Login
Login

Möbius–Kantor polygon

class=wikitable style="float:right; margin-left:8px; width:240px"

!bgcolor=#e7dcc3 colspan=2|Möbius–Kantor polygon

align=center

|colspan=2|240px
The 8 3-edges (4 in red, 4 in green) projected symmetrically into 8 vertices of a square antiprism.

bgcolor=#e7dcc3|Shephard symbol3(24)3
bgcolor=#e7dcc3|Schläfli symbol3{3}3
bgcolor=#e7dcc3|Coxeter diagram{{CDD|3node_1|3|3node}}
bgcolor=#e7dcc3|Edges8 3{} 30px
bgcolor=#e7dcc3|Vertices8
bgcolor=#e7dcc3|Petrie polygonOctagon
bgcolor=#e7dcc3|Shephard group3[3]3, order 24
bgcolor=#e7dcc3|Dual polyhedronSelf-dual
bgcolor=#e7dcc3|PropertiesRegular

In geometry, the Möbius–Kantor polygon is a regular complex polygon 3{3}3, {{CDD|3node_1|3|3node}}, in \mathbb{C}^2. 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 3[3]3, isomorphic to the binary tetrahedral group, order 24.

Coordinates

The 8 vertex coordinates of this polygon can be given in \mathbb{C}^3, as:

class=wikitable

| (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 \omega = \tfrac{-1+i\sqrt3}{2} .

As a configuration

The configuration matrix for 3{3}3 is:Coxeter, Complex Regular polytopes, p.117, 132 \left [\begin{smallmatrix}8&3\\3&8\end{smallmatrix}\right ]

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, {{CDD|node_1|3|node|3|node|4|node}}, 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.

class=wikitable

|+ orthographic projections

style="text-align:center;"

!Plane

!colspan=2|B4

!F4

style="text-align:center;"

!Graph

|120px

|120px

|120px

style="text-align:center;"

!Symmetry

|colspan=2|[8]

|[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.

:480px

Related polytopes

class=wikitable style="float:right; margin-left:8px; width:300px"

|150px
This graph shows the two alternated polygons as a compound in red and blue 3{3}3 in dual positions.

|150px
3{6}2, {{CDD|3node_1|6|node}} or {{CDD|3node_1|3|3node_1}}, 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 {{CDD|node_1|6|3node}}, represented as {{CDD|node_h|6|3node}}. {{CDD|node_1|6|3node}} has 16 vertices, and 24 edges. A compound of two, in dual positions, {{CDD|3node_1|3|3node}} and {{CDD|3node|3|3node_1}}, can be represented as {{CDD|node_h3|6|3node}}, contains all 16 vertices of {{CDD|node_1|6|3node}}.

The truncation {{CDD|3node_1|3|3node_1}}, is the same as the regular polygon, 3{6}2, {{CDD|3node_1|6|node}}. Its edge-diagram is the cayley diagram for 3[3]3.

The regular Hessian polyhedron 3{3}3{3}3, {{CDD|3node_1|3|3node|3|3node}} has this polygon as a facet and vertex figure.

Notes

{{reflist}}

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 [https://www.jstor.org/stable/1575843?seq=1#page_scan_tab_contents stable version]

{{DEFAULTSORT:Mobius-Kantor polygon}}

Category:Polytopes

Category:Complex analysis