATH project: text analysis in mathematics / Ulf Sch{\"o}nberg and Wolfram Sperber", xxseries = "Lecture notes in artificial intelligence; LNCS sublibrary. SL 7, Artificial intelligence", } @Proceedings{Hong:2014:MSI, editor = "Hoon Hong and Chee-Keng Yap", booktitle = "{Mathematical Software --- ICMS 2014: 4th International Conference, Seoul, South Korea, August 5--9, 2014. Proceedings}", title = "{Mathematical Software --- ICMS 2014: 4th International Conference, Seoul, South Korea, August 5--9, 2014. Proceedings}", volume = "8592", publisher = pub-SV, address = pub-SV:adr, pages = "xxxi + 735", year = "2014", DOI = "https://doi.org/10.1007/978-3-662-44199-2", ISBN = "3-662-44198-5, 3-662-44199-3 (e-book)", ISBN-13 = "978-3-662-44198-5, 978-3-662-44199-2 (e-book)", LCCN = "QA76.95", bibdate = "Sat Aug 13 13:49:23 MDT 2016", bibsource = "fsz3950.oclc.org:210/WorldCat; http://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; http://www.math.utah.edu/pub/tex/bib/kepler.bib", series = "Lecture Notes in Computer Science", abstract = "This book constitutes the proceedings of the 4th International Conference on Mathematical Software, ICMS 2014, held in Seoul, South Korea, in August 2014. The 108 papers included in this volume were carefully reviewed and selected from 150 submissions. The papers are organized in topical sections named: invited; exploration; group; coding; topology; algebraic; geometry; surfaces; reasoning; special; Gr{\"o}bner; triangular; parametric; interfaces and general.", acknowledgement = ack-nhfb, subject = "Mathematik.; Software.", tableofcontents = "Experimental Computation and Visual Theorems \\ Soft Math --- Math Soft \\ Flyspecking Flyspeck \\ Symbolic Computing Package for Mathematica for Versatile Manipulation of Mathematical Expressions \\ Representing, Archiving, and Searching the Space of Mathematical Knowledge \\ Early Examples of Software in Mathematical Knowledge Management \\ Discourse-Level Parallel Markup and Meaning Adoption in Flexiformal Theory Graphs \\ Complexity Analysis of the Bivariate Buchberger Algorithm in Theorema \\ Theorema 2.0: A System for Mathematical Theory Exploration \\ New Approaches in Black Box Group Theory \\ A GAP Package for Computing with Real Semisimple Lie Algebras \\ Bacterial Genomics and Computational Group Theory: The BioGAP Package for GAP \\ SgpDec: Cascade (De)Compositions of Finite Transformation Semigroups and Permutation Groups \\ Approximating Generators for Integral Arithmetic Groups \\ Software for Groups: Theory and Practice \\ Computation of Genus 0 Belyi Functions \\ On Computation of the First Baues-Wirsching Cohomology of a Freely-Generated Small Category \\ Codes over a Non Chain Ring with Some Applications \\ On the Weight Enumerators of the Projections of the 2-adic Golay Code of Length 24 to Z2e \\ Computer Based Reconstruction of Binary Extremal Self-dual Codes of Length 32 \\ Magma Implementation of Decoding Algorithms for General Algebraic Geometry Codes \\ Reversible Codes and Applications to DNA \\ javaPlex: A Research Software Package for Persistent (Co)Homology \\ PHAT \\ Persistent Homology Algorithms Toolbox \\ Computing Persistence Modules on Commutative Ladders of Finite Type \\ Heuristics for Sphere Recognition \\ CAPD::RedHom v2 \\ Homology Software Based on Reduction Algorithms \\ The Gudhi Library: Simplicial Complexes and Persistent Homology \\ Bertini real: Software for One- and Two-Dimensional Real Algebraic Sets \\ Hom4PS-3: A Parallel Numerical Solver for Systems of Polynomial Equations Based on Polyhedral Homotopy Continuation Methods \\ Geometry CGAL \\ Reliable Geometric Computing for Academia and Industry \\ Implementing the L[infinity] Segment Voronoi Diagram in CGAL and Applying in VLSI Pattern Analysis \\ BULL! \\ The Molecular Geometry Engine Based on Voronoi Diagram, Quasi-Triangulation, and Beta-Complex \\ Integrating Circumradius and Area Formulae for Cyclic Pentagons \\ Computer Aided Geometry \\ The Sustainability of Digital Educational Resources \\ A Touch-Operation-Based Dynamic Geometry System: Design and Implementation \\ OpenGeo: An Open Geometric Knowledge Base \\ On Computing a Cell Decomposition of a Real Surface Containing Infinitely Many Singularities \\ Robustly and Efficiently Computing Algebraic Curves and Surfaces \\ Computing the Orthogonal Projection of Rational Curves onto Rational Parameterized Surface by Symbolic Methods \\ Isotopic Approximation of Algebraic Curves \\ Isotopic Arrangement of Simple Curves: An Exact Numerical Approach Based on Subdivision \\ Real Quantifier Elimination in the