Möbius–Kantor polygon

Möbius–Kantor polygon

The 8 3-edges (4 in red, 4 in green) projected symmetrically into 8 vertices of a square antiprism.
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 . 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 , 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]

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.

orthographic projections
Plane B4 F4
Graph
Symmetry [8] [12/3]

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.


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]

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

  1. ^ Coxeter and Shephard, 1991, p.30 and p.47
  2. ^ Coxeter and Shephard, 1992
  3. ^ Coxeter, Complex Regular polytopes, p.117, 132
  4. ^ 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 [1]