# 2 51 honeycomb

251 honeycomb
(No image)
Type Uniform tessellation
Family 2k1 polytope
Schläfli symbol {3,3,35,1}
Coxeter symbol 251
Coxeter-Dynkin diagram
8-face types 241
{37}
7-face types 231
{36}
6-face types 221
{35}
5-face types 211
{34}
4-face type {33}
Cells {32}
Faces {3}
Edge figure 051
Vertex figure 151
Edge figure 051
Coxeter group ${\tilde{E}}_8$, [35,2,1]

In 8-dimensional geometry, the 251 honeycomb is a space-filling uniform tessellation. It is composed of 241 polytope and 8-simplex facets arranged in a 8-demicube vertex figure. It is the final figure in the 2k1 family.

## Construction

It is created by a Wythoff construction upon a set of 9 hyperplane mirrors in 8-dimensional space.

The facet information can be extracted from its Coxeter-Dynkin diagram.

Removing the node on the short branch leaves the 8-simplex.

Removing the node on the end of the 5-length branch leaves the 241.

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the 8-demicube, 151.

The edge figure is the vertex figure of the vertex figure. This makes the rectified 7-simplex, 051.

## Related polytopes and honeycombs

2k1 figures in n dimensions
Space Finite Euclidean Hyperbolic
n 3 4 5 6 7 8 9 10
Coxeter
group
E3=A2×A1 E4=A4 E5=D5 E6 E7 E8 E9 = ${\tilde{E}}_{8}$ = E8+ E10 = E8++
Coxeter
diagram
Symmetry
(order)
[3-1,2,1]
(12)
[30,2,1]
(120)
[[31,2,1]]
(384)
[32,2,1]
(51,840)
[33,2,1]
(2,903,040)
[34,2,1]
(696,729,600)
[35,2,1]
(∞)
[36,2,1]
(∞)
Graph
Name 2-1,1 201 211 221 231 241 251 261