Möbius–Kantor polygon


In geometry, the Möbius–Kantor polygon is a regular complex polygon 33,, in. 33 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,.
Discovered by G.C. Shephard in 1952, he represented it as 33, with its symmetry, Coxeter called as 33, isomorphic to the binary tetrahedral group, order 24.

Coordinates

The 8 vertex coordinates of this polygon can be given in, as:

where.

As a Configuration

The configuration matrix for 33 is:

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.

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, 32,. Its edge-diagram is the cayley diagram for 33.
The regular Hessian polyhedron 333, has this polygon as a facet and vertex figure.