Möbius–Kantor polygon

In geometry, the Möbius–Kantor polygon is a regular complex polygon 3{3}3,, in $$\mathbb{C}^2$$. 3{3}3 has 8 vertices, and 8 edges. It is self-dual. Every vertex is shared by 3 triangular edges. Coxeter named it a Möbius–Kantor polygon for sharing the complex configuration structure as the Möbius–Kantor configuration, (83).

Discovered by G.C. Shephard in 1952, he represented it as 3(24)3, with its symmetry, Coxeter called as 3[3]3, isomorphic to the binary tetrahedral group, order 24.

Coordinates
The 8 vertex coordinates of this polygon can be given in $$\mathbb{C}^3$$, as: where $$\omega = \tfrac{-1+i\sqrt3}{2} $$.

As a configuration
The configuration matrix for 3{3}3 is: $$\left [\begin{smallmatrix}8&3\\3&8\end{smallmatrix}\right ]$$

Its structure can be represented as a hypergraph, connecting 8 nodes by 8 3-node-set hyperedges.

Real representation
It has a real representation as the 16-cell,, in 4-dimensional space, sharing the same 8 vertices. The 24 edges in the 16-cell are seen in the Möbius–Kantor polygon when the 8 triangular edges are drawn as 3-separate edges. The triangles are represented 2 sets of 4 red or blue outlines. The B4 projections are given in two different symmetry orientations between the two color sets.

The 3{3}3 polygon can be seen in a regular skew polyhedral net inside a 16-cell, with 8 vertices, 24 edges, 16 of its 32 faces. Alternate yellow triangular faces, interpreted as 3-edges, make two copies of the 3{3}3 polygon.
 * 16-cell 8-ring net4.png

Related polytopes
It can also be seen as an alternation of, represented as. has 16 vertices, and 24 edges. A compound of two, in dual positions, and, can be represented as , contains all 16 vertices of.

The truncation, is the same as the regular polygon, 3{6}2,. Its edge-diagram is the cayley diagram for 3[3]3.

The regular Hessian polyhedron 3{3}3{3}3, has this polygon as a facet and vertex figure.