1 33 honeycomb

133 honeycomb
(no image)
Type Uniform tessellation
Schläfli symbol {3,33,3}
Coxeter symbol 133
Coxeter-Dynkin diagram
7-face type 132
6-face types 122
131
5-face types 121
{34}
4-face type 111
{33}
Cell type 101
Face type {3}
Cell figure Square
Face figure Triangular duoprism
Edge figure Tetrahedral duoprism
Vertex figure Trirectified 7-simplex
Coxeter group {\tilde{E}}_7, [33,3,1]
Properties vertex-transitive, facet-transitive

In 7-dimensional geometry, 133 is a uniform honeycomb, also given by Schlafli symbol {31,3,3}, is composed of 132 facets.

Contents

Construction

It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 7-dimensional space.

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

Removing a node on the end of one of the 3-length branch leaves the 132, its only facet type.

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the trirectified 7-simplex, 033.

The edge figure is determined by removing the ringed nodes of the vertex figure and ringing the neighboring node. This makes the tetrahedral duoprism, {3,3}×{3,3}.

The face figure is determined by removing the ringed nodes of the edge figure and ringing the neighboring node. This makes the triangular duoprism, {3}×{3}.

Kissing number

Each vertex of this polytope corresponds to the center of a 6-sphere in a moderately dense sphere packing, in which each sphere is tangent to 70 others; the best known for 7 dimensions (the kissing number) is 126.

Geometric folding

The {\tilde{E}}_7 group is related to the {\tilde{F}}_4 by a geometric folding, so this honeycomb can be projected into the 4-dimensional demitesseractic honeycomb.

{\tilde{E}}_7 {\tilde{F}}_4
{3,33,3} {3,3,4,3}

See also

References