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,

R

2

{\displaystyle \mathbb {R} ^{2}}

, while a complex polygon exists in two complex dimensions,

C

2

{\displaystyle \mathbb {C} ^{2}}

, which can be given real representations in 4 dimensions,

R

4

{\displaystyle \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

C

n

{\displaystyle \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.

Notations

Shephard's modified Schläfli notation

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