1_{ 33} honeycomb
In 7-dimensional geometry, 1_{33} is a uniform honeycomb, also given by Schläfli symbol {3,3^{3,3}}, and is composed of 1_{32} facets.
1_{33} honeycomb | |
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(no image) | |
Type | Uniform tessellation |
Schläfli symbol | {3,3^{3,3}} |
Coxeter symbol | 1_{33} |
Coxeter-Dynkin diagram | or |
7-face type | 1_{32} |
6-face types | 1_{22} 1_{31} |
5-face types | 1_{21} {3^{4}} |
4-face type | 1_{11} {3^{3}} |
Cell type | 1_{01} |
Face type | {3} |
Cell figure | Square |
Face figure | Triangular duoprism |
Edge figure | Tetrahedral duoprism |
Vertex figure | Trirectified 7-simplex |
Coxeter group | , [[3,3<sup>3,3</sup>]] |
Properties | vertex-transitive, facet-transitive |
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 1_{32}, 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, 0_{33}.
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}.
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 group is related to the by a geometric folding, so this honeycomb can be projected into the 4-dimensional demitesseractic honeycomb.
{3,3^{3,3}} | {3,3,4,3} |
E_{7}^{*} lattice
contains as a subgroup of index 144.[1] Both and can be seen as affine extension from from different nodes:
The E_{7}^{*} lattice (also called E_{7}^{2})[2] has double the symmetry, represented by [[3,3<sup>3,3</sup>]]. The Voronoi cell of the E_{7}^{*} lattice is the 1_{32} polytope, and voronoi tessellation the 1_{33} honeycomb.[3] The E_{7}^{*} lattice is constructed by 2 copies of the E_{7} lattice vertices, one from each long branch of the Coxeter diagram, and can be constructed as the union of four A_{7}^{*} lattices, also called A_{7}^{4}:
∪ = ∪ ∪ ∪ = dual of .
Related polytopes and honeycombs
The 1_{33} is fourth in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 1_{3k} series. The final is a noncompact hyperbolic honeycomb, 1_{34}.
Space | Finite | Euclidean | Hyperbolic | |||
---|---|---|---|---|---|---|
n | 4 | 5 | 6 | 7 | 8 | 9 |
Coxeter group |
A_{3}A_{1} | A_{5} | D_{6} | E_{7} | =E_{7}^{+} | =E_{7}^{++} |
Coxeter diagram |
||||||
Symmetry | [3^{−1,3,1}] | [3^{0,3,1}] | [3^{1,3,1}] | [3^{2,3,1}] | [[3<sup>3,3,1</sup>]] | [3^{4,3,1}] |
Order | 48 | 720 | 23,040 | 2,903,040 | ∞ | |
Graph | - | - | ||||
Name | 1_{3,-1} | 1_{30} | 1_{31} | 1_{32} | 1_{33} | 1_{34} |
Rectified 1_{33} honeycomb
Rectified 1_{33} honeycomb | |
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(no image) | |
Type | Uniform tessellation |
Schläfli symbol | {3^{3,3,1}} |
Coxeter symbol | 0_{331} |
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 | , [[3,3<sup>3,3</sup>]] |
Properties | vertex-transitive, facet-transitive |
The rectified 1_{33} or 0_{331}, Coxeter diagram
See also
Notes
- N.W. Johnson: Geometries and Transformations, (2018) 12.4: Euclidean Coxeter groups, p.294
- http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/Es7.html
- The Voronoi Cells of the E6* and E7* Lattices, Edward Pervin
References
- H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
- Coxeter The Beauty of Geometry: Twelve Essays, Dover Publications, 1999, ISBN 978-0-486-40919-1 (Chapter 3: Wythoff's Construction for Uniform Polytopes)
- Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3–45]
- Klitzing, Richard. "7D Heptacombs o3o3o3o3o3o3o *d3x - linoh".
- Klitzing, Richard. "7D Heptacombs o3o3o3x3o3o3o *d3o - rolinoh".
Fundamental convex regular and uniform honeycombs in dimensions 2-9 | ||||||
---|---|---|---|---|---|---|
Space | Family | / / | ||||
E^{2} | Uniform tiling | {3^{[3]}} | δ_{3} | hδ_{3} | qδ_{3} | Hexagonal |
E^{3} | Uniform convex honeycomb | {3^{[4]}} | δ_{4} | hδ_{4} | qδ_{4} | |
E^{4} | Uniform 4-honeycomb | {3^{[5]}} | δ_{5} | hδ_{5} | qδ_{5} | 24-cell honeycomb |
E^{5} | Uniform 5-honeycomb | {3^{[6]}} | δ_{6} | hδ_{6} | qδ_{6} | |
E^{6} | Uniform 6-honeycomb | {3^{[7]}} | δ_{7} | hδ_{7} | qδ_{7} | 2_{22} |
E^{7} | Uniform 7-honeycomb | {3^{[8]}} | δ_{8} | hδ_{8} | qδ_{8} | 1_{33} • 3_{31} |
E^{8} | Uniform 8-honeycomb | {3^{[9]}} | δ_{9} | hδ_{9} | qδ_{9} | 1_{52} • 2_{51} • 5_{21} |
E^{9} | Uniform 9-honeycomb | {3^{[10]}} | δ_{10} | hδ_{10} | qδ_{10} | |
E^{n-1} | Uniform (n-1)-honeycomb | {3^{[n]}} | δ_{n} | hδ_{n} | qδ_{n} | 1_{k2} • 2_{k1} • k_{21} |