Английская Википедия:1 33 honeycomb
| 133 honeycomb | |
|---|---|
| (no image) | |
| Type | Uniform tessellation |
| Schläfli symbol | {3,33,3} |
| Coxeter symbol | 133 |
| Coxeter-Dynkin diagram | Шаблон:CDD or Шаблон:CDD |
| 7-face type | 132 Файл:Gosset 1 32 petrie.svg |
| 6-face types | 122Файл:Gosset 1 22 polytope.svg 131Файл:Demihexeract ortho petrie.svg |
| 5-face types | 121Файл:Demipenteract graph ortho.svg {34}Файл:5-simplex t0.svg |
| 4-face type | 111Файл:Cross graph 4.svg {33}Файл:4-simplex t0.svg |
| Cell type | 101Файл:3-simplex t0.svg |
| Face type | {3}Файл:2-simplex t0.svg |
| Cell figure | Square |
| Face figure | Triangular duoprism Файл:3-3 duoprism.png |
| Edge figure | Tetrahedral duoprism |
| Vertex figure | Trirectified 7-simplex Файл:7-simplex t3.svg |
| Coxeter group | <math>{\tilde{E}}_7</math>, [[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.
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}.
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 <math>{\tilde{E}}_7</math> group is related to the <math>{\tilde{F}}_4</math> by a geometric folding, so this honeycomb can be projected into the 4-dimensional demitesseractic honeycomb.
| <math>{\tilde{E}}_7</math> | <math>{\tilde{F}}_4</math> |
|---|---|
| Шаблон:CDD | Шаблон:CDD |
| {3,33,3} | {3,3,4,3} |
E7* lattice
<math>{\tilde{E}}_7</math> contains <math>{\tilde{A}}_7</math> as a subgroup of index 144.[1] Both <math>{\tilde{E}}_7</math> and <math>{\tilde{A}}_7</math> can be seen as affine extension from <math>A_7</math> from different nodes: Файл:Affine A7 E7 relations.png
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:
- Шаблон:CDD ∪ Шаблон:CDD = Шаблон:CDD ∪ Шаблон:CDD ∪ Шаблон:CDD ∪ Шаблон:CDD = dual of Шаблон:CDD.
Related polytopes and honeycombs
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. Шаблон:1 3k polytopes
Rectified 133 honeycomb
| Rectified 133 honeycomb | |
|---|---|
| (no image) | |
| Type | Uniform tessellation |
| Schläfli symbol | {33,3,1} |
| Coxeter symbol | 0331 |
| Coxeter-Dynkin diagram | Шаблон:CDD or Шаблон:CDD |
| 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 | <math>{\tilde{E}}_7</math>, [[3,33,3]] |
| Properties | vertex-transitive, facet-transitive |
The rectified 133 or 0331, Coxeter diagram Шаблон:CDD has facets Шаблон:CDD and Шаблон:CDD, and vertex figure Шаблон:CDD.
See also
Notes
References
- H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
- Coxeter The Beauty of Geometry: Twelve Essays, Dover Publications, 1999, Шаблон:ISBN (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, Шаблон:ISBN [1]
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3–45]
- Шаблон:KlitzingPolytopes
- Шаблон:KlitzingPolytopes
- ↑ N.W. Johnson: Geometries and Transformations, (2018) 12.4: Euclidean Coxeter groups, p.294
- ↑ Шаблон:Cite web
- ↑ The Voronoi Cells of the E6* and E7* Lattices Шаблон:Webarchive, Edward Pervin