RegularChains Library \\ Software for Quantifier Elimination in Propositional Logic \\ Quantifier Elimination for Linear Modular Constraints \\ Skolemization Modulo Theories \\ Incremental QBF Solving by DepQBF \\ NLCertify: A Tool for Formal Nonlinear Optimization \\ Developing Linear Algebra Packages on Risa/Asir for Eigenproblems \\ Mathematical Software for Modified Bessel Functions \\ BetaSCP2: A Program for the Optimal Prediction of Side-Chains in Proteins \\ Computation of an Improved Lower Bound to Giuga's Primality Conjecture \\ An Extension and Efficient Calculation of the Horner's Rule for Matrices \\ What Is New in CoCoA? \\ Maximizing Likelihood Function for Parameter Estimation in Point Clouds via Gr{\"o}bner Basis \\ Gr{\"o}bner Basis in Geodesy and Geoinformatics \\ Gr{\"o}bner Bases in Theorema \\ Effective Computation of Radical of Ideals and Its Application to Invariant Theory \\ Generic and Parallel Gr{\"o}bner Bases in JAS \\ Application of Gr{\"o}bner Basis Methodology to Nonlinear Mechanics Problems \\ Software for Discussing Parametric Polynomial Systems: The Gr{\"o}bner Cover \\ An Algorithm for Computing Standard Bases by Change of Ordering via Algebraic Local Cohomology \\ Verification of Gr{\"o}bner Basis Candidates \\ Cylindrical Algebraic Decomposition in the RegularChains Library \\ Hierarchical Comprehensive Triangular Decomposition \\ A Package for Parametric Matrix Computations \\ Choosing a Variable Ordering for Truth-Table Invariant Cylindrical Algebraic Decomposition by Incremental Triangular Decomposition \\ Using the Regular Chains Library to Build Cylindrical Algebraic Decompositions by Projecting and Lifting \\ An Improvement of Rosenfeld-Gr{\"o}bner Algorithm \\ Doing Algebraic Geometry with the RegularChains Library \\ On Multivariate Birkhoff Rational Interpolation \\ Computing Moore-Penrose Inverses of Ore Polynomial Matrices \\ Software Using the Gr{\"o}bner Cover for Geometrical Loci Computation and Classification \\ Using Maple's RegularChains Library to Automatically Classify Plane Geometric Loci \\ Solving Parametric Polynomial Systems by RealComprehensiveTriangularize \\ QE Software Based on Comprehensive Gr{\"o}bner Systems \\ SyNRAC: A Tolbox for Solving Real Algebraic Constraints.-An Algorithm for Computing Tjurina Stratifications of [mu]-Constant Deformations by Using Local Cohomology Classes with Parameters \\ An Implementation Method of Boolean Gr{\"o}bner Bases and Comprehensive Boolean Gr{\"o}bner Bases on General Computer Algebra Systems \\ A Method to Determine if Two Parametric Polynomial Systems Are Equal \\ An Implementation Method of a CAS with a Handwriting Interface on Tablet Devices \\ New Way of Explanation of the Stochastic Interpretation of Wave Functions and Its Teaching Materials Using KETpic \\ IFSGen4LATEX: Interactive Graphical User Interface For Generation and Visualization of Iterated Function Systems in LATEX \\ GNU TEXMACS Towards a Scientific Office Suite \\ Computer Software Program for Representation and Visualization of Free-Form Curves through Bio-inspired Optimization Techniques \\ On Some Attempts to Verify the Effect of Using High-Quality Graphics in Mathematics Education \\ Math Web Search Interfaces and the Generation Gap of Mathematicians \\ Practice with Computer Algebra Systems in Mathematics Education and Teacher Training Courses \\ Development of Visual Aid Materials in Teaching the Bivariate Normal Distributions \\ Creating Interactive Graphics for Mathematics Education Utilizing KETpic \\ A Tablet-Compatible Web-Interface for Mathematical Collaboration \\ Development and Evaluation of a Web-Based Drill System to Master Basic Math Formulae Using a New Interactive Math Input Method \\ Generating Data of Mathematical Figures for 3D Printers with KETpic and Educational Impact of the Printed Models \\ A Touch-Based Mathematical Expression Editor \\ Establishment of KETpic Programming Styles for Drawing \\ Integration of Libnormaliz in CoCoALib and CoCoA 5 \\ Elements of Design for Containers and Solutions in the LinBox Library \\ Recent Developments in Normaliz \\ The Basic Polynomial Algebra Subprograms \\ Function Interval Arithmetic \\ Generating Optimized Sparse Matrix Vector Product over Finite Fields \\ swMATH \\ An Information Service for Mathematical Software \\ MathLibre: Modifiable Desktop Environment for Mathematics \\ Software Packages for Holonomic Gradient Method \\ Metalibm: A Mathematical Functions Code Generator \\ From Calculus to Algorithms without Errors \\ Dense Arithmetic over Finite Fields with the CUMODP Library", }