Möbius–Kantor polygon

In geometry, the Möbius–Kantor polygon is a regular complex polygon 3{3}3, , in

C

2

{\displaystyle \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

C

3

{\displaystyle \mathbb {C} ^{3}}

, as:

where

ω

=

−

1

+

i

3

2

{\displaystyle \omega ={\tfrac {-1+i{\sqrt {3}}}{2}}}

.

As a configuration

The configuration matrix for 3{3}3 is:

[

8

3

3

8

]

{\displaystyle \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.