Möbius–Kantor polygon
| Möbius–Kantor polygon | |
|---|---|
Orthographic projection shown here with 4 red and 4 blue 3-edge triangles. | |
| Shephard symbol | 3(24)3 |
| Schläfli symbol | 3{3}3 |
| Coxeter diagram |   
|
| Edges | 8 3{} 
|
| Vertices | 8 |
| Petrie polygon | Octagon |
| Shephard group | 3[3]3, order 24 |
| Dual polyhedron | Self-dual |
| Properties | Regular |
In geometry, the Möbius–Kantor polygon is a regular complex polygon 3{3}3, , in [math]\displaystyle{ \mathbb{C}^2 }[/math]. 3{3}3 has 8 vertices, and 8 edges. It is self-dual. Every vertex is shared by 3 triangular edges.[1] Coxeter named it a Möbius–Kantor polygon for sharing the complex configuration structure as the Möbius–Kantor configuration, (83).[2]
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 [math]\displaystyle{ \mathbb{C}^3 }[/math], as:
| (ω,−1,0) | (0,ω,−ω2) | (ω2,−1,0) | (−1,0,1) |
| (−ω,0,1) | (0,ω2,−ω) | (−ω2,0,1) | (1,−1,0) |
where [math]\displaystyle{ \omega = \tfrac{-1+i\sqrt3}{2} }[/math].
As a configuration
The configuration matrix for 3{3}3 is:[3] [math]\displaystyle{ \left [\begin{smallmatrix}8&3\\3&8\end{smallmatrix}\right ] }[/math]
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.
| Plane | B4 | F4 | |
|---|---|---|---|
| Graph | 
|
|
|
| Symmetry | [8] | [12/3] | |
Related polytopes
 This graph shows the two alternated polygons as a compound in red and blue 3{3}3 in dual positions. |
 3{6}2,  or , with 24 vertices in black, and 16 3-edges colored in 2 sets of 3-edges in red and blue.[4]
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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.
Notes
References
- Shephard, G.C.; Regular complex polytopes, Proc. London math. Soc. Series 3, Vol 2, (1952), pp 82–97.
- Coxeter, H. S. M. and Moser, W. O. J.; Generators and Relations for Discrete Groups (1965), esp pp 67–80.
- Coxeter, H. S. M.; Regular Complex Polytopes, Cambridge University Press, (1974), second edition (1991).
- Coxeter, H. S. M. and Shephard, G.C.; Portraits of a family of complex polytopes, Leonardo Vol 25, No 3/4, (1992), pp 239–244 [1]
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