\input blue.tex \loadindexmacros \report \hfuzz=25pt
\newif\ifmac
\macfalse%\mactrue
\ifmac\def\smiley{\cs{S}}\let\sadsmiley\smiley
\else
\loadsmileys \unitlength.2pt\unitlengthy.2pt\unitlengthx.2pt
\fi
\bluechapter Pictures

\beginsummary
\bluetex{} contains graphics facilities 
collected from \TB, manmac and gkpmac.
The gkpmac picture macros 
provide a subset of La\TeX's functionality.
I introduced a database of pictures to cooperate with \TeX.
Two main issues have been addressed: 
using a picture from the picturebase, and 
creating a picture.
No interaction with META\-FONT (or META\-post) nor
the incorporation of encapsulated Post\-Script have
been treated.
As appetizer a section about turtle graphics and fractals
has been inserted.
\endsummary

\bluepictures \bluemodelpic \flowchartlooppic
   \furutapic \linespic \ntgpiepic \ovalpic \overviewtexpic
    \sgmltexpic \simplepic\systemtheorypic
    \bintreepic\chartpic\cantordustpic\dragonpic
    \hfractalpic\kochfractalpic\levyfractalpic
    \pythpic\pythtreepic\rotatedbintreepic
    \sierpinskitripic\spiralpic
    \trimodelpic\tritreepic\windpic


\bluehead Introduction

\bluetex{} allows pictures to be used from the picture database pic.dat. 
For creating pictures the 
turtle graphics approach\Dash the graphics body language\Dash
manmac's \cs{makeblankbox}\Dash \TeX's boxing approach\Dash and
gkpmac's  (distilled from \LaTeX) picture environment\Dash Descartes
                                            coordinate approach\Dash 
are detailed with.\ftn{% 
   {\smc METAfont} or {\smc Metapost} are not part of \bluetex,
   nor is the use of encapsulated {\smc PostScript} considered.
   BLUe's format approach entails a high degree of portability at the expense
   of creating  picturebase entries for reuse, which can be tedious.}
The three ways of creating pictures are loosely connected.
 
\bluehead The roots

In \TB{} Appendix D, Knuth states
\beginquote
\llap{`}\TeX{} is designed to put boxes together either horizontally
or vertically, not diagonally. But that is not a serious
limitation, because the use of negative spacing makes it
possible to put things anywhere on the page.'
\endquote

Let us start with \cs{putatxy}, 
to put a dot\Dash a small \cs{hrule}\Dash
in the position $(x, y)$ relative to the reference point, and 
exercise a little `^{pointillisme} en \TeX.'
I defined \cs{x} and \cs{y} as dimen variables.
\beginverbatim
\def\putatxy{\hbox to0pt{\kern\x
                   \vbox to0pt{\vss\hrule width.3pt\kern\y}\hss}}
!endverbatim

\blueexample Sierpi\'nski carpets, via gambling

^^{Sierpi\'nski carpet}
These carpets display the point sets which emerge when we  `average' 
a point $k$ with an arbitrary corner of the (enclosing) polygon to obtain point 
$k+1$, $\forall k$.\ftn{Courtesy Alan Hoenig.}
\bgroup
%Hoenig's probabilistic/stochastic fractal,  Sierpinski's carpets, TUGboat 1989.
%Point set defined by the sequence
%(x,y)_{k+1} := [(x, y)_k +  `a corner at random']/(n-1), k = 0, 1, ... ,
%                                                     n number of corners
%cgl@rc.service.rug.nl June 1995.
%\newdimen\unitlength 
%\newcount\k \newcount\kmax 
%\newcount\throw 
\newcount\nmone
%\newdimen\x\newdimen\y
\newdimen\xi   \newdimen\yi
\newdimen\xii  \newdimen\yii
\newdimen\xiii \newdimen\yiii
\newdimen\xiv  \newdimen\yiv
\newdimen\xv   \newdimen\yv
\def\nextpoint{\throwdie
   \advance\x\ifcase\throw \xi\or\xii\or\xiii\or\xiv\or\xv\fi
   \advance\y\ifcase\throw \yi\or\yii\or\yiii\or\yiv\or\yv\fi 
   \divide\x\nmone \divide\y\nmone }             
\def\putatxy{\hbox to0pt{\kern\x
         \vbox to0pt{\vss\hrule width.3pt\kern\y
         }\hss}}
%blue.tex contains \dieiii, \dieiv, \diev, next to \rnd
%\def\dieiii{\rnd\throw=
%   \ifnum\rndval<34 0\else
%      \ifnum\rndval<67 1\else2
%   \fi\fi}

\unitlength.1pt %Scale factor
\null\kern700\unitlength %v-positioning

$$\hbox to4cm{\unitlength.075pt 
\k0 \kmax=500
\x500\unitlength    \y500\unitlength        %Initial point
\nmone2
\xi0pt      \yi0pt                          %Corners
\xii1000\unitlength   \yii0pt 
\xiii500\unitlength  \yiii1000\unitlength   
\let\throwdie\dieiii
\loop \nextpoint \putatxy \advance\k1
   \ifnum\k<\kmax 
\repeat
\hss}\hbox to4.25cm{\unitlength.15pt 
\k0 \kmax400
\x500\unitlength     \y500\unitlength       %Initial point
\nmone3
\xi0pt               \yi0pt                 %Corners
\xii1000\unitlength  \yii0pt 
\xiii1000\unitlength \yiii1000\unitlength  
\xiv0pt              \yiv1000\unitlength  
\let\throwdie\dieiv
\loop \nextpoint \putatxy \advance\k1
   \ifnum\k<\kmax
\repeat
\hss}\hbox to2cm{\unitlength.045pt 
\k0 \kmax=400
\x500\unitlength    \y500\unitlength        %Initial point
\nmone4
\xi0pt      \yi0pt                          %Corners
\xii3000\unitlength  \yii0pt 
\xiii4000\unitlength \yiii2800\unitlength  
\xiv1500\unitlength  \yiv5000\unitlength  
\xv-1000\unitlength  \yv2800\unitlength  
\let\throwdie\diev
\loop \nextpoint \putatxy \advance\k1
   \ifnum\k<\kmax 
\repeat
\hss}$$

\kern70\unitlength
\egroup
Amazing isn't it? 
The figures are essentially obtained via\ftn{For
   the coding details consult the file pwt.grf.}
\thisverbatim{\unmc}
\beginverbatim
\loop \nextpoint \putatxy \advance\k1 \ifnum\k<\kmax \repeat
!endverbatim
The following more general \cs{point}, from \TB{} {\oldstyle389},  
is the basis for \LaTeX's picture environment.
\begincenterverbatim
\hbox{\unitlength\baselineskip\point 0 0 \point 1 1.5 }%etc.
%with
\newdimen\unitlength
\def\point#1 #2 {\rlap{\kern#1\unitlength\raise#2\unitlength\hbox{$
   \scriptstyle\bullet\;(#1,#2)$}}}
!endcenterverbatim

The examples with quarter circles, \TB{} {\oldstyle389}--{\oldstyle390},
the dragon figures, remind me of Papert's `Turtle' graphics.
The latter approach has been implemented by Gurari in 
`\TeX{} \& \LaTeX\Dash Drawing \& Literate Programming.'
The displayed fractals below are small, of low order, because of the memory needed,
and \TeX's limitations. However, the programs can be used  stand-alone for larger
orders.

\everyverbatim{\unmc}

\bluehead Turtle graphics digression

^^{Turtle graphics}%
Knuth showed how to cope with turtle graphics in the infamous 
Appendix~D by the so-called dragon figures.
Possibly because of the popularity of the \LaTeX{} picture environment next to 
the ubiquitous (PC) displays and laser printers, that is, raster and vector graphics, 
the Turtle graphics approach\Dash a turtle paints its movements, a pedagogical\ftn{See for 
        example Seymour Papert's Mindstorms, which builds on the pedagogical principles
        of Piaget. Gurari has implemented a turtle graphics suite in Dra\TeX.} 
     synonym for the pen of the plotter, very close to handdrawing\Dash
has faded. 
For highly systematic pictures like (simple, deterministic) fractals,\ftn{Geometric figures 
   in endless repetition. A teaser for WYSIWYG oriented people.}
the turtle graphics approach can be beneficial.\ftn{Much of the work in this section
   was incented by Gurari's Drawings and Literate programming work. 
   Details and algorithms for drawing fractals
   have been borrowed from Lauwerier's `Fractals\Dash meetkundige figuren in eindeloze
   herhaling.' (It contains  programs in BASIC to draw the fractals.)}
To quote Papert
\begincenter
`Turtle geometry is a {\em computationally\/} style of geometry.'
\endcenter 

Let us concentrate on the four directions of the wind north, east, south, west, and
I'll show what can be obtained by this.
Then we will extend the movements to {\oldstyle45}$^\circ$ directions, i.e., the halfwinds 
northeast, southeast, southwest, and northwest. The argument of these wind directions
is assumed to expand to a `factor,' i.e. an integer or decimal constant, 
see \TB{} {\oldstyle270} for the syntax of factor. 

The basic idea is that the position of the turtle is maintained in \cs{x}, \cs{y},
with the reference point left invariant. The accompanying figure shows the effect of \cs{N1}.
%
%Perhaps I better put it in pic.dat
\vskip2.5\bigskipamount
$$\unitlength.5ex
  \y=0pt\x-10\unitlength
  {\W{4}\E{40}}\S{4}%
  {\advance\y-2\unitlength\xytxt{ Ref.\ point}}%
  \N{20}\whiteE{20}\S4\whiteS4\xytxt{ (x,y)}$$
\vskip1.5\bigskipamount
%
The movements\Dash our first steps in the turtle graphics world\Dash 
can be achieved by the control sequences
\bitem \cs{N}1 means: draw the line (\cs{x}, \cs{y}))\dash(\cs{x}, \cs{y}+1)),  that is 
       the turtle moves up.
       After completion \cs{y} has been increased by \cs{unitlength}.
\bitem \cs{N}, \cs{E}, \cs{S}, \cs{W} mean draw north, east, south and west.
\bitem \cs{whiteN}, \cs{whiteE}, \cs{whiteS}, \cs{whiteW} mean
      {\em  white}-draw north, east, south and west, i.e., the turtle just moves.
\smallbreak

\blueexample A line bundle

Important is that the length of the lines in each direction is given by the argument.
This is in contrast with \LaTeX's (and gkp's) picture environment.
\begindemo 
\thispicture{\unitlength5ex}
$$\windpic$$
!yields
\thispicture{\unitlength5ex}
$$\windpic$$
\enddemo

Let us look at an example, where \cs{E} is used only\Dash just moving in one direction\Dash 
to get the so-called Cantor dust ^{fractal}.

\blueexample  Cantor dust fractal

^^{Cantor dust fractal}
^^{Cantor dust fractal, two-dimensional}
^^{Sierpi\'nski carpet}
The idea is that each line is replaced by its left and right third parts.
In the picture the upper line is {\oldstyle243}pt, and the {\oldstyle32} 
lines at the bottom are each {\oldstyle1}pt.
This pattern can be repeated yielding what is called a fractal.\ftn{\cs{cantordustpic}
    is included in pic.dat. Generalization to {\oldstyle2} dimensions yield the so-called
    Sierpi\'nski carpets. For example subdivide a square in $3^2$ equal subsquares
    and delete the middle, or the middle cross, 
    and this repeatedly for the remaining squares.
    pic.dat contains \cs{sierpinskisquarepic} and \cs{sierpinskicrosspic}.}
\begindemo
\def\cf{{\E{\the\k}}\divide\k3
   \ifnum\k=0 \fc\fi   
   \advance\y-3\unitlength{\cf}%
   \advance\x\the\k\unitlength
   \advance\x\the\k\unitlength
   \cf\relax}
\def\fc#1\relax{\fi}
$$\k243 \unitlength1pt  
  \x-\the\k\unitlength\divide\x2 
  \cf$$
%with auxiliary
\def\E#1{\hbox to0pt{\kern\x
   \vbox to0pt{\vss\hrule width
      #1\unitlength\kern\y}\hss
   }\advance\x#1\unitlength}
!yields
\thispicture{\xoffset{10}}
$$\cantordustpic$$
\enddemo

The maze example of Salomon\Dash see also the Tabular Material chapter\Dash 
can be obtained straightforward as follows.

\blueexample Maze

^^{Maze}
\begindemo
$$\unitlength3ex\x-2\unitlength\y0pt
  \N1\E2%
   \S1\W1\S2\E2\N1\W1%
   \whiteN1\E1\whiteN1\whiteW1% 
  \E2\S4\W2\whiteW1\W1\N2$$
!yields
$$\unitlength3ex\x-2\unitlength\y0pt
  \N1\E2\S1\W1%
    \S2\E2\N1\W1\whiteN1\E1\whiteN1\whiteW1%inner 
  \E2\S4\W2\whiteW1\W1\N2$$
\enddemo

\blueexample Spiral

\begindemo
$$\unitlength.5ex \y0pt \k30 
 \x-\the\k\unitlength\divide\x2
 \loop\E{\the\k}\advance\k-1
      \S{\the\k}\advance\k-1
      \W{\the\k}\advance\k-1
      \N{\the\k}\advance\k-1
 \ifnum\k>4 \repeat$$
!yields
$$\spiralpic$$
\enddemo
The \cs{spiralpic} has been added to pic.dat.

\exercise How can we let the turtle draw nested squares?
\answer The nested squares can be obtained as follows.
\beginverbatim
$$\unitlength.5ex  \k32 
\loop\x-\the\k\unitlength 
   \divide\x2 \y-\x 
   \E{\the\k}\S{\the\k}\W{\the\k}\N{\the\k}%
\advance\k-2 \ifnum\k>1
\repeat$$
!endverbatim

%endanswer

\blueexample Binary tree

^^{Binary tree}
%Turtle-draw the binary (fractal) tree.
%\def\bintree{\S1\ifnum\kk=2 \eertnib\fi
%   \divide\kk2
%   {\W{\the\kk}\bintree}%
%   \E{\the\kk}\bintree\relax}
%\def\eertnib#1\relax{\fi}
\begindemo
\def\bintree{\S1\ifnum\kk=2 \eertnib\fi
   \divide\kk2 {\W{\the\kk}\bintree}%
   \E{\the\kk}\bintree\relax}
\def\eertnib#1\relax{\fi}
$$\def\eertnib#1\relax{\fi}
  \unitlength.5ex\x0pt\y0pt
  \kk32 \bintree$$
!yields
$$\bintreepic$$
\enddemo
\exercise How can we add leaves?
\answer To give you an idea how to add leaves  the following.
\beginverbatim
\def\eertnib#1\relax{\fi\global\advance\k1
   \whiteS3\trttxt{\hss\csname\the\k\endcsname}}
%with
\def\1{x} \def\2{y} %etc.
!endverbatim

%endanswer 

\bgroup
\blueexample Rotated tree

Once we understand turtle graphics, rotating a tree can be done easily 
via shifting the meaning of the directions, and adjusting
positioning of the leaves.\ftn{A white lie. The tree is actually mirrored
   because I like the leaves to be numbered from the top. 
   In general we can rotate via \PS.}
%\def\bintree{\S1\ifnum\kk=2 \eertnib\fi
%   \divide\kk2
%   {\W{\the\kk}\bintree}%
%   \E{\the\kk}\bintree\relax}
\begindemo
%rotation via shifting
\let\W\S \let\S\E \let\E\N
%the leaves
\def\1{x}\def\2{y}\def\3{a}\def\4{b}
\def\5{piet}\def\6{hans}\def\7{etc.}
$$\k0\unitlength1ex\x0pt\y0pt
  \kk16\bintree$$
%with
\def\eertnib#1\relax{\fi\global\advance\k1
 \whiteS1\xytxt{ \csname\the\k\endcsname}}
!yields
$$\thispicture{\yoffset{-16}}
  \rotatedbintreepic$$
\enddemo
\egroup
Because of its usefulness the above 
\cs{bintree} and \cs{eertnib} are in blue.tex.

%chart TB 22.14, 248 cgl@rc.service.rug.nl
%\input blue.tex \report
The chart example\Dash see also the Tabular Material chapter\Dash
can be done too via a variant binary tree implementation.
\bgroup\newcount\ewidth
\blueexample Chart, \TB{} ex{\oldstyle22}.{\oldstyle14} {\oldstyle248}

\begindemo
%labels in preorder
\def\1{LMB, 1912}
\def\2{MJHB, 1882}\def\5{PAME, 1884}
\def\3{JHB, 1838}\def\4{MDB, 1840}
\def\6{EFE, 1845}\def\7{CLW, 1850}
\ekk8\k0\unitlength2ex
\x0pt\y0pt\kk8\hbox{\bintree}
%with auxiliaries
\def\bintree{\E{\the\ekk}%
   \ifnum\kk=2 \eertnib\fi
   \divide\kk2
   {\N{\the\kk}\bintree}%
   \S{\the\kk}\bintree\relax}
\def\eertnib#1\relax{\fi}
%
\let\Eold\E
\def\E{\global\advance\k1
  \trttext{ \csname\the\k
  \endcsname}\Eold}
!yields
\chartpic
\enddemo
\egroup
Remark. An aid in finding the numbers of the branches is
to delete \cs{csname} and \cs{endcsname} in \cs{E} .
The traversal at hand is called preorder.
% i.e., root, one branch (left or up),  and 
%other branch (right or down),  and this recursively.

\vfill\eject
\blueexample Trinary tree

^^{Trinary tree}
Here we have to correct for the direction and also adjust the drawing element,
which is a rectangular block of size $.348{\times}.2$pt.
%
\begindemo
%ESE= east-south-east etc
\def\tritree{\ifnum1=\level \eertirt\fi
    \advance\level-1 \divide\kk2
    {\N{\the\kk}\tritree}%
    {\ESE{\the\kk}\tritree}%
    \WSW{\the\kk}\tritree\relax}
\def\eertirt#1\relax{\fi}
$$\x0pt\y0pt\level6 \kk128\tritree$$
!yields
$$\N{128}\tritreepic$$
\enddemo
Remark. The \cs{unitlength}s are default and 
        equal to the sides of the elementary rectangular block.
        The size of the tree can be controlled by \cs{kk}.
        If the \cs{tritree} is preceded by |\N{128}| a Christmas tree will
        emerge. The picture is included in pic.dat with name \cs{tritreepic}.

Remark. Another class of fractals\Dash the so-called star fractals\Dash
emerge if the {\em change\/} of direction is constant, while the length of each step varies
according to a regular pattern.
These fractals can be drawn by one (broken) line and are therefore suited for the
turtle graphics approach. 
%If not for the triangular case.
Because these are memory intensive I have omitted an example.

\bgroup
\exercise How can we obtain the following ^{H-fractal}?
$$\hfractalpic$$
\answer The H-fractal can be obtained via
\thisverbatim{\unmc}
\beginverbatim
\let\0\N \let\1\E \let\2\S \let\3\W
\newcount\dir
\def\mod#1{\advance\dir#1 \ifnum3<\dir \advance\dir-4 \fi}%mod 4 
\def\nxt{\csname\the\dir\endcsname{\the\kk}\multiply\kk17\divide\kk24
         \advance\level-1 \hf}
\def\hf{\ifnum\level=0 \fh\fi {\mod1\nxt}\mod3\nxt\relax}
\def\fh#1\relax{\fi}
$$\unitlength.1ex\x-10\unitlength\y0pt 
  \kk48 \dir0 \level6 \hf$$
!endverbatim
To use it is even simpler because it comes with pic.dat.

%end answer
\egroup

\ifmac\vfill\eject\fi

\exercise How can we get the following ^{Sierpi\'nski carpet} fractal?
$$\sierpinskitripic$$
\answer The triangle can be obtained via
\beginverbatim
\def\triangle{\ifnum\kk=0 \elgnairt\fi
        {\N{\the\kk}}{\E{\the\kk}}\divide\kk2
        {\triangle}%
        {\advance\x\the\kk\unitlength\triangle}%
         \advance\y\the\kk\unitlength\triangle\relax}
\def\elgnairt#1\relax{\fi}
$$\unitlength1ex\x-8\unitlength\kk16 \triangle$$
!endverbatim

%end answer

\bgroup
\exercise How can we extend the turtle movements? 
          For example to {\oldstyle45}$^\circ$ directions?
Draw the following binary tree with these edges.
\def\E#1{\hbox to0pt{\kern\x\vbox to0pt{\vss
           \hrule width#1\unitlengthx\kern\y
         }\hss}\advance\x#1\unitlengthx
}
\def\N#1{\yy\y \advance\y#1\unitlengthy
         \hbox to0pt{\kern\x
         \vbox{\hrule height\y
                      depth-\yy
                      width.2pt}\hss}}
\def\W#1{\advance\x-#1\unitlengthx
         \hbox to0pt{\kern\x\vbox to0pt{\vss
           \hrule width#1\unitlengthx\kern\y
         }\hss}}
\def\S#1{\yy\y \advance\y-#1\unitlengthy
         \hbox to0pt{\kern\x
         \vbox{\hrule height\yy
                      depth-\y
                      width.2pt}\hss}}
%
\font\tenln=line10 
\setbox\nel=\hbox{\tenln\char'045}
\setbox\nwl=\hbox{\tenln\char'145}
\unitlengthy\ht\nel
\unitlengthx\wd\nel
\def\NE#1{\auxdim#1\unitlengthx 
   \loop\advance\auxdim-\unitlengthx
      \hbox to0pt{\kern\x
         \vbox to0pt{\vss\copy\nel
                 \kern\y}\hss}%
   \advance\x\unitlengthx\advance\y\unitlengthy 
   \ifdim\auxdim>0pt
   \repeat}
\def\NW#1{\auxdim#1\unitlengthx 
   \loop\advance\auxdim-\unitlengthx
      \advance\x-\unitlengthx
      \hbox to0pt{\kern\x
         \vbox to0pt{\vss\copy\nwl
            \kern\y}\hss}%
   \advance\y\unitlengthy
   \ifdim\auxdim>0pt
   \repeat}
%
\def\tree{\ifnum\kk=1 \eert\fi
    \N1 \divide\kk2 
   {\NW{\the\kk}\tree}%
    \NE{\the\kk}\tree\relax}
\def\eert#1\relax{\fi}
\bigskip
$$\x0pt \y0pt \kk4 \tree$$
\answer We can extend it in a limited sense by using two lines from \LaTeX's
line fonts.
\thisverbatim{\unmc}
\beginverbatim
Draw the following binary tree with these edges.
\def\E#1{\hbox to0pt{\kern\x\vbox to0pt{\vss
           \hrule width#1\unitlengthx\kern\y
         }\hss}\advance\x#1\unitlengthx
}
\def\N#1{\yy\y \advance\y#1\unitlengthy
         \hbox to0pt{\kern\x
         \vbox{\hrule height\y
                      depth-\yy
                      width.2pt}\hss}}
\def\W#1{\advance\x-#1\unitlengthx
         \hbox to0pt{\kern\x\vbox to0pt{\vss
           \hrule width#1\unitlengthx\kern\y
         }\hss}}
\def\S#1{\yy\y \advance\y-#1\unitlengthy
         \hbox to0pt{\kern\x
         \vbox{\hrule height\yy
                      depth-\y
                      width.2pt}\hss}}
%
\font\tenln=line10 
\setbox\nel=\hbox{\tenln\char'045}
\setbox\nwl=\hbox{\tenln\char'145}
\unitlengthy\ht\nel
\unitlengthx\wd\nel
\def\NE#1{\auxdim#1\unitlengthx 
   \loop\advance\auxdim-\unitlengthx
      \hbox to0pt{\kern\x
         \vbox to0pt{\vss\copy\nel
                 \kern\y}\hss}%
   \advance\x\unitlengthx\advance\y\unitlengthy 
   \ifdim\auxdim>0pt
   \repeat}
\def\NW#1{\auxdim#1\unitlengthx 
   \loop\advance\auxdim-\unitlengthx
      \advance\x-\unitlengthx
      \hbox to0pt{\kern\x
         \vbox to0pt{\vss\copy\nwl
            \kern\y}\hss}%
   \advance\y\unitlengthy
   \ifdim\auxdim>0pt
   \repeat}
%
\def\tree{\ifnum\kk=1 \eert\fi
    \N1 \divide\kk2 
   {\NW{\the\kk}\tree}%
    \NE{\the\kk}\tree\relax}
\def\eert#1\relax{\fi}
\bigskip
$$\x0pt \y0pt \kk4 \tree$$
!endverbatim
The better approach is to adhere to {\em square\/} atom elements, which is adopted
in BLUe's format turtle graphics.

%end answer
\egroup

The  picture at the beginning of this section about the turtle movement
could easily have been obtained via gkppic. Via turtle graphics
the markup reads as follows.
\beginverbatim
\def\N{...}\def\E{...}\def\S{...}\def\W{...}
%white writing i.e. just move, don't draw
\def\whiteE#1{\advance\x#1\unitlength}%etc 
\vskip2.5\bigskipamount
$$\unitlength.5ex \y=0pt\x-10\unitlength{\W{4}\E{40}}\S{4}%
  {\advance\y-2\unitlength\xytxt{ Ref.\ point}}%
  \N{20}\whiteE{20}\S4\whiteS4\xytxt{ (x,y)}$$\vskip1.5\bigskipamount
!endverbatim

Prompted by the above the turtle movements are extended to the directions of the wind
and the halfwind. 

\vfill\eject
\blueexample Pythagorean tree
  
With the above directions\Dash and a little more especially when the lines
are getting smaller and smaller\Dash we can obtain the classical 
^{Pythagorean fractal}s.

\begindemo
\def\pythtree{\ifnum1=\level\eerthtyp\fi
    \advance\level-1 
    \multiply\kk23\divide\kk32
    {\leftbranche\draw\pythtree}%
     \rightbranche\draw\pythtree\relax}
\def\eerthtyp#1\relax{\fi}
$$\unitlength.2pt\x0pt \y0pt 
  \level7 \dir0 \kk90 
  \N{\the\kk} \pythtree$$
%with auxiliaries
\let\0\N \let\1\NE etc.
\def\leftbranche{\advance\dir7 
 \ifnum\dir>7 \advance\dir-8 \fi}
\def\rightbranche{\advance\dir1 etc.}
\def\draw{\csname\the\dir\endcsname
   {\the\kk}}
!yields
$$\pythtreepic$$
\enddemo

\bgroup
\exercise How can we get the asymmetric Pythagorean trees?
\answer The asymmetric Pythagorean trees can be obtained as follows.
\beginverbatim
\input blue.tex \report
\def\asympythtree{\ifnum\level=1 \eerthtyp\fi
    \advance\level-1 
    {\multiply\kk3 \divide\kk5 
     \leftbranche \draw\asympythtree}%
    \multiply\kk5 \divide\kk6 
    \rightbranche\draw\asympythtree\relax}
\def\eerthtyp#1\relax{\fi}
%with auxiliaries
\let\0\N \let\1\NE \let\2\E \let\3\SE \let\4\S \let\5\SW \let\6\W \let\7\NW
\def\draw{\csname\the\dir\endcsname{\the\kk}}
\def\leftbranche{\advance\dir7 \ifnum\dir>7 \advance\dir-8 \fi}%mod 8
\def\rightbranche{\advance\dir1 \ifnum\dir>7 \advance\dir-8 \fi}%mod 8
%
\unitlength0.2ex
$$\x20\unitlength \y0pt \level9\dir0 \kk50 \N{\the\kk} \asympythtree$$
%\level9 \kk50 needed 60K of memory
%\level10 \kk50 needed 95K of memory
%$$\x20\unitlength \y0pt \level10 \dir0 \kk60 \N{\the\kk} \asympythtree$$
\tracingstats1
\bye
!endverbatim

%endanswer
\egroup

\ifmac\vfill\eject\fi
\blueexample Pythagorean fractal with squares

The Pythagorean fractal with squares is more complex, 
because of how the turtle should draw the left and right squares.
\begindemo
$$\unitlength0.2ex \kk20 
  \rightx\the\kk\unitlength
  \divide\rightx2 \x-\rightx \y0pt 
  \level5 \pyth$$
\let\drawsq\ldrawsq
\def\pyth{\ifnum\level=1 \htyp\fi
    \drawsq\advance\level-1 
    \multiply\kk18\divide\kk25
    {\turn7\x\leftx  \y\lefty 
      \let\drawsq\ldrawsq\pyth}%
     \turn1\x\rightx \y\righty
      \let\drawsq\rdrawsq\pyth\relax}
\def\htyp#1\relax{\fi}
!yields
$$\thispicture{\xoffset{10}}\pythpic$$
\enddemo
The Pythagorean pictures have been added to pic.dat with names
\cs{pythtreepic},  and \cs{pythpic}.

\ifmac\vfill\eject\fi
\blueexample Minkowski sausage

\bgroup
^^{Minkowski sausage}
This fractal emerges from replacing repeatedly a line by a
 block-wave, i.e., \dir1 \kk4 \unitlength1ex 
\hbox{\x0pt\y0pt {\E4}\kern5\unitlength$\longrightarrow$% 
                \kochfractalpic \kern5\unitlength}.
\thisverbatim{\unmc}
\begindemo
\def\koch{\ifnum\kk=1 \draw\hcok\fi
   {\divide\kk4
    \koch\left\koch\right\koch\right\koch\koch
         \left\koch\left\koch\right\koch}\relax}
\def\hcok#1\relax{\fi}
%
$$\x0pt\y0pt\dir1 \kk4  \koch$$
$$\x0pt\y0pt\dir1 \kk16 \koch$$
%with auxiliaries
\def\left{\advance\dir3 \ifnum\dir>3
                  \advance\dir-4 \fi}
\def\right{\advance\dir1 \ifnum\dir>3
                  \advance\dir-4 \fi}
\def\draw{\csname\the\dir\endcsname1}
!yields
\def\draw{\csname\the\dir\endcsname1}
$$\x0pt\y0pt\dir1 \kk4  \kochfractalpic$$
$$\thispicture{\kk16 
  \ydim{10}\yoffset{-5}}
  \kochfractalpic$$
\enddemo
\egroup

\ifmac\vfill\eject\fi

\blueexample L\'evy fractal 

^^{L\'evy fractal}
This fractal emerges from replacing repeatedly a line by a roof.
Two replacements  in a row have the effect of a line replaced by a hat, i.e., 
\unitlength1ex 
\hbox{\x0pt\y0pt {\E2}\kern3\unitlength$\longrightarrow\,$\ 
{\N1\E2\S1}\kern3\unitlength }.

As shown by Lauwerier we don't need recursion to obtain the regular patterns. 
Just a little (ahummm, \smiley)  
analysis might yield the following.
\thisverbatim{\unmc}
\begindemo
\def\levyfractal{\loop \summodfour 
   \draw \advance\k1
   \ifnum\k<\order \repeat}
$$\unitlength1ex\x0pt\y0pt 
  \k0 \dir0 \order256 \levyfractal$$
%with auxiliaries
\let\0\E\let\1\S\let\2\W\let\3\N
%
\def\summodfour{{\modsum0 
  \loop \ifodd\k \advance\modsum1 \fi
   \divide\k2 \ifnum\k>0
  \repeat \global\dir\modsum
  \divide\modsum4 \multiply\modsum4
  \global\advance\dir-\modsum}}
\def\draw{\csname\the\dir\endcsname
  \the\kk}
!yields
$$\let\0\E\let\1\S\let\2\W\let\3\N
  \levyfractalpic$$
\enddemo

\blueexample Dragon figures

To conclude this digression a simplified variant of Knuth's dragon figure, \TB{}
{\oldstyle391}.
Just straight lines 
as used in the original {\em dragon\/} figures of Heighway.
The figures emerge when we repeatedly fold a slip of paper\Dash double, double etc.\Dash
and then unfold it. 
The change of the directions of the line elements obey a regular pattern. 
Because of this we can draw the curve starting 
from the first element and calculate the directions for the subsequent elements.
\thisverbatim{\unmc}
\begindemo
\def\dragon{\k0
\loop \draw 
  \advance\k1 \kk\k \calcdir
\ifnum\k<\kmax
\repeat}
\unitlength2ex 
$$\dir0\x0pt\y0pt \kmax4 \dragon$$
$$\dir0\x0pt\y0pt \kmax64 \dragon$$
!yields
\unitlength2ex 
$$\dragonpic$$
$$\thispicture{\kmax64
   \ydim{7}\yoffset{-4}} 
   \dragonpic$$
\enddemo

\exercise How can we draw curves, for example a spline relative
          to the reference point?
\answer Quintic spline \TeX nique.   gkppic provides \cs{squines}, 
        which is the quadratic variant of the \cs{mfs} given below.
        Another promising approach is given by Gurari's work
        `\TeX{} \& \LaTeX--- Drawing \& Literate programming,' 
        especially the (Al)Dra\TeX{} macros.
        Within the \cs{Draw}\dots\cs{EndDraw} environment among other things 
        Bezier curves can be drawn
        via \cs{Curve($p_1, q_1, q_2, p_2$)}. 
        However, I have provided \cs{mfs} a macro which
        provides for the spline as given in {\it The METAfont\/} book. %\MFB. 
\beginverbatim
%\mfs macro that creates a spline through
%(x1, y1), (x4, y4) with control points
%(x2, y2), (x3, y3).
%The method is the same as given in the METAfont book, p13.
%\S<size> means the southbound moving in turtle graphics.
%As examples some smileys are added.
%cgl@rc.service.rug.nl, July 1995
\let\loadsmileys\relax%\input blue.tex
\let\ea\expandafter \newdimen\unitlength
\newdimen\x \newdimen\y \newdimen\yy 
\newcount\k \newcount\kmax
\ea\newdimen\csname X1\endcsname
\ea\newdimen\csname X2\endcsname
\ea\newdimen\csname X3\endcsname
\ea\newdimen\csname X4\endcsname
\ea\newdimen\csname X12\endcsname
\ea\newdimen\csname X23\endcsname
\ea\newdimen\csname X34\endcsname
\ea\newdimen\csname X123\endcsname
\ea\newdimen\csname X234\endcsname
\ea\newdimen\csname X1234\endcsname
\def\X#1{\csname X#1\endcsname}
%
\ea\newdimen\csname Y1\endcsname
\ea\newdimen\csname Y2\endcsname
\ea\newdimen\csname Y3\endcsname
\ea\newdimen\csname Y4\endcsname
\ea\newdimen\csname Y12\endcsname
\ea\newdimen\csname Y23\endcsname
\ea\newdimen\csname Y34\endcsname
\ea\newdimen\csname Y123\endcsname
\ea\newdimen\csname Y234\endcsname
\ea\newdimen\csname Y1234\endcsname
\def\Y#1{\csname Y#1\endcsname}
%
\newbox\smileybox
%Next the METAfont spline implementation
\def\mfs{\ifnum\k=\kmax \sfm\fi
\advance\k1
\midpoints\x\X{1234}\y\Y{1234}\putatxy
{\X2\X{12}\X3\X{123}\X4\X{1234}%
 \Y2\Y{12}\Y3\Y{123}\Y4\Y{1234}\mfs}%
{\X1\X{1234}\X2\X{234}\X3\X{34}% 
 \Y1\Y{1234}\Y2\Y{234}\Y3\Y{34}\mfs}\relax}%end \mfs
%
\def\sfm#1\relax{\fi}
%
\def\midpoints{%z1,...,z4 -> z1, z12, z23,...,z1234 
\X{12}\X1\advance\X{12}by\X2\divide\X{12}by2
\X{23}\X2\advance\X{23}by\X3\divide\X{23}by2
\X{34}\X3\advance\X{34}by\X4\divide\X{34}by2
\X{123}\X{12}\advance\X{123}by\X{23}\divide\X{123}by2
\X{234}\X{23}\advance\X{234}by\X{34}\divide\X{234}by2
\X{1234}\X{123}\advance\X{1234}by\X{234}\divide\X{1234}by2
%
\Y{12}\Y1\advance\Y{12}by\Y2\divide\Y{12}by2
\Y{23}\Y2\advance\Y{23}by\Y3\divide\Y{23}by2
\Y{34}\Y3\advance\Y{34}by\Y4\divide\Y{34}by2
\Y{123}\Y{12}\advance\Y{123}by\Y{23}\divide\Y{123}by2
\Y{234}\Y{23}\advance\Y{234}by\Y{34}\divide\Y{234}by2
\Y{1234}\Y{123}\advance\Y{1234}by\Y{234}\divide\Y{1234}by2 }
%\putatxy already in blue.tex
\def\putatxy{\hbox to0pt{\kern\x\kern-.15pt
                   \vbox to0pt{\vss\hrule width.3pt
                                          height.3pt\relax
                   \kern\y}\hss}}
%
\def\S#1{\yy\y \advance\y-#1\unitlength
   \hbox to0pt{\kern\x
      \vbox{\hrule height\yy depth-\y width.2pt}\hss}}
%
\def\initialize#1#2#3#4#5#6#7#8{%points: (#1,#2),...(#7,#8)
   \csname X1\endcsname#1\unitlength
   \csname X2\endcsname#3\unitlength
   \csname X3\endcsname#5\unitlength
   \csname X4\endcsname#7\unitlength 
   \csname Y1\endcsname#2\unitlength
   \csname Y2\endcsname#4\unitlength
   \csname Y3\endcsname#6\unitlength
   \csname Y4\endcsname#8\unitlength}
% 
%
\def\eyes{{\x250\unitlength\y150\unitlength 
           \putatxy}%l-eye
          {\x750\unitlength\y150\unitlength 
           \putatxy}%r-eye
}
\def\lefteyes{{\x200\unitlength\y150\unitlength
               \putatxy}%l-eye
              {\x700\unitlength\y150\unitlength
               \putatxy}%r-eye
}
\def\squinteyes{{\x300\unitlength\y150\unitlength
                \putatxy}%l-eye
                {\x700\unitlength\y150\unitlength
                \putatxy}%r-eye
}
\def\winkeye{\initialize{100}{150}{250}{75}{250}{75}{400}{150}%
              {\k0\kmax2\x0pt\y0pt\mfs}%l-winkeye
              {\x750\unitlength\y150\unitlength
              \putatxy}%r-eye
}
\def\mouth{\initialize{250}{-250}{500}{-350}{500}{-350}{750}{-250}%
           \k0\x0pt\y0pt\mfs
}
\def\sadmouth{\initialize{250}{-350}{500}{-250}{500}{-250}{750}{-350}%
   \k0\x0pt\y0pt\mfs}
%
\kmax5 %level
\setbox\smileybox=\hbox{\unitlength.01pt
  \initialize000{650}{1000}{650}{1000}0%
  \k0\putatxy\mfs
  \initialize 000{-650}{1000}{-650}{1000}0%
  \k0\mfs\y0pt\x\X4\putatxy
  \kmax3
  \initialize{100}{200}{250}{250}{250}{250}{400}{200}%
  \k0\x0pt\y0pt\mfs%l-brow
  \initialize{600}{200}{750}{250}{750}{250}{900}{200}%
  \k0\x0pt\y0pt\mfs%r-brow
  {\x500\unitlength\y150\unitlength\S{250}}%Nose
}
\def\smiley{\loadsmileys\raise.5ex\hbox{\unitlength.01pt
   \copy\smileybox\eyes\mouth\kern1000\unitlength} }
\def\winksmiley{\loadsmileys\raise.5ex\hbox{\unitlength.01pt
   \copy\smileybox\winkeye\mouth\kern1000\unitlength} }
\def\sadsmiley{\loadsmileys\raise.5ex\hbox{\unitlength.01pt
   \copy\smileybox\squinteyes\sadmouth\kern1000\unitlength} }
% 
$$pretext\ \smiley\ posttext$$
%
$$pretext\ \winksmiley\ posttext$$
%
$$pretext\ \sadsmiley\ posttext$$
\tracingstats1
\bye
!endverbatim
 
For the moment I'll stop `\dots fooling around making pictures, 
                          instead of typesetting ordinary text\dots'

One thing is crystal clear, however. Fractals are a `mer \`a boire,' to exercise recursive
programming, while on the outlook for straightforward alternatives.

\everyverbatim{\emc}

\bluehead Using pictures

pic.dat contains definitions of prefab pictures. 
In order to use these, the picture must be loaded via\ftn{I use myself 
               \cs{bluepictures}%
               $\cs{}\langle\,name_1\rangle\hbox{\tt pic}\dots
               \cs{}\langle\,name_n\rangle\hbox{\tt pic}$, followed by a blank line
               or \cs{par}. \smiley.} 
^^|\beginpictures|%^^|\pictures|
\begincenterblock
\cs{beginpictures}
$\cs{}\langle\,name_1\rangle\hbox{\tt pic}\dots%
 \cs{}\langle\,name_n\rangle\hbox{\tt pic}\qquad%
 \hbox{or}\qquad%
\cs{pictures}\{\cs{}\langle\,name_1\rangle\hbox{\tt pic}\dots%
                \cs{}\langle\,name_n\rangle\hbox{\tt pic}\}$
\cs{endpictures}
\endcenterblock

After this each picture loaded can be used by its name, i.e., 
via the invocation 
\displaycenterline{$\cs{}\langle\,name_x\rangle\hbox{\tt pic}$}
if necessary preceded by ^|\thispicture||{...}|, ^^|\everypicture| 
with actual values for
^|\unitlength|, and the offsets.
The result is packed as an |\hbox to#1\unitlength{...}| with depth
reset to zero.

Note that although the picture database is a file of defs,
alphabetically sorted on definition name, 
the names of the pictures to be selected
can be supplied in any order to \cs{pictures}.

\exercise What is the effect of |\pictures{}|?
\answer The common-for-pictures macro, \cs{commonforpic}, is loaded, and
        invoked. The latter means that you can use for example the macros
        \cs{corners},  i.e., the macros which
        are common for several pictures, and these can be used over and over.


\blueexample Simple picture

\begindemo
\thispicture{\markorigintrue}
\simplepic
!yields
\thispicture{\markorigintrue}
\qquad\qquad\qquad\qquad\simplepic
\enddemo

\bluesubhead Pictures from tools.dat

The database of tools also contains pictures especially my PMS, 
i.e.\ poor man's smileys. 
\blueexample Smileys

\begindemo
\input blue.tex 
\centerline{\smiley\quad
            \sadsmiley\quad 
            \winksmiley}
!yields
\centerline{\smiley\quad
            \sadsmiley\quad 
            \winksmiley}
\enddemo

Similar for \cs{ntglogo} and \cs{xyz}-axes.

\vfill\eject
\bluehead Creating pictures

I'll discuss two approaches:
one for simple schemes, and
the ^{picture environment} as used by Graham, Knuth, and Pastashnik. 

\bluesubhead Stacking boxes

For simple schemes we can use ^{framed boxes}. \bluetex{} provides
\begincenterverbatim
\def\fbox#1{\setbox0\hbox{\strut$\;$#1$\,$}\leavevmode
 \rlap{\copy0}\makeblankbox{.2pt}{.2pt}}
!endcenterverbatim
The idea is that the argument is set in box 0. 
This box is framed by manmac's \cs{makeblankbox}.  

\blueexample Simple scheme

\begindemo
\stack{\fbox{amsppt.sty}}     
\vconnector
\stack{\fbox{amstex.tex}}
\vconnector
\stack{\fbox{\TeX}}}
%with
\def\stack#1{\hbox to10ex{\hss
                      #1\hss}}
\def\vconnector{\stack{\vrule 
   height2.5exdepth1exwidth.2pt}}
!yields\qquad\qquad\qquad\qquad\qquad
\vbox{\offinterlineskip 
\def\stack#1{\hbox to 10ex{\hss#1\hss}}
\def\vconnector{\stack{\vrule 
   height2.5exdepth1exwidth.2pt}}
\stack{\fbox{amsppt.sty}}      
\vconnector
\stack{\fbox{amstex.tex}}
\vconnector
\stack{\fbox{\TeX}}}
\enddemo

\bluesubhead Samplebox diversions

A lot of details can be learned from Knuth's markup of the \cs{samplebox}
picture, \TB{} chapter {\oldstyle11}:  
composing long vertical arrows,
to measure the axis height,
samplebox coding. 
For daily use this markup is too cumbersome. 
However, the ^|\samplebox| macro is amply used in 
\TB{} chapter {\oldstyle12}.

\blueexample Manmac's samplebox

Knuth attaches three dimensions to these boxes: height, width,
and depth. 
\begindemo
%Vertical arrows, similar to 
%\arrows, but now vertically.
\setbox0=\hbox{$\uparrow$}
\setbox1=\hbox to \wd0{$\hss\mid
\hss$}%with luck,they'll line up
\def\vline{\nointerlineskip
  \kern-.5pt\copy1}
\setbox2=\vbox{\copy0
  \vline\vline
  \moveleft 1em\hbox{height}
  \copy1\vline 
  \nointerlineskip\kern-.5pt
  \hbox{$\downarrow$}\kern.2pt}
%To measure axis height
%\ht0 is the axis height
\setbox0=\hbox{$\vcenter{}$}
!yields
\qquad\vbox{\eightpoint
\setbox0=\hbox{$\uparrow$}
\setbox1=\hbox to \wd0{$\hss\mid\hss$} % with luck, they'll line up
\setbox2=\vbox{\copy0
  \nointerlineskip \kern-.5pt \copy1
  \nointerlineskip \kern-.5pt \copy1
  \moveleft 1em\hbox{height}
  \copy1 \nointerlineskip \kern-.5pt
  \copy1 \nointerlineskip \kern-.5pt
  \hbox{$\downarrow$}
  \kern.2pt}
\setbox3=\vbox{\kern.2pt\copy0
  \moveleft 1em\hbox{depth}
  \hbox{$\downarrow$}
  \kern0pt}
\setbox4=\vtop{\kern-3pt % this cancels the null text above the samplebox
  \hbox{\samplebox{\ht2}{\ht3}{6em}{}%
    \kern-6em
    \raise3pt\hbox to 6em{\hss Baseline\hss}}
  \kern3pt
  \arrows{6em}{width}}
\medskip\indent
\setbox0=\hbox{$\vcenter{}$}% \ht0 is the axis height
\lower\ht0\hbox{Reference point$-$\kern-.2em$\rightarrow$\kern2pt}%
\raise\ht2\box4
\kern1.5em
\raise\ht2\vtop{\kern0pt\box2\nointerlineskip\box3}}
\enddemo
From the above I distilled the following ^|\gbox| 
macro\Dash generalized samplebox\Dash 
which draws a box similar to sample box, 
with depth, height and width, and a hook
to put in material relative to the reference point.
This is a generalization of a framed box, because
the frame has its own dimensions independent of
the material to be framed.
In order to be in full control of the positioning of the elements, 
I chose the parameters ^|\xshift| and ^|\yshift|\Dash 
dimen vars\Dash to shift material relative to the reference point.
%The above used with |\setbox<value>=\hbox{<some of the above>}| and 
%\cs{makelightbox} can put elements anywhere within a framed box.
%For the framing I used \cs{gbox} a modification of \cs{samplebox}.

\blueexample Generalized samplebox macro

\begindemo
pretext 
\xshift1ex \yshift2ex
\gbox{7ex}{3ex}{5ex}{abc}
posttext
%with the generalized sample box
\def\gbox#1#2#3#4{%#1=ht,#2=dp
%#3=wd, #4=text. 
%Assumed: the dimen vars \xshift, 
%\yshift have been initialized.
 {\setbox0=\vtop{\vbox to#1{%
            \hbox to#3{}\vss}
  \nointerlineskip
  \vbox to#2{}}
%now \box0 has the desired 
%ht, dp, and wd
 \hbox{\makelightbox\kern-#3%
  \raise\yshift\hbox to#3{%
        \kern\xshift#4\hss}}}}
!yields
\def\gbox#1#2#3#4{%#1=ht,#2=dp,#3=wd
%#4=text. Assumed: the dimen vars
%\xshift, \yshift have been initialized.
%position of begin of text.
 {\setbox0=\vtop{\vbox to#1{%
  \hbox to#3{}\vss}\nointerlineskip
  \vbox to#2{}}
%now \box0 has the desired 
%ht, dp, and wd
 \hbox{\makelightbox\kern-#3%
  \raise\yshift\hbox to#3{%
        \kern\xshift#4\hss}
 }}}
\qquad\qquad\qquad pretext
\xshift1ex \yshift2ex
\gbox{7ex}{3ex}{5ex}{abc} 
posttext
\enddemo

The markup of the picture of astronomy data, \TB{} {\oldstyle389},
can be simplified.
Also ^|\point| can be generalized as follows.
\def\ftntxt{\cs{prepoint}=$\{$\$\cs{scriptstyle}
            \cs{bullet}\cs{;}\$$\}$ was hardwired
             in \TB.}
\begincenterverbatim
\def\point#1 #2 #3{\kern#1\unitlength\raise#2\unitlength\hbox to0pt{\the\prefil
   \the\prepoint#3\the\postfil}\kern-#1\unitlength}
%with parameters
\prepoint to allow material to be inserted (toks var)!ftn!bgroup!ftntxt!egroup
\prefil   to insert the fill               (toks var)
\postfil  to insert the fill               (toks var, default \hss)
!endcenterverbatim

\blueexample Atlas \TB{} {\oldstyle389}

Use \cs{gbox} of the right size with the data in a box. 
{\prepoint{$\scriptstyle\bullet\;$}
\begindemo
\unitlength3ex
\prepoint{$\scriptstyle\bullet\;$}
\setbox1=
\hbox{\point 0 0 {Alioth} 
      \point 0 8 {Dubhe}
      \point 0 -8 {Alkaid}
      \point -1 -2.5 {Miza}
      \point 4 7 {Merak} 
      \point 4 2 {Phekda}
      \point 1 1.5 {Megrez}
}
\xshift2\unitlength 
pretext \gbox{10\unitlength}%height
             {9\unitlength} %depth
             {11\unitlength}%width
             {\copy1} %material
posttext
!yields
%\newdimen\unitlength 
\unitlength=3ex
\setbox1=\hbox{\point 0 0 {Alioth} 
      \point 0 8 {Dubhe}
      \point 0 -8 {Alkaid}
      \point -1 -2.5 {Miza}
      \point 4 7 {Merak} 
      \point 4 2 {Phekda}
      \point 1 1.5 {Megrez}
}
\xshift2\unitlength%\yshift0pt
pretext \gbox{10\unitlength}%height
             {9\unitlength} %depth
             {11\unitlength}%width
             {\copy1} %material
posttext
\enddemo}

\vfill\eject
\blueexample Layout pages

\cs{gbox} and \cs{point} can be combined for drawing a rough design 
of pages, what the pages should look like. 
As an example a sketch of the Inside Cover Page and the Title Page 
as implemented in the \cs{report} format has been given below.
^^{design pages}
\begindemo
\unitlength1cm
\prepoint{\small}
\noindent\setbox1=\hbox{%
\point1.3 1 {{\tiny\copyright}%
         \thinspace signature}% 
\point.3 1.5 {acknowledgements}%
\point.3 3.75 {bibliographydata}%
\point.3 4.25 {keywords}%
{\prefil{\hss}\point1.5 5.1 
            {Inside Cover}}}
\gbox{5\unitlength} %height
     {0pt}    %depth
     {3\unitlength} %width
     {\copy1} %material
%
\prefil{\hss}
\setbox1=\hbox{%
\point1.5 2.85 {abstractname}% 
\point1.5 1.5 {\gbox{1.25\unitlength}
            {0pt}{2.5\unitlength}{}}% 
\point1.5 3.5 {author}%
\point1.5 4.05 {\tiny subtitle}%
\point1.5 3.8 {\tiny by}%
\point1.5 4.25 {title}%
\point1.5 5.1 {Title Page}}
\gbox{5\unitlength} %height
     {0pt}    %depth
     {3\unitlength} %width
     {\copy1} %material
!yields
\unitlength1cm
\prepoint{\small}
\noindent\setbox1=\hbox{%
\point 1.3 1 {{\tiny\copyright}%
     \thinspace signature}% 
\point  .3 1.5 {acknowledgements}%
\point  .3 3.75 {bibliographydata}%
\point  .3 4.25 {keywords}%
{\prefil{\hss}\point 1.5 5.1 
 {Inside Cover}}}
\gbox{5\unitlength} %height
     {0pt}    %depth
     {3\unitlength} %width
     {\copy1} %material
%
\prefil{\hss}
\setbox1=\hbox{%
\point 1.5 2.85 {abstractname}% 
\point 1.5 1.5 {\gbox{1.25\unitlength}{0pt}{2.5\unitlength}{}}% 
\point 1.5 3.5 {author}%
\point 1.5 4.05 {\tiny subtitle}%
\point 1.5 3.8 {\tiny by}%
\point 1.5 4.25 {title}%
\point 1.5 5.1 {Title Page}}
\gbox{5\unitlength} %height
     {0pt}    %depth
     {3\unitlength} %width
     {\copy1} %material
\enddemo

\blueexample Typesetting outlines, \TB{} ex{\oldstyle11}.{\oldstyle5}

Just an outline of characters is also used 
in the logo of the GUST bulletin. ^^{outline}
Below is my solution to \TB{} ex11.5\Dash via nested FIFO,
   processing: words, via \cs{fifow},
   and characters, via \cs{fifo}\Dash 
to this `tough problem.'
\begindemo
\fifow Tough  exercise
       \wofif \unskip.
%with
\def\fifow#1 {\ifx\wofif#1\wofif\fi
   \processw{#1}\fifow}
\def\processw#1{\fifo#1 \ofif\ }
\def\process#1{\boxit#1}
\def\boxit#1{\setbox0=\hbox{#1}%
  \hbox{\lower\dp0\vbox{\hrule
   \hbox{\vrule\phantom#1\vrule}%
   \hrule}}}
!yields
\def\fifow#1 {\ifx\wofif#1\wofif\fi
   \processw{#1}\fifow}
\def\processw#1{\fifo#1\ofif\ }
\def\process#1{\boxit#1}
\def\boxit#1{\setbox0=\hbox{#1}%
  \hbox{\lower\dp0\vbox{\hrule
   \hbox{\vrule\phantom#1\vrule}%
   \hrule}}}
\qquad\qquad\qquad
\fifow Tough exercise \wofif \unskip.~
\enddemo

\blueexample Sierpi\'nski carpet again

Salomon\ftn{Advanced \TeX, Springer, {\oldstyle1995}.} provides another variant
of one of ^{Sierpi\'nski carpet}s, done by just stacking boxes.
\begindemo
\setbox0=\vbox{%
   \hrule width2pt height0pt
   \hrule width0pt height1pt
   \hrule width1pt height1pt}
\def\tri#1{{\count0=#1
\loop\ifnum\count0>0
   \setbox0=\vbox{\offinterlineskip
      \hbox{\kern.5\wd0\copy0}%
      \hbox{\copy0\copy0}}%
   \advance\count0-1
\repeat\copy0}}
\hbox{\tri1 \tri2 \tri3 \tri4 \tri5}
!yields
\setbox0=\vbox{\hrule width2pt height0pt
               \hrule width0pt height1pt
               \hrule width1pt height1pt}
\def\tri#1{{\count0=#1
\loop\ifnum\count0>0
   \setbox0=\vbox{\offinterlineskip
      \hbox{\kern.5\wd0\copy0}%
      \hbox{\copy0\copy0}}%
   \advance\count0-1
\repeat\copy0}}
\hbox{\tri1 \tri2 \tri3 \tri4 \tri5}
\enddemo

\bluesubhead Picture environment

It has been called in the file of gkpmac 
`Pictures (a subset of \LaTeX's conventions).'
\LaTeX's ^{line\ font}s are used.
Macros are provided for  
lines, vectors, curves, disks, circles, and ovals.
The lines and vectors have discrete orientations and can vary
in thickness ({\oldstyle2} states).
The implementation is simpler than \LaTeX's, because the
options have been omitted.
Dimensionless units get their value via \cs{unitlength}.
%The user must initialize the latter. 
The documentation of the ^{picture environment} in pseudo-code
comes with \LaTeX, and applies to this set too.
In \bluetex{} a macro for rhombus-framed elements has been added
to the facilities provided by gkpmac.

\thissubsubhead{\runintrue}
\bluesubsubhead The  markup for the environment \par reads
^|\gkpbeginpicture| and |\gkpendpicture|.\ftn{Not conform to 
   \LaTeX, because the offset parameters are obligatory. 
   These two outer level names have been  prefixed by gkp,
   to preserve \cs{beginpicture} and \cs{endpicture}.}

\beginverbatim
\def\gkpbeginpicture(#1,#2)(#3,#4){%#1,#2 x,y-dimensions; #3,#4 x,y-offsets
...}
!endverbatim

\bluesubsubhead Line thickness

Default is ^|\thinlines| which can be toggled via
the use of ^|\thicklines|. This does not apply to curves.

\bluesubsubhead (multi)put

These put control sequences are used for the positioning of 
picture objects. ^^|\multiput|^^|\put|
\begincenterverbatim
\def\put(#1,#2)#3{%#1,#2 x,y-coordinates   
   %#3 picture object, printed at #1,#2 to the right
   \raise#2\unitlength
     \rlap{\kern#1\unitlength #3}\ignorespaces}
\def\multiput(#1,#2)(#3,#4)#5#6{%
%#1,#2 x,y-coordinates; #3,#4 x,y-increases;
%#5 number of times; #6 picture object
...}
!endcenterverbatim

\bluesubsubhead Picture objects

Provided are: (h)box, line, vector, disk, circle, oval, and rhombus.
^^|\makebox|^^|\line|^^|\vector|^^|\disk|^^|\circle|^^|\oval|^^|\rhombus|
\thisverbatim{\unmc}
\begincenterverbatim
\def\makebox(#1,#2)#3{%#1,#2 x,y-sizes (dimensionless)
%#3 is put in hbox to#1\unitlength{\hss#3\hss}
...}
%
\def\framebox(#1,#2)#3{%#1,#2 x,y-sizes (dimensionless)
%#3 is put in framed hbox to#1\unitlength{\hss#3\hss}
...}
%
\def\line(#1,#2)#3{%abs(#1,#2)<=6
%#1,#2 horizontal,vertical-slopes; #3 horizontal size
...}
%
\def\vector(#1,#2)#3{%abs(#1,#2)<=4
%#1,#2 horizontal, vertical slopes; #3 horizontal size
...}
%
\def\disk#1{%#1 diameter <=.2inch
...}
%
\def\circle#1{%#1 diameter <=.5inch
...}
%
\def\oval(#1,#2){%#1,#2 x,y-sizes (dimensionless)
                 %centre put at reference point
...}
\def\rhombus(#1,#2)#3#4{%#1,#2 hor. respectively vert. ratio <=6
   %#3 the size in unitlengths (half of total width)
   %#4 (text) put in the middle, that is the reference point.
...}
!endcenterverbatim
\noindent Remark. Quarters of an oval are set and 
controlled via 
\cs{ifovalbl} (bottom left),
\cs{ifovalbr} (bottom right),
\cs{ifovaltl} (top left),
\cs{ifovaltr} (top right). 
By default all four are true, but we can set one quarter only
if we wish.

\blueexample Simple picture markup
 
\begincenterverbatim
\unitlength=1cm 
\begingkppicture(3,1.25)(0,0)
   \ifmarkorigin\put(0,0)\markorigin\fi
   \put(1.5,.5){\oval(2,1)}
   \put(0.5,0){\makebox(2,1){Kees}}
\endgkppicture
!endcenterverbatim

\bluesubsubhead Curves

The method has been explained in the \MF{} book page {\oldstyle13}.
The functionality  provided is similar to Lamport's
Bezier.sty, although here a ^{quadratic spline} is used. ^^|\squine|
\begincenterverbatim
\def\squine(#1,#2,#3,#4,#5,#6){%
%#1, #2, #3 x-coordinates of consecutive points
%#4, #5, #6 y-coordinates of consecutive points
%To plot f(x) between x0 and x1
%\put(0, 0){\squine(x0,xm,x1,y0,ym,y1)}, with
%y0=f(x0), y1=f(x1)
%xm=(y0-y1+s1x1-s0x0)/(s1-s0)
%ym=(s0(s1x1-y1)-s1(s0x0-y0))/(s1-s0)
%s0=f'(x0), s1=f'(x1)
!endcenterverbatim
\noindent The `line thickness' can be varied by redefining
^|\squineelem|. Default: |\def\squineelem{.}|.


\bluehead Adding a picture to the database pic.dat
 
Why a picturebase? ^^{database,\ pic.dat}^^{add\ a\ picture}
To begin with, it is convenient to have all your pictures at hand and 
well-organized.
Furthermore, it is nice to load the picture from the picturebase,
and use it with `arguments,' especially with the value for \cs{unitlength}, 
or actual values for the offsets, and if necessary with
values for  the x,y-dimensions. 
It also nice to specify the `level' of the picture via \cs{fulltrue},
which by default is set to false.
The defaults can be changed via providing 
values in |\thispicture{...}|.
\cs{pictures} load all the specified names as defs.
The above ideas resulted in the following approach.
 
\blueexample Entry for pic.dat

\cs{unitlength} is a dimension variable, ^^{entry\ for\ pic.dat}
and \cs{xdim}, \cs{ydim}, \cs{xoffset}, and \cs{yoffset}
are token variables. Each entry is a list element preceded by the
list element tag, \cs{lst}, and its name.
\begincenterverbatim
\lst\simplepic{\bgroup%next supply defaults
  \unitlength=1cm\xdim{3}\ydim{1.25}%offsets are 0 by default
\beginpicture
   \ifmarkorigin\put(0,0)\markorigin\fi
   \put(1.5,.5){\oval(2,1)}
   \put(0.5,0){\makebox(2,1){Kees}}
\endpicture\egroup}%end simplepic
%with in \bluetex{} the shortcuts
\def\beginpicture{\the\everypicture\the\thispicture
 \gkpbeginpicture(\the\xdim,\the\ydim)(\the\xoffset,\the\yoffset)}
\def\endpicture{\gkpendpicture\global\thispicture{}}
!endcenterverbatim

A little more abstract an entry for  pic.dat reads as follows.
\begincenterverbatim
\lst\<name>pic{\bgroup%next the defaults
 \unitlength<dimen value>\xdim{<number>}\ydim{<number>} 
 \xoffset{<number>}\yoffset{<number>}
\beginpicture         %next the picture elements
   <picture elements>
\endpicture\egroup}
!endcenterverbatim
^^{list\ of\ database}
\exercise How can we get a list of names of the pictures in the database?
\answer \bluetex{} contains a macro \cs{contentsdatabase}
        that writes the list of names of
         a lit, pic, or address database to the file contentslit, 
        contentspic, or contentsaddress.\\
Let us assume that we wish the list of names in a toks
variable \cs{namelst}. To facilitate processing of \cs{namelst}
\cs{lst} has been inserted before each name in the list.  
\beginverbatim
\newtoks\namelst
\def\lst#1#2{\namelst\ea{\the\namelst\lst#1}}
\input pic.dat
%and to verify the contents
\baselineskip12pt plus2pt minus1pt
\lineskip1pt plus 1pt minus1pt
\def\lst#1{{\tt\string#1}\endgraf}
\the\namelst
!endverbatim


\bluesubhead Check for data integrity

After a picture has been added one can check whether the file can be scanned by
\TeX{} via making a table of contents as follows.
\begincenterverbatim
\input blue.tex
\contentsdatabase{pic}
\bye
!endcenterverbatim
This script produces no pages of output (that is OK). 
The file contentspic will be created, though.

\bluehead Examples

These  are added to illustrate what can be achieved
with the picture environment. It is true that creating
the markup is not as simple as drawing by the mouse.
But once you have got the template, (parts of) it can be reused.
I explained the approach in broad terms.
%The origin has been printed  via a + now and then.
^^{flow chart}

\blueexample Flow chart loop: flowchartlooppic

\thisverbatim{\unmc}
\begindemo
Interesting is the rhombus.
Similar to \oval and \framebox.
\def\rhombus(#1,#2)#3#4{%
%#1,#2 ratio of hor./vert. size, <=6
%#3 the size in unitlengths 
%   (half of total width)
%#4 text to be put in the middle.
%The centre is the reference point,
%similar as with \oval.
\setbox0=\hbox{\line(#1,#2){#3}}%
\setbox1=\hbox{\line(#1,-#2){#3}}%
\hbox to0pt{\hss
   \vbox to0pt{\offinterlineskip\vss
               \copy0\copy1\vss}%
   \vbox to0pt{\vss\copy1\copy0\vss
               }\hss}%
   \vbox to0pt{\vss
    \hbox to 0pt{\hss#4\hss}\vss}}
!yields
\thispicture{\xoffset{-4}}
\flowchartlooppic
\enddemo

\blueexample Flow chart due to Furuta: furutapic

Note the use of PMA---poor man's arrow heads.
\begindemo
Interesting here are the arcs and 
the arrow heads. 
The latter have been provided 
separately for reuse. 
The reference point is at 
the head. 
%lu is pointing left up,
%rd is pointing right down
\def\rdarrowhead{\hbox 
  to0pt{\hss\vrule height.1pt
            depth.1pt
            width.5ex
   \vrule height.5ex\relax}}
\def\luarrowhead{\hbox 
  to0pt{\vrule depth.5ex
   \vrule height.1pt
          depth.1pt
          width.5ex}}
!yields
\furutapic
\enddemo

\newpage
\blueexample Salomon's overviewtexpic

^^{overviewtexpic}
\begindemo
\thispicture{\fulltrue
             \xoffset{.25}}
\overviewtexpic

The right upper corners, 
suggesting multiple copies, 
are nice.
Recursive use of \put has 
been applied. Use 
\put(<x>,<y>){\corners{<size>}}
The reference point is <x>,<y>.
\def\corners#1{%#1 the size
 \put(.1,.1){\line(-1,0){#1}}%
 \put(.1,.1){\line(0,-1){#1}}%
 \put(.2,.2){\line(-1,0){#1}}%
 \put(.2,.2){\line(0,-1){#1}}%
}

The quarter ovals are specified
via e.g. \put(1.1, 2.1){
    \ovaltlfalse%TopLeft
    \ovalblfalse%BottomLeft
    \ovalbrfalse%BottomRight
    \oval(2, 1)}
!yields
\thispicture{\xoffset{.25}\fulltrue}
\overviewtexpic
\enddemo
\exercise Apply the same approach for a macro \cs{card}, i.e.,
for the figure to denote multiples,
as used with \TeX, \LaTeX, \AmS-\AllTeX, to denote some of \TeX's flavours. 
\answer The important parameters for this macro are the coordinates 
        of the upper right corner of the card. 
        Given that point we can draw lines, put text somewhere et cetera.
        The macro has been included in the `common-for-pictures'
        part of pic.dat to be used in pictures. 
\beginverbatim
\def\card#1#2#3{%#1,#2 x,y coord.corner; #3 label
   \put(#1,#2){\line(-1,0){1.25}}
   \put(#1,#2){\line(0,-1){ .45}}
   \put(#1,#2){\lower.45\unitlength
     \hbox{\line(-1,0){ .25}}}
  \put(#1,#2){\hbox to0pt{\kern-1.25\unitlength
     \hbox{\line(0,-1){ .25}}\hss}}
  \put(#1,#2){\lower.5\unitlength\hbox
  {\hskip.05\unitlength{\tiny #3}}}
}%end card
!endverbatim

\exercise And what about the diagonal dots?
\answer The important parameters for this macro are the coordinates 
        of the first point of the right\-upper diagonal ellipsis.
        The macro has been included in the `common-for-pictures'
        part of pic.dat to be used in pictures. The basic part from below
        is borrowed from \TB{} ex18.45. 
\beginverbatim
\def\diagdots#1#2{\put(#1,#2){$.^{.^.}$}}
!endverbatim

\blueexample SGML-\TeX{} relation

^^{sgmltexpic}
\begindemo
The interesting thing in the 
markup is the abstraction of 
the level in the picture via 
the counter \leg.
Instead of explicit numbers 
the counter is used via 
\the\leg<.fraction>, and
advanced for the next level.

