Dom::DihedralGroup
--
dihedral groupsDom::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.
Dom::DihedralGroup(n)
n |
- | positive integer |
Dom::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.
Dom::DihedralGroup
n(l)
l |
- | list or array of two integers |
Cat::Group
the 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.Define 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.
Ax::canonicalRep