Möbius–Kantor polygon
In geometry, the Möbius–Kantor polygon is a regular complex polygon 3{3}3,
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 |
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 , 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 .
As a Configuration
The configuration matrix for 3{3}3 is:[3]
Real representation
It has a real representation as the 16-cell,
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, |
It can also be seen as an alternation of
The truncation
The regular Hessian polyhedron 3{3}3{3}3,
Notes
- Coxeter and Shephard, 1991, p.30 and p.47
- Coxeter and Shephard, 1992
- Coxeter, Complex Regular polytopes, p.117, 132
- Coxeter, Regular Complex Polytopes, p. 109
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