Möbius–Kantor polygon

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Möbius–Kantor polygon
Orthographic projection
Complex polygon 3-3-3.png
shown here with 4 red and 4 blue 3-edge triangles.
Shephard symbol 3(24)3
Schläfli symbol 3{3}3
Coxeter diagram CDel 3node 1.pngCDel 3.pngCDel 3node.png
Edges 8 3{} Complex trion.png
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, CDel 3node 1.pngCDel 3.pngCDel 3node.png, 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]

Real representation[]

It has a real representation as the 16-cell, CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png, 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 Complex polygon 3-3-3-B4.svg Complex polygon 3-3-3-B4b.svg Complex polygon 3-3-3.png
Symmetry [8] [12/3]

Related polytopes[]

Compound of two complex polygon 3-3-3.png
This graph shows the two alternated polygons as a compound in red and blue 3{3}3 in dual positions. 		Complex polygon 3-6-2.png
3{6}2, CDel 3node 1.pngCDel 6.pngCDel node.png or CDel 3node 1.pngCDel 3.pngCDel 3node 1.png, 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 CDel node 1.pngCDel 6.pngCDel 3node.png, represented as CDel node h.pngCDel 6.pngCDel 3node.png. CDel node 1.pngCDel 6.pngCDel 3node.png has 16 vertices, and 24 edges. A compound of two, in dual positions, CDel 3node 1.pngCDel 3.pngCDel 3node.png and CDel 3node.pngCDel 3.pngCDel 3node 1.png, can be represented as CDel node h3.pngCDel 6.pngCDel 3node.png, contains all 16 vertices of CDel node 1.pngCDel 6.pngCDel 3node.png.

The truncation CDel 3node 1.pngCDel 3.pngCDel 3node 1.png, is the same as the regular polygon, 3{6}2, CDel 3node 1.pngCDel 6.pngCDel node.png. Its edge-diagram is the cayley diagram for 3[3]3.

The regular Hessian polyhedron 3{3}3{3}3, CDel 3node 1.pngCDel 3.pngCDel 3node.pngCDel 3.pngCDel 3node.png 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]
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