Möbius–Kantor polygonThe 8 3-edges (4 in red, 4 in green) projected symmetrically into 8 vertices of a square antiprism.Shephard symbol3(24)3Schläfli symbol3{3}3Coxeter diagramEdges8 3{} Vertices8Petrie polygonOctagonShephard group33, order 24Dual polyhedronSelf-dualPropertiesRegular 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.Coxeter and Shephard, 1991, p. 30 and p. 47 Coxeter named it a Möbius–Kantor polygon for sharing the complex configuration structure as the Möbius–Kantor configuration, (83).Coxeter and Shephard, 1992
Discovered by G.C. Shephard in 1952, he represented it as 3(24)3, with its symmetry, Coxeter called as 33, 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:Coxeter, Complex Regular polytopes, p. 117, 132
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 projectionsPlaneB4F4GraphSymmetry[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.
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.Coxeter, Regular Complex Polytopes, p. 109
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 33.
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, stable version.