Supriya Ghosh (Editor)

Möbius–Kantor polygon

Updated on
Edit
Like
Comment
Share on FacebookTweet on TwitterShare on LinkedInShare on Reddit
Möbius–Kantor polygon

In geometry, the Möbius–Kantor polygon is a regular complex polygon 3{3}3, , in C 2 . 3{3}3 has 8 vertices, and 8 edges. It is self-dual. Every vertex is shared by 3 triangular edges. Coxeter named it a Möbius–Kantor polygon for sharing the complex configuration structure as the Möbius–Kantor configuration, (83).

Contents

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 C 3 , as:

where ω = − 1 + i 3 2 .

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.

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.

References

Möbius–Kantor polygon Wikipedia
(Text) CC BY-SA