6-simplex honeycomb

6-simplex honeycomb
(No image)
Type Uniform honeycomb
Family Simplectic honeycomb
Schläfli symbol {3[7]}
Coxeter–Dynkin diagrams
6-face types {3,3,3,3,3}
t1{3,3,3,3,3}
t2{3,3,3,3,3}
5-face types {3,3,3,3}
t1{3,3,3,3}
t2{3,3,3,3}
4-face types {3,3,3}
t1{3,3,3}
Cell types {3,3}
t1{3,3}
Face types {3}
Vertex figure t05{35}
Coxeter groups {\tilde{A}}_6, [3[7]]
Properties vertex-transitive

In six-dimensional Euclidean geometry, the 6-simplex honeycomb is a space-filling tessellation (or honeycomb). The tessellation fills space by 6-simplex, rectified 6-simplex, and birectified 6-simplex facets. These facet types occur in proportions of 1:1:1 respectively in the whole honeycomb.

This vertex arrangement is called the A6 lattice or 6-simplex lattice. The 42 vertices of the expanded 6-simplex vertex figure represent the 42 roots of the {\tilde{A}}_6 Coxeter group.[1]

Contents

Related polytopes and honeycombs

This honeycomb is one of 17 unique uniform honycombs[2] constructed by the {\tilde{A}}_6 Coxeter group. The Coxeter–Dynkin diagrams of the others are:

See also

Notes

  1. ^ http://www2.research.att.com/~njas/lattices/A6.html
  2. ^ * Weisstein, Eric W., "Necklace" from MathWorld., A000029 18-1 cases, skipping one with zero marks

References