| 133 honeycomb | |
|---|---|
| (no image) | |
| Type | Uniform tessellation |
| Schläfli symbol | {3,33,3} |
| Coxeter symbol | 133 |
| Coxeter-Dynkin diagram | 👁 Image 👁 Image 👁 Image 👁 Image 👁 Image 👁 Image 👁 Image 👁 Image 👁 Image 👁 Image 👁 Image 👁 Image 👁 Image or 👁 Image 👁 Image 👁 Image 👁 Image 👁 Image 👁 Image 👁 Image 👁 Image 👁 Image |
| 7-face type | 132 👁 Image |
| 6-face types | 122 👁 Image 131 👁 Image |
| 5-face types | 121 👁 Image {34} 👁 Image |
| 4-face type | 111 👁 Image {33} 👁 Image |
| Cell type | 101 👁 Image |
| Face type | {3} 👁 Image |
| Cell figure | Square |
| Face figure | Triangular duoprism 👁 Image |
| Edge figure | Tetrahedral duoprism |
| Vertex figure | Trirectified 7-simplex 👁 Image |
| Coxeter group | 👁 {\displaystyle {\tilde {E}}_{7}} , [[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 also named pentacontahexa-hecatonicosihexa-exic heptacomb and Jonathan Bowers gives it acronym linoh[1]
Construction
[edit]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
[edit]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
[edit]The 👁 {\displaystyle {\tilde {E}}_{7}}
group is related to the 👁 {\displaystyle {\tilde {F}}_{4}}
by a geometric folding, so this honeycomb can be projected into the 4-dimensional demitesseractic honeycomb.
E7* lattice
[edit]👁 {\displaystyle {\tilde {E}}_{7}}
contains 👁 {\displaystyle {\tilde {A}}_{7}}
as a subgroup of index 144.[2] Both 👁 {\displaystyle {\tilde {E}}_{7}}
and 👁 {\displaystyle {\tilde {A}}_{7}}
can be seen as affine extension from 👁 {\displaystyle A_{7}}
from different nodes: 👁 Image
The E7* lattice (also called E72)[3] 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.[4] 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:
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.
Related polytopes and honeycombs
[edit]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 | 👁 {\displaystyle {\tilde {E}}_{7}} =E7+ |
👁 {\displaystyle {\bar {T}}_{8}} =E7++ |
| Coxeter diagram |
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| 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 | 👁 Image |
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- | - | |
| Name | 13,-1 | 130 | 131 | 132 | 134 | |
Rectified 133 honeycomb
[edit]| Rectified 133 honeycomb | |
|---|---|
| (no image) | |
| Type | Uniform tessellation |
| Schläfli symbol | {33,3,1} |
| Coxeter symbol | 0331 |
| Coxeter-Dynkin diagram | 👁 Image 👁 Image 👁 Image 👁 Image 👁 Image 👁 Image 👁 Image 👁 Image 👁 Image 👁 Image 👁 Image 👁 Image 👁 Image or 👁 Image 👁 Image 👁 Image 👁 Image 👁 Image 👁 Image 👁 Image 👁 Image 👁 Image |
| 7-face type | Trirectified 7-simplex Rectified 132 |
| 6-face types | Birectified 6-simplex Birectified 6-cube Rectified 122 |
| 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 | 👁 {\displaystyle {\tilde {E}}_{7}} , [[3,33,3]] |
| Properties | vertex-transitive, facet-transitive |
The rectified 133 or 0331, Coxeter diagram 👁 Image
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has facets 👁 Image
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and 👁 Image
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, and vertex figure 👁 Image
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Alternative names
[edit]- Pentacontahexa-hecatonicosihexa-pentacosiheptacontahexa-exic heptacomb
- Rectified pentacontahexa-hecatonicosihexa-exic heptacomb
- Acronym: lanquoh (Jonathan Bowers)[5]
See also
[edit]Notes
[edit]- ^ Klitzing, (o3o3o3o3o3o3o *d3x - linoh).
- ^ N.W. Johnson: Geometries and Transformations, (2018) 12.4: Euclidean Coxeter groups, p.294
- ^ "The Lattice E7".
- ^ The Voronoi Cells of the E6* and E7* Lattices Archived 2016-01-30 at the Wayback Machine, Edward Pervin
- ^ Klitzing, (o3o3o3x3o3o3o *d3o - lanquoh).
References
[edit]- H.S.M. Coxeter, Regular Polytopes, 1973, 3rd edition, Dover, New York, ISBN 0-486-61480-8
- 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, wiley.com, 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, o3o3o3x3o3o3o *d3o - lanquoh
Fundamental convex regular and uniform honeycombs in dimensions 2–9
| ||||||
|---|---|---|---|---|---|---|
| Space | Family | 👁 {\displaystyle {\tilde {A}}_{n-1}} |
👁 {\displaystyle {\tilde {C}}_{n-1}} |
👁 {\displaystyle {\tilde {B}}_{n-1}} |
👁 {\displaystyle {\tilde {D}}_{n-1}} |
👁 {\displaystyle {\tilde {G}}_{2}} / 👁 {\displaystyle {\tilde {F}}_{4}} / 👁 {\displaystyle {\tilde {E}}_{n-1}} |
| E2 | Uniform tiling | 0[3] | Hexagonal | |||
| E3 | Uniform convex honeycomb | 0[4] | ||||
| E4 | Uniform 4-honeycomb | 0[5] | 24-cell honeycomb | |||
| E5 | Uniform 5-honeycomb | 0[6] | ||||
| E6 | Uniform 6-honeycomb | 0[7] | 222 | |||
| E7 | Uniform 7-honeycomb | 0[8] | • 331 | |||
| E8 | Uniform 8-honeycomb | 0[9] | 152 • 251 • 521 | |||
| E9 | Uniform 9-honeycomb | 0[10] | ||||
| E10 | Uniform 10-honeycomb | 0[11] | ||||
| En−1 | Uniform (n−1)-honeycomb | 0[n] | 1k2 • 2k1 • k21 | |||