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,

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.

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 originally devised a modified form of Schläfli's notation for regular polytopes.

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.

A more modern notation p1{q}p2 is due to Coxeter,[2] and is based on group theory.

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.

If they do not, the group will create "starry" polygons, with overlapping element.

Coxeter enumerated this list of regular complex polygons in

A regular complex polygon, p{q}r or , has p-edges, and r-gonal vertex figures.

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

A regular complex polygon can be drawn in orthogonal projection with h-gonal symmetry.

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

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.

[5] The group order, g, is used to compute the total number of vertices and edges.

[7] Polygons of the form p{2r}q can be visualized by q color sets of p-edge.

Polygons of the form 2{4}q are called generalized orthoplexes.

The perspective is distorted slightly for odd dimensions to move overlapping vertices from the center.

Polygons of the form p{r}p have equal number of vertices and edges.

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.

The quasiregular polygon has p vertices on the p-edges of the regular form.