133 honeycomb | |
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(no image) | |
Type | Uniform tessellation |
Schläfli symbol | {3,33,3} |
Coxeter symbol | 133 |
Coxeter-Dynkin diagram | or |
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 | , [[3,33,3]] |
Properties | vertex-transitive, facet-transitive |
In 7-dimensional geometry, 133 is a uniform honeycomb, also given by Schläfli symbol {3,33,3}, and is composed of 132 facets.
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}.
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.
The group is related to the by a geometric folding, so this honeycomb can be projected into the 4-dimensional demitesseractic honeycomb.
{3,33,3} | {3,3,4,3} |
contains as a subgroup of index 144.[1] Both and can be seen as affine extension from from different nodes:
The E7* lattice (also called E72)[2] has double the symmetry, represented by [[3,33,3]]. The Voronoi cell of the E7* lattice is the 132 polytope, and voronoi tessellation the 133 honeycomb.[3] The E7* lattice is constructed by 2 copies of the E7 lattice vertices, one from each long branch of the Coxeter diagram, and can be constructed as the union of four A7* lattices, also called A74:
The 133 is fourth in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 13k series. The final is a noncompact hyperbolic honeycomb, 134.
Space | Finite | Euclidean | Hyperbolic | |||
---|---|---|---|---|---|---|
n | 4 | 5 | 6 | 7 | 8 | 9 |
Coxeter group |
A3A1 | A5 | D6 | E7 | =E7+ | =E7++ |
Coxeter diagram |
||||||
Symmetry | [3−1,3,1] | [30,3,1] | [31,3,1] | [32,3,1] | [[33,3,1]] | [34,3,1] |
Order | 48 | 720 | 23,040 | 2,903,040 | ∞ | |
Graph | - | - | ||||
Name | 13,-1 | 130 | 131 | 132 | 133 134 |
Rectified 133 honeycomb | |
---|---|
(no image) | |
Type | Uniform tessellation |
Schläfli symbol | {33,3,1} |
Coxeter symbol | 0331 |
Coxeter-Dynkin diagram | or |
7-face type | Trirectified 7-simplex Rectified 1_32 |
6-face types | Birectified 6-simplex Birectified 6-cube Rectified 1_22 |
5-face types | Rectified 5-simplex Birectified 5-simplex Birectified 5-orthoplex |
4-face type | 5-cell Rectified 5-cell 24-cell |
Cell type | {3,3} {3,4} |
Face type | {3} |
Vertex figure | {}×{3,3}×{3,3} |
Coxeter group | , [[3,33,3]] |
Properties | vertex-transitive, facet-transitive |
The rectified 133 or 0331, Coxeter diagram has facets and , and vertex figure .
Space | Family | / / | ||||
---|---|---|---|---|---|---|
E2 | Uniform tiling | 0[3] | δ3 | hδ3 | qδ3 | Hexagonal |
E3 | Uniform convex honeycomb | 0[4] | δ4 | hδ4 | qδ4 | |
E4 | Uniform 4-honeycomb | 0[5] | δ5 | hδ5 | qδ5 | 24-cell honeycomb |
E5 | Uniform 5-honeycomb | 0[6] | δ6 | hδ6 | qδ6 | |
E6 | Uniform 6-honeycomb | 0[7] | δ7 | hδ7 | qδ7 | 222 |
E7 | Uniform 7-honeycomb | 0[8] | δ8 | hδ8 | qδ8 | 133 • 331 |
E8 | Uniform 8-honeycomb | 0[9] | δ9 | hδ9 | qδ9 | 152 • 251 • 521 |
E9 | Uniform 9-honeycomb | 0[10] | δ10 | hδ10 | qδ10 | |
E10 | Uniform 10-honeycomb | 0[11] | δ11 | hδ11 | qδ11 | |
En-1 | Uniform (n-1)-honeycomb | 0[n] | δn | hδn | qδn | 1k2 • 2k1 • k21 |