Sylvester equation
| Demipenteractic honeycomb | |
|---|---|
| (No image) | |
| Type | uniform honeycomb |
| Family | Alternated hypercubic honeycomb |
| Schläfli symbol | h{4,3,3,3,4} |
| Coxeter diagram |
Template:CDD or Template:CDD
|
| Facets | {3,3,3,4}  h{4,3,3,3} 
|
| Vertex figure | t1{3,3,3,4} 
|
| Coxeter group | [4,3,3,31,1] [31,1,3,31,1] |
The 5-demicube honeycomb, or demipenteractic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 5-space. It is constructed as an alternation of the regular 5-cube honeycomb.
It is the first tessellation in the demihypercube honeycomb family which, with all the next ones, is not regular, being composed of two different types of uniform facets. The 5-cubes become alternated into 5-demicubes h{4,3,3,3} and the alternated vertices create 5-orthoplex {3,3,3,4} facets.
D5 lattice
The vertex arrangement of the 5-demicubic honeycomb is the D5 lattice which is the densest known sphere packing in 5 dimensions.[1] The 40 vertices of the rectified 5-orthoplex vertex figure of the 5-demicubic honeycomb reflect the kissing number 40 of this lattice.[2]
The DTemplate:Sup sub packing (also called DTemplate:Sup sub) can be constructed by the union of two D5 lattices. The analogous packings form lattices only in even dimensions. The kissing number is 24=16 (2n-1 for n<8, 240 for n=8, and 2n(n-1) for n>8).[3]
The DTemplate:Sup sub[4] lattice (also called DTemplate:Sup sub and CTemplate:Sup sub) can be constructed by the union of all four 5-demicubic lattices:[5] It is also the 5-dimensional body centered cubic, the union of two 5-cube honeycombs in dual positions.
The kissing number of the DTemplate:Sup sub lattice is 10 (2n for n≥5) and it Voronoi tessellation is a tritruncated 5-cubic honeycomb, Template:CDD, containing all with bitruncated 5-orthoplex, Template:CDD Voronoi cells.[6]
Symmetry constructions
There are three uniform construction symmetries of this tessellation. Each symmetry can be represented by arrangements of differened colors on the 32 5-demicube facets around each vertex.
| Coxeter group | Schläfli symbol | Coxeter-Dynkin diagram | Vertex figure Symmetry |
Facets/verf |
|---|---|---|---|---|
| = [31,1,3,3,4] = [1+,4,3,3,4] |
{31,1,3,3,4} = h{4,3,3,3,4} |
Template:CDD = Template:CDD | Template:CDD [3,3,3,4] |
32: 5-demicube 10: 5-orthoplex |
| = [31,1,3,31,1] = [1+,4,3,31,1] |
{31,1,3,31,1} | Template:CDD = Template:CDD | Template:CDD [32,1,1] |
16+16: 5-demicube 10: 5-orthoplex |
| = ([[4,3,3,4,2+]]) | ht0,4{4,3,3,4} | Template:CDD | 16+8+8: 5-demicube 10: 5-orthoplex |
Related honeycombs
| Cantellated 4-simplex honeycomb | |
|---|---|
| (No image) | |
| Type | Uniform 4-honeycomb |
| Schläfli symbol | t0,2{3[5]} |
| Coxeter diagram | Template:CDD |
| 4-face types | t1{33}  t0,2{33}  t0,3{33} 
|
| Cell types | Tetrahedron  Octahedron  Cuboctahedron  Triangular prism 
|
| Vertex figure | triangular elongated-antiprismatic prism |
| Symmetry | ×2, [[3[5]]] |
| Properties | vertex-transitive |
In four-dimensional Euclidean geometry, the cantellated 4-simplex honeycomb or cantellated 5-cell honeycomb is a space-filling tessellation honeycomb.
Alternate names
- small cyclorhombated pentachoric tetracomb
- small prismatodispentachoric tetracomb
Related polytopes and honeycombs
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See also
Regular and uniform honeycombs in 4-space:
- tesseractic honeycomb
- 16-cell honeycomb
- 24-cell honeycomb
- Truncated 24-cell honeycomb
- Snub 24-cell honeycomb
- 5-cell honeycomb
- Truncated 5-cell 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]
- George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs) Model 136
- Template:KlitzingPolytopes x3o3x3o3o3*a - scyropot - O136
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See also
Regular and uniform honeycombs in 5-space:
- 5-cube honeycomb
- 5-demicube honeycomb
- 5-simplex honeycomb
- Truncated 5-simplex honeycomb
- Omnitruