Dom::DihedralGroup --
dihedral groups
IntroductionDom::DihedralGroup(n) creates the group of
all congruent mappings of the plane that induce a bijective mapping of
the set of corners of a regular n-angle to itself.
DomainDom::DihedralGroup(n)
Parametersn |
- | positive integer |
IntroductionDom::DihedralGroup(n)([a,b]) represents
the group element ``ta carried out after
rb'', where r is a rotation that maps
each corner to its left neighbor, and t is a reflection
w.r.t. some fixed central diagonal.
Creating
ElementsDom::DihedralGroupn(l)
Parametersl |
- | list or array of two integers |
Cat::Group
Related
Domainsthe number of elements, which equals 2n.
the mapping leaving each point fixed.
_mult(dom a...)Dom::DihedralGroup is
defined as their functional composition, with the factors applied from
right to left._mult._invert(dom a)a is defined to be the mapping that
sends every corner to its pre-image under a. (This agrees
with the usual notion of the inverse of a bijective mapping.)_invert._power(dom a, integer
n)_power.order(dom a)random()expr(dom a)Dom::DihedralGroup.TeX(dom a)a is returned as a TeXstring generated
from its list representation. This avoids using fixed names for the
generators, as there is no standard for them in the literature.
Example
1Define the group D_6, i.e., the group of congruence mappings of the hexagon:
>> G := Dom::DihedralGroup(6)
Dom::DihedralGroup(6)
Then elements may be created as follows:
>> a := G([7, 19]);
[1, 1]
This means that 19 rotations--mapping each corner to its left neighbor--and 7 reflections have the same effect as one operation of either type.
Super-DomainAx::canonicalRep