5-simplex honeycomb

5-simplex honeycomb
(No image)
Type Uniform honeycomb
Family Simplectic honeycomb
Schläfli symbol {3[6]}
Coxeter–Dynkin diagrams
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 t04{34}
Coxeter groups {\tilde{A}}_5, [3[6]]
Properties vertex-transitive

In five-dimensional Euclidean geometry, the 5-simplex honeycomb or hexateric honeycomb is a space-filling tessellation (or honeycomb or pentacomb). Each vertex is shared by 12 5-simplexes, 30 rectified 5-simplexes, and 20 birectified 5-simplexes. These facet types occur in proportions of 2:2:1 respectively in the whole honeycomb.

This vertex arrangement is called the A5 lattice or 5-simplex lattice. The 30 vertices of the stericated 5-simplex vertex figure represent the 30 roots of the {\tilde{A}}_5 Coxeter group.[1]

Contents

Related polytopes and honeycombs

This honeycomb is one of 12 unique uniform honycombs[2] constructed by the {\tilde{A}}_5 Coxeter group. The Coxeter–Dynkin diagrams of the other 11 are: , , , , , , , , , , .

Projection by folding

The 5-simplex honeycomb can be projected into the 3-dimensional cubic honeycomb by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same vertex arrangement:

{\tilde{A}}_5
{\tilde{C}}_3

See also

Notes

  1. ^ http://www2.research.att.com/~njas/lattices/A5.html
  2. ^ [1], A000029 13-1 cases, skipping one with zero marks

References