3_{ 31} honeycomb
In 7-dimensional geometry, the 3_{31} honeycomb is a uniform honeycomb, also given by Schläfli symbol {3,3,3,3^{3,1}} and is composed of 3_{21} and 7-simplex facets, with 56 and 576 of them respectively around each vertex.
3_{31} honeycomb | |
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(no image) | |
Type | Uniform tessellation |
Schläfli symbol | {3,3,3,3^{3,1}} |
Coxeter symbol | 3_{31} |
Coxeter-Dynkin diagram | |
7-face types | 3_{21} {3^{6}} |
6-face types | 2_{21} {3^{5}} |
5-face types | 2_{11} {3^{4}} |
4-face type | {3^{3}} |
Cell type | {3^{2}} |
Face type | {3} |
Face figure | 0_{31} |
Edge figure | 1_{31} |
Vertex figure | 2_{31} |
Coxeter group | , [3^{3,3,1}] |
Properties | vertex-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 the node on the short branch leaves the 6-simplex facet:
Removing the node on the end of the 3-length branch leaves the 3_{21} facet:
The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes 2_{31} polytope.
The edge figure is determined by removing the ringed node and ringing the neighboring node. This makes 6-demicube (1_{31}).
The face figure is determined by removing the ringed node and ringing the neighboring node. This makes rectified 5-simplex (0_{31}).
The cell figure is determined by removing the ringed node of the face figure and ringing the neighboring nodes. This makes tetrahedral prism {}×{3,3}.
Kissing number
Each vertex of this tessellation is the center of a 6-sphere in the densest known packing in 7 dimensions; its kissing number is 126, represented by the vertices of its vertex figure 2_{31}.
E7 lattice
The 3_{31} honeycomb's vertex arrangement is called the E_{7} lattice.[1]
contains as a subgroup of index 144.[2] Both and can be seen as affine extension from from different nodes:
The E_{7} lattice can also be expressed as a union of the vertices of two A_{7} lattices, also called A_{7}^{2}:
= ∪
The E_{7}^{*} lattice (also called E_{7}^{2})[3] 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.[4] 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 honeycombs
It is in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 3_{k1} series. A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral hosohedron.
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<sup>1,3,1</sup>]] = [4,3,3,3,3] |
[3^{2,3,1}] | [3^{3,3,1}] | [3^{4,3,1}] |
Order | 48 | 720 | 46,080 | 2,903,040 | ∞ | |
Graph | - | - | ||||
Name | 3_{1,-1} | 3_{10} | 3_{11} | 3_{21} | 3_{31} | 3_{41} |
See also
References
- http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/E7.html
- N.W. Johnson: Geometries and Transformations, (2018) Chapter 12: Euclidean symmetry groups, p 177
- http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/Es7.html
- The Voronoi Cells of the E6* and E7* Lattices, Edward Pervin
- 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 GoogleBook
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3–45]
- R. T. Worley, The Voronoi Region of E7*. SIAM J. Discrete Math., 1.1 (1988), 134-141.
- Conway, John H.; Sloane, Neil J. A. (1998). Sphere Packings, Lattices and Groups ((3rd ed.) ed.). New York: Springer-Verlag. ISBN 0-387-98585-9. p124-125, 8.2 The 7-dimensinoal lattices: E7 and E7*
- Klitzing, Richard. "7D Heptacombs x3o3o3o3o3o3o *d3o - naquoh".
Fundamental convex regular and uniform honeycombs in dimensions 2-9 | ||||||
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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} |