7-simplex honeycomb
In seven-dimensional Euclidean geometry, the 7-simplex honeycomb is a space-filling tessellation (or honeycomb). The tessellation fills space by 7-simplex, rectified 7-simplex, birectified 7-simplex, and trirectified 7-simplex facets. These facet types occur in proportions of 2:2:2:1 respectively in the whole honeycomb.
7-simplex honeycomb | |
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(No image) | |
Type | Uniform 7-honeycomb |
Family | Simplectic honeycomb |
Schläfli symbol | {3^{[8]}} |
Coxeter diagram | |
6-face types | {3^{6}} t_{2}{3^{6}} |
6-face types | {3^{5}} t_{2}{3^{5}} |
5-face types | {3^{4}} t_{2}{3^{4}} |
4-face types | {3^{3}} |
Cell types | {3,3} |
Face types | {3} |
Vertex figure | t_{0,6}{3^{6}} |
Symmetry | ×2_{1}, <[3^{[8]}]> |
Properties | vertex-transitive |
A7 lattice
This vertex arrangement is called the A7 lattice or 7-simplex lattice. The 56 vertices of the expanded 7-simplex vertex figure represent the 56 roots of the Coxeter group.[1] It is the 7-dimensional case of a simplectic honeycomb. Around each vertex figure are 254 facets: 8+8 7-simplex, 28+28 rectified 7-simplex, 56+56 birectified 7-simplex, 70 trirectified 7-simplex, with the count distribution from the 9th row of Pascal's triangle.
contains as a subgroup of index 144.[2] Both and can be seen as affine extensions from from different nodes:
The A^{2}
_{7} lattice can be constructed as the union of two A_{7} lattices, and is identical to the E7 lattice.
The A^{4}
_{7} lattice is the union of four A_{7} lattices, which is identical to the E7* lattice (or E^{2}
_{7}).
The A^{*}
_{7} lattice (also called A^{8}
_{7}) is the union of eight A_{7} lattices, and has the vertex arrangement to the dual honeycomb of the omnitruncated 7-simplex honeycomb, and therefore the Voronoi cell of this lattice is an omnitruncated 7-simplex.
Related polytopes and honeycombs
This honeycomb is one of 29 unique uniform honeycombs[3] constructed by the Coxeter group, grouped by their extended symmetry of rings within the regular octagon diagram:
A7 honeycombs | ||||
---|---|---|---|---|
Octagon symmetry |
Extended symmetry |
Extended diagram |
Extended group |
Honeycombs |
a1 |
[3^{[8]}] |
| ||
d2 |
<[3^{[8]}]> | ×2_{1} |
| |
p2 |
[[3^{[8]}]] | ×2_{2} |
| |
d4 |
<2[3^{[8]}]> | ×4_{1} |
| |
p4 |
[2[3^{[8]}]] | ×4_{2} |
| |
d8 |
[4[3^{[8]}]] | ×8 | ||
r16 |
[8[3^{[8]}]] | ×16 |
Projection by folding
The 7-simplex honeycomb can be projected into the 4-dimensional tesseractic honeycomb by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same vertex arrangement:
See also
Regular and uniform honeycombs in 7-space:
Notes
- http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A7.html
- N.W. Johnson: Geometries and Transformations, (2018) 12.4: Euclidean Coxeter groups, p.294
- Weisstein, Eric W. "Necklace". MathWorld., OEIS sequence A000029 30-1 cases, skipping one with zero marks
References
- Norman Johnson Uniform Polytopes, Manuscript (1991)
- 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 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380–407, MR 2,10] (1.9 Uniform space-fillings)
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3–45]
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} |