Power diagram
| 7-simplex honeycomb | |
|---|---|
| (No image) | |
| Type | Uniform honeycomb |
| Family | Simplectic honeycomb |
| Schläfli symbol | {3[8]} |
| Coxeter–Dynkin diagram | Template:CDD |
| 6-face types | {36}  , t1{36}  t2{36}  , t3{36} 
|
| 6-face types | {35}  , t1{35}  t2{35} 
|
| 5-face types | {34}  , t1{34}  t2{34} 
|
| 4-face types | {33}  , t1{33} 
|
| Cell types | {3,3}  , t1{3,3} 
|
| Face types | {3} 
|
| Vertex figure | t0,6{36} 
|
| Symmetry | ×2, <[3[8]]> |
| Properties | vertex-transitive |
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.
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 ATemplate:Sup sub lattice can be constructed as the union of two A7 lattices, and is identical to the E7 lattice.
Template:CDD + Template:CDD = Template:CDD.
The ATemplate:Sup sub lattice is the union of four A7 lattices, which is identical to the E7* lattice (or ETemplate:Sup sub).
Template:CDD + Template:CDD + Template:CDD + Template:CDD.
The ATemplate:Sup sub lattice (also called ATemplate:Sup sub) is the union of eight A7 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.
Template:CDD + Template:CDD + Template:CDD + Template:CDD + Template:CDD + Template:CDD + Template:CDD + Template:CDD = dual of Template:CDD.
Related polytopes and honeycombs
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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:
| Template:CDD | |
| Template:CDD |
See also
Regular and uniform honeycombs in 7-space:
- 7-cubic honeycomb
- 7-demicubic honeycomb
- Truncated 7-simplex honeycomb
- Omnitruncated 7-simplex honeycomb
- E7 honeycomb
Notes
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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 [1]
- (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]
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- ↑ http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A7.html
- ↑ N.W. Johnson: Geometries and Transformations, Manuscript, (2011) Chapter 12: Euclidean symmetry groups, p 177