geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, the Möbius–Kantor polygon is a

regular complex polygon In geometry, a regular complex polygon is a generalization of a regular polygon in real space to an analogous structure in a complex Hilbert space, where each real dimension is accompanied by an imaginary one. A regular polygon exists in 2 real ...

₃₃, , in

\mathbb^2

. ₃₃ 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 In geometry, the Möbius–Kantor configuration is a configuration consisting of eight points and eight lines, with three points on each line and three lines through each point. It is not possible to draw points and lines having this pattern of i ...

, (8₃). Discovered by G.C. Shephard in 1952, he represented it as 3(24)3, with its symmetry, Coxeter called as ₃ sub>3, isomorphic to the

binary tetrahedral group In mathematics, the binary tetrahedral group, denoted 2T or , Coxeter&Moser: Generators and Relations for discrete groups: : Rl = Sm = Tn = RST is a certain nonabelian group of order 24. It is an extension of the tetrahedral group T or (2,3,3) of ...

, order 24.

Coordinates

The 8 vertex coordinates of this polygon can be given in

\mathbb^3

, as: where

\omega = \tfrac

As a configuration

The configuration matrix for ₃₃ is:Coxeter, Complex Regular polytopes, p.117, 132

\left begin8&3\\3&8\end\right /math>

Real representation

It has a real representation as the

16-cell In geometry, the 16-cell is the regular convex 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the mi ...

, , 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, ₃₂, . Its edge-diagram is the

cayley diagram In mathematics, a Cayley graph, also known as a Cayley color graph, Cayley diagram, group diagram, or color group is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayley's theorem (named after Arthur Cayle ...

for ₃ sub>3. The regular

Hessian polyhedron In geometry, the Hessian polyhedron is a regular complex polyhedron 333, , in \mathbb^3. It has 27 vertices, 72 3 edges, and 27 33 faces. It is self-dual. Coxeter named it after Ludwig Otto Hesse for sharing the ''Hessian configuration'' \left ...

₃₃₃, has this polygon as a

facet Facets () are flat faces on geometric shapes. The organization of naturally occurring facets was key to early developments in crystallography, since they reflect the underlying symmetry of the crystal structure. Gemstones commonly have facets cut ...

and

vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connect ...

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 239–24

{{DEFAULTSORT:Mobius-Kantor polygon Polytopes Complex analysis