Bennett's inequality
| 5-simplex honeycomb | |
|---|---|
| (No image) | |
| Type | Uniform honeycomb |
| Family | Simplectic honeycomb |
| Schläfli symbol | {3[6]} |
| Coxeter–Dynkin diagrams | Template:CDD |
| 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 | t0,4{34} 
|
| Coxeter groups | ×2, <[3[6]]> |
| Properties | vertex-transitive |
In five-dimensional Euclidean geometry, the 5-simplex honeycomb or hexateric honeycomb is a space-filling tessellation (or honeycomb or pentacomb). Each vertex is shared by 12 5-simplexes, 30 rectified 5-simplexes, and 20 birectified 5-simplexes. These facet types occur in proportions of 2:2:1 respectively in the whole honeycomb.
A5 lattice
This vertex arrangement is called the A5 lattice or 5-simplex lattice. The 30 vertices of the stericated 5-simplex vertex figure represent the 30 roots of the Coxeter group.[1] It is the 5-dimensional case of a simplectic honeycomb.
The ATemplate:Sup sub lattice can be constructed as the union of two A5 lattices:
The ATemplate:Sup sub is the union of three A5 lattices:
Template:CDD + Template:CDD + Template:CDD.
The ATemplate:Sup sub lattice (also called ATemplate:Sup sub) is the union of six A5 lattices, and is the dual vertex arrangement to the omnitruncated 5-simplex honeycomb, and therefore the Voronoi cell of this lattice is an omnitruncated 5-simplex.
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 5-simplex honeycomb can be projected into the 3-dimensional cubic 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 5-space:
- 5-cubic honeycomb
- 5-demicube honeycomb
- Truncated 5-simplex honeycomb
- Omnitruncated 5-simplex 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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