\thispicture{\small
    \unitlength=3ex
    \xoffset{-9}
}\sgmltexpic
!yields
\thispicture{\small
    \unitlength=2.75ex
    \xoffset{-9}
}\sgmltexpic
\enddemo

\newpage
\blueexample Systemtheory picture

\begindemo
%These kinds of pictures occur often.
%Just the template from pic.dat
\lst\systemtheorypic
{\bgroup
\unitlength1cm\xoffset{-1.5}
\xdim{7}\ydim{2.25}
\beginpicture
\ifmarkorigin\put(0,0){\markorigin}\fi
\put(-1,1.75){$u_1$}
\put(-.4,1.5){$^{\fivesy+}$}
\put(-1,1.5){\vector(1,0){.9}}
\put(0,0){\vector(0,1){1.4}}
\put(-.3,1.125){$^{\fivesy+}$}
\put(0,1.5){\circle{.1}}
%
\put(.1,1.5){\vector(1,0){1.65}}
\put(.5,1.25){$e_1$}
\put(1.75,1.125){\framebox(1,.75){$G$}}
\put(2.75,1.5){\line(1,0){1.75}}
%
\put(4.5,1.5){\vector(0,-1){1.4}}
\put(4.6,.2){$^{\fivesy+}$}
\put(4.5,0){\circle{.1}}
%
\put(1.75,0){\line(-1,0){1.75}}
\put(1.75,-.375){\framebox(1,.75){$K$}}
\put(4.4,0){\vector(-1,0){1.65}}
\put(3.25,-.25){$e_1$}
%
\put(5.5,0){\vector(-1,0){.9}}
\put(4.7,-.375){$^{\fivesy+}$}
\put(5.5,-.25){\llap{$u_2$}}
\endpicture\egroup}
!yields
$$\systemtheorypic$$
\enddemo

\blueexample Pie-chart: ntgpiepic

^^{ntgpiepic}
\begindemo
The challenge was to create a 
`circle' via splines. 
Another thing is that the top 
of the candles is done by \line,
which does not show in print 
when too small.
The origin is close to the middle
of the chart.
!yields
\ntgpiepic
\enddemo
\exercise How to generalize the above picture in such a way 
          that a business pie-chart can be represented?
\answer It is not a problem to complete the circle. An open problem is
        how to specify  and implement the parts of the pie-chart, 
        especially each size. If you have found a graceful
        solution, let me know. Perhaps we should let it go because
        of \PS, where it can be done easily. 
        However,  Gurari in `\TeX{} \& \LaTeX--- Drawing \& Literate programming,' 
        provides a solution within \TeX, especially the AlDra\TeX{} 
        macros for drawing pie, bar and XY charts.

The picture on the cover\Dash BLUe's format in a nutshell\Dash
is nice too. Consult pic.dat for the markup.

\bluehead Advanced fractals
 
On the edge of order and chaos, the limit cycles of various non-linear models, 
there is a fascinating, inspiring  world for all those who love nature
the mother of all art. 
Julia researched the mapping $z^2,  z\in\C$, perhaps the simplest nonlinear mapping. 
Nevertheless its limit cycle  consists of what is called today the Mandelbrot 
fractals.\ftn{I had to kludge a little because a dimen${\times}$dimen is forbidden,
   yields arithmetic overflow.
   To circumvent this \cs{x} and \cs{y} are counters.}

\blueexample Mira model

^^{Mira}
\bgroup
%Mira dust fractals, without much detail, too course arithmetic. Iteration with TeX!
%cgl@rc.service.rug.nl, June 1995.
%Iteration (in psuedo-TeX notation
%   x_{n+1}=b*y_n + f(x_n)
%   y_{n+1}= -x_n + f(x_{n+1}), f(x)= a*x + 2(1-a)x^2/(1+x^2), 
%   or one of your choice :-)
%Very primitive (integer) arithmetic, but immediate formatting.
%\newdimen\unitlength \newcount\k
\newcount\x   \newcount\y      
\newcount\oldx\newcount\oldy
\newcount\huna\newcount\scaledb \newcount\scale   \newcount\hunc
\newcount\sqx \newcount\sqxone
\newcount\wx
\newcount\npoints
%Typeset a `dot' at (x, y). Ref point left invariant.
\def\putatxy{\hbox to0pt{\kern\x\unitlength
         \vbox to0pt{\vss\hrule width.3pt\kern\y\unitlength
         }\hss}}
%
%Perhaps I should take over Gurari's *, /, + and - for assigning arithmetic
\def\w{\sqx\x \multiply\sqx\x \sqxone\sqx \advance\sqxone1 %\sqxone:= x^2 + 1 
       \wx\x \multiply\wx\huna \multiply\sqx\hunc \divide\sqx\sqxone
       \advance\wx\sqx \divide\wx100              %\wx:= a*x + c*x^2/(x^2 + 1)
}
\def\mira#1%100*a
         #2%\scale*b
         #3%100*x
         #4%100*y
{\huna#1 \scaledb#2 \hunc\huna \multiply\hunc-2 \advance\hunc200
                                                            %c:= 2-2a
\x#3 \y#4 \w  %\huna means 100 times a (simple scaling to compensate for integer arithmetic)
\k0 \loop \putatxy
   \oldx\x \x\y \multiply\x\scaledb \divide\x\scale \advance\x\wx %\x:= y*b + w
   \w \y\wx \advance\y-\oldx                                      %\y:= w - x
\ifnum\x=0 \ifnum\y=0 \message{x and y are 0, at k=\the\k.}\k\npoints \fi\fi
\advance\k1 \ifnum\k<\npoints
\repeat}%end mira
%
\unitlength.2pt \scale100 \npoints400
%
%MIRA mappings, cgl, from Lauwerier's Fractals
\begindemo
%x_{n+1}=b*y_n + f(x_n)
%y_{n+1}= -x_n + f(x_{n+1}), 
%   f(x)= a*x + 2(1-a)x^2/(1+x^2)
%a=-.48 b=.93 x=4 y=0 \scale100
$$\mira{-48}{93}{400}{0}$$
!yields
$$\scale100 \mira{-48}{93}{400}{0}$$
\enddemo

\vskip4\bigskipamount
\begindemo
%a=-.25 b=.99 x=3 y=0
$$\scale100 \mira{-25}{99}{300}{0}$$
!yields
$$\scale100 \mira{-25}{99}{300}{0}$$
\enddemo
\vskip2\bigskipamount
%Three paths nested:
\begindemo
%a=.32, b=1, x=-5, y=0
%a=.31, b=1, x=12, y=0
%a=.30, b=1, x= 7, y=-12
\unitlength.02pt \scale100
$$\mira{32}{100}{-500}0%
  \mira{31}{100}{1200}0%
  \mira{30}{100}{700}{-1200}$$
!yields
\unitlength.02pt \scale100
$$\mira{32}{100}{-500}0%
  \mira{31}{100}{1200}0%
  \mira{30}{100}{700}{-1200}$$
\enddemo

\kern-1\medskipamount
\blueexample H\'enon model

^^{H\'enon}
%H\'enon's  model. 
%cgl@rc.service.rug.nl June 1995.
%\newdimen\unitlength \unitlength1pt
%\newcount\k\newcount\maxk
%\newcount\x\newcount\y\newcount\oldx\newcount\oldy
\newcount\thoa \newcount\thob
\def\putatxy{\hbox to0pt{\kern\x\unitlength
         \vbox to0pt{\vss\hrule width.5pt height.5pt\kern\y\unitlength
         }\hss}}
\def\henon#1#2{%#1=1000*a #2=1000*b. a, b have a factor 1000
%Transforms  (x) (a -b) (x)   (b*x*x)
%            |  |=|    | |  | + |       |
%            (y) (b  a) (y)   (-a*x*x)
\loop%\showthe\x\showthe\y;
   \oldx\x \multiply\oldx\oldx \multiply\y100 \advance\y-\oldx 
   \oldy\y \oldx\x
   \multiply\x#1 \multiply\x100       \multiply\oldy#2 \advance\x-\oldy \divide\x100000 
   \multiply\oldx#2 \multiply\oldx100 \multiply\y#1    \advance\y\oldx  \divide\y100000
   %\ifnum\k>500 
   \putatxy%\fi
   %\ifnum\x=0 \ifnum\y=0 \message{x and y are 0, at k=\the\k.} \k\kmax\fi\fi
\advance\k1
\ifnum\k<\kmax
\repeat}%end \henon
\begindemo
%x_{k+1} = ax_k - by_k + b*x_k^2
%y_{k+1} = bx_k + ay_k - a*x_k^2
%a=cos(alfa) b=sin(alfa)
\x50 \y99 \unitlength1pt
\thoa240 \thob=971 %alfa=~76 degrees
\maxk200 \vskip2\bigskipamount
$$\henon{\the\thoa}{\the\thob}$$
!yields
\x50 \y99 \unitlength1pt
\thoa=240 \thob=971 %alfa=~76 degrees
%a=\the\thoa/1000, b=\the\thob/1000, x=\the\x, y=\the\y 
\kmax200 \vskip2\bigskipamount
$$\henon{\the\thoa}{\the\thob}$$
\enddemo
\egroup
\vskip5\bigskipamount
Afterthought. 
To program  fractals is fun. 
With \TeX{} it is tedious because of the limited arithmetic facilities
and the consumed memory. 
This can be a nuisance for authors writing about fractals in \TeX.

The more complex fractals emerging from for example looking into the details
of the structure of limit cycles, such as the Mira islands, are beyond the scope 
of this work and \TeX's arithmetic capabilities. 
In the last picture the islands emerge at the horizon, at the end of the spirals, 
with a little imagination.
%\tracingstats1\endinput 
\endinput