User:Tomruen/Complex polygons

Rank 2


The symmetry of a regular complex polygon is p[q]r, called a Shephard group, analogous to a Coxeter group, while also allowing real and unitary reflections.

The rank 2 solutions that generate complex polygons are: 2[q]2, p[4]2, 3[3]3, 3[6]2, 3[4]3, 4[3]4, 3[8]2, 4[6]2, 4[4]3, 3[5]3, 5[3]5, 3[10]2, 5[6]2, and 5[4]3 or, , , , , , , , , , , ,.

Starry groups with whole q are, , , , ,.

Enumeration of regular complex polygons
Coxeter enumerated this list of regular complex polygons in $$\mathbb{C}^2$$. Groups of the form p[2q]2 have a half symmetry p[q]p, so a regular polygon is the same as quasiregular.

Enumeration of quasiregular complex polygons
The truncation of a regular complex polygon, is. It is quasiregular, alternating two types of edges. A regular polyhedron with v vertices and e edges has qe vertices, and v+e edges of two types: v q{} edges, and e p{} edges.

Regular complex apeirogons
Coxeter expresses them as &delta; where q is constrained to satisfy $q = 2/(1 – (p + r)/pr)$.

Among the aperiogons, four are self-dual (when p = r), while eight exist as dual polytope pairs. Only one, {&infin;}, is real. The 12 pairs (p, r) as corresponding to aperiogons are (2,2), (3,2), (2,3), (3,3), (4,2), (2,4), (4,4), (6,2), (2,6), (6,3), (3,6), and (6,6).

Rank 2 complex apeirogons have symmetry p[q]r, where 1/p + 2/q + 1/r = 1. There are 8 solutions: 2[&infin;]2, 3[12]2, 4[8]2, 6[6]2, 3[6]3, 6[4]3, 4[4]4, and 6[3]6 or, , , , , ,.

Including affine nodes, there are 3 more infinite solutions:, , , the first is an index 2 subgroup of the second, while the last is starry.

A regular complex apeirogon p{q}r has p-edges and q-gonal vertex figures. The dual apeirogon of p{q}r is r{q}p. An apeirogon of the form p{q}p is self-dual. Groups of the form p[2q]2 have a half symmetry p[q]p, so a regular apeirogon is the same as quasiregular.

Apeirogons can be represented on the Argand plane share four different vertex arrangements. Apeirogons of the form 2[q]r have a vertex arrangement as {q/2,p}. The form p[q]2 have vertex arrangement as r{p,q/2}.

Quasiregular
There are 7 quasiregular complex apeirogons which alternate edges between two dual complex apeirogons.

Rank 3

 * - 27
 * - 54
 * - 64
 * - 96
 * - 125
 * - 150
 * - 162
 * - 336
 * - 384
 * - 648
 * - 750
 * - 1296
 * - 2160
 * - 2160