Regular complex polygon

In geometry, a regular complex polygon is a generalization of a regular polygon in real space to an analogous structure in a complex Hilbert space, where each real dimension is accompanied by an imaginary one. A regular polygon exists in 2 real dimensions, $$\mathbb{R}^2$$, while a complex polygon exists in two complex dimensions, $$\mathbb{C}^2$$, which can be given real representations in 4 dimensions, $$\mathbb{R}^4$$, which then must be projected down to 2 or 3 real dimensions to be visualized. A complex polygon is generalized as a complex polytope in $$\mathbb{C}^n$$.

A complex polygon may be understood as a collection of complex points, lines, planes, and so on, where every point is the junction of multiple lines, every line of multiple planes, and so on.

The regular complex polygons have been completely characterized, and can be described using a symbolic notation developed by Coxeter.

A regular complex polygon with all 2-edges can be represented by a graph, while forms with k-edges can only be related by hypergraphs. A k-edge can be seen as a set of vertices, with no order implied. They may be drawn with pairwise 2-edges, but this is not structurally accurate.

Regular complex polygons
While 1-polytopes can have unlimited p, finite regular complex polygons, excluding the double prism polygons p{4}2, are limited to 5-edge (pentagonal edges) elements, and infinite regular apeirogons also include 6-edge (hexagonal edges) elements.

Shephard's modified Schläfli notation
Shephard originally devised a modified form of Schläfli's notation for regular polytopes. For a polygon bounded by p1-edges, with a p2-set as vertex figure and overall symmetry group of order g, we denote the polygon as p1(g)p2.

The number of vertices V is then g/p2 and the number of edges E is g/p1.

The complex polygon illustrated above has eight square edges (p1=4) and sixteen vertices (p2=2). From this we can work out that g = 32, giving the modified Schläfli symbol 4(32)2.

Coxeter's revised modified Schläfli notation
A more modern notation p 1 {q}p 2 is due to Coxeter, and is based on group theory. As a symmetry group, its symbol is p 1 [q]p 2.

The symmetry group p 1 [q]p 2 is represented by 2 generators R1, R2, where: R1p1 = R2p2 = I. If q is even, (R2R1)q/2 = (R1R2)q/2. If q is odd, (R2R1)(q−1)/2R2 = (R1R2)(q−1)/2R1. When q is odd, p1=p2.

For 4[4]2 has R14 = R22 = I, (R2R1)2 = (R1R2)2.

For 3[5]3 has R13 = R23 = I, (R2R1)2R2 = (R1R2)2R1.

Coxeter–Dynkin diagrams
Coxeter also generalised the use of Coxeter–Dynkin diagrams to complex polytopes, for example the complex polygon p{q}r is represented by and the equivalent symmetry group, p[q]r, is a ringless diagram. The nodes p and r represent mirrors producing p and r images in the plane. Unlabeled nodes in a diagram have implicit 2 labels. For example, a real regular polygon is 2{q}2 or {q} or.

One limitation, nodes connected by odd branch orders must have identical node orders. If they do not, the group will create "starry" polygons, with overlapping element. So and  are ordinary, while  is starry.

12 Irreducible Shephard groups
Coxeter enumerated this list of regular complex polygons in $$\mathbb{C}^2$$. A regular complex polygon, p{q}r or, has p-edges, and r-gonal vertex figures. p{q}r is a finite polytope if (p + r)q > pr(q − 2).

Its symmetry is written as p[q]r, called a Shephard group, analogous to a Coxeter group, while also allowing unitary reflections.

For nonstarry groups, the order of the group p[q]r can be computed as $$g = 8/q \cdot (1/p+2/q+1/r-1)^{-2}$$.

The Coxeter number for p[q]r is $$h = 2/(1/p+2/q+1/r-1)$$, so the group order can also be computed as $$g = 2h^2/q$$. A regular complex polygon can be drawn in orthogonal projection with h-gonal symmetry.

The rank 2 solutions that generate complex polygons are:

Excluded solutions with odd q and unequal p and r are: 6[3]2, 6[3]3, 9[3]3, 12[3]3, ..., 5[5]2, 6[5]2, 8[5]2, 9[5]2, 4[7]2, 9[5]2, 3[9]2, and 3[11]2.

Other whole q with unequal p and r, create starry groups with overlapping fundamental domains:, , , , , and.

The dual polygon of p{q}r is r{q}p. A polygon 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 polygon is the same as quasiregular. As well, regular polygon with the same node orders,, have an alternated construction , allowing adjacent edges to be two different colors.

The group order, g, is used to compute the total number of vertices and edges. It will have g/r vertices, and g/p edges. When p=r, the number of vertices and edges are equal. This condition is required when q is odd.

Matrix generators
The group p[q]r,, can be represented by two matrices: With
 * $$ k = \sqrt \frac{ \cos(\frac{\pi}{p}-\frac{\pi}{r})+\cos(\frac{2\pi}{q}) }{2\sin\frac{\pi}{p}\sin\frac{\pi}{r} } $$


 * Examples

Enumeration of regular complex polygons
Coxeter enumerated the complex polygons in Table III of Regular Complex Polytopes.

2D graphs
Polygons of the form p{2r}q can be visualized by q color sets of p-edge. Each p-edge is seen as a regular polygon, while there are no faces.


 * Complex polygons 2{r}q:

Polygons of the form 2{4}q are called generalized orthoplexes. They share vertices with the 4D q-q duopyramids, vertices connected by 2-edges.

3D perspective
3D perspective projections of complex polygons p{4}2 can show the point-edge structure of a complex polygon, while scale is not preserved.

The duals 2{4}p: are seen by adding vertices inside the edges, and adding edges in place of vertices.

Quasiregular polygons
A quasiregular polygon is a truncation of a regular polygon. A quasiregular polygon contains alternate edges of the regular polygons  and. The quasiregular polygon has p vertices on the p-edges of the regular form.