Complex polytope

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

However, there is no natural complex analogue of the ordering of points on a real line (or of the associated combinatorial properties).

More fully, say that a collection P of affine subspaces (or flats) of a complex unitary space V of dimension n is a regular complex polytope if it meets the following conditions:[1][2] (Here, a flat of dimension −1 is taken to mean the empty set.)

The regular complex polytopes were discovered by Shephard (1952), and the theory was further developed by Coxeter (1974).

In the left image, the outlined squares are not elements of the polytope but are included merely to help identify vertices lying in the same complex line.

The octagonal perimeter of the left image is not an element of the polytope, but it is a petrie polygon.

So we may assume (after a suitable choice of scale) that the vertices on the edge satisfy the equation

Thus, in the Argand diagram of the edge, the vertex points lie at the vertices of a regular polygon centered on the origin.

Three real projections of regular complex polygon 4{4}2 are illustrated above, with edges a, b, c, d, e, f, g, h. It has 16 vertices, which for clarity have not been individually marked.

The sides of the square are not parts of the polygon but are drawn purely to help visually relate the four vertices.

(Note that the diagram looks similar to the B4 Coxeter plane projection of the tesseract, but it is structurally different).

A regular real 1-dimensional polytope is represented by an empty Schläfli symbol {}, or Coxeter-Dynkin diagram .

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.

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

As well, regular polygon with the same node orders, , have an alternated construction , allowing adjacent edges to be two different colors.

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

or higher exist in three families, the real simplexes and the generalized hypercube, and orthoplex.

Coxeter enumerated this list of nonstarry regular complex apeirotopes or honeycombs.

There are 22 regular complex apeirohedra, of the form p{a}q{b}r. 8 are self-dual (p=r and a=b), while 14 exist as dual polytope pairs.

For example, the van Oss polygons of a real octahedron are the three squares whose planes pass through its center.

[36] If it exists, the van Oss polygon of regular complex polytope of the form p{q}r{s}t... has p-edges.

The dual of a product polytope can be written as a sum p{}+q{} and have real representations as the 4-dimensional p-q duopyramid.

The vertex arrangements of these apeirogon have real representations with the regular and uniform tilings of the Euclidean plane.

The symmetry of the self-dual families can be doubled, so creating an identical geometry as the regular forms: = Like real polytopes, a complex quasiregular polyhedron can be constructed as a rectification (a complete truncation) of a regular polyhedron.

The middle quasiregular form p-generalized cuboctahedron, , has 3p2 vertices, 3p3 edges, and 3p+p3 faces.

Other nonregular complex polytopes can be constructed within unitary reflection groups that don't make linear Coxeter graphs.

As with all Wythoff constructions, polytopes generated by reflections, the number of vertices of a single-ringed Coxeter diagram polytope is equal to the order of the group divided by the order of the subgroup where the ringed node is removed.

Some groups have the same order, but a different structure, defining the same vertex arrangement in complex polytopes, but different edges and higher elements, like and with p≠3.

The second is a fractional generalized cube, reducing p-edges into single vertices leaving ordinary 2-edges.

Complex 1-polytopes represented in the Argand plane as regular polygons for p = 2, 3, 4, 5, and 6, with black vertices. The centroid of the p vertices is shown seen in red. The sides of the polygons represent one application of the symmetry generator, mapping each vertex to the next counterclockwise copy. These polygonal sides are not edge elements of the polytope, as a complex 1-polytope can have no edges (it often is a complex edge) and only contains vertex elements.
A real edge is generated as the line between a point and its reflective image across a mirror. A unitary reflection order 2 can be seen as a 180 degree rotation around a center. An edge is inactive if the generator point is on the reflective line or at the center.
12 irreducible Shephard groups with their subgroup index relations. [ 8 ] Subgroups index 2 relate by removing a real reflection:
p [2 q ] 2 → p [ q ] p , index 2.
p [4] q → p [ q ] p , index q .
p [4] 2 subgroups: p=2,3,4...
p [4] 2 → [ p ], index p
p [4] 2 → p []× p [], index 2
Some rank 3 Shephard groups with their group orders, and the reflective subgroup relations
Some subgroups of the apeirogonal Shephard groups
11 complex apeirogons p { q } r with edge interiors colored in light blue, and edges around one vertex are colored individually. Vertices are shown as small black squares. Edges are seen as p -sided regular polygons and vertex figures are r -gonal.
A quasiregular apeirogon is a mixture of two regular apeirogons and , seen here with blue and pink edges. has only one color of edges because q is odd, making it a double covering.
A red square van Oss polygon in the plane of an edge and center of a regular octahedron.
Example truncation of 3-generalized octahedron, 2 {3} 2 {4} 3 , , to its rectified limit, showing outlined-green triangles faces at the start, and blue 2 {4} 3 , , vertex figures expanding as new faces.