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In geometry, the Möbius–Kantor polygon is a regular complex polygon 3{3}3, , in
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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
where
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.
Related polytopes
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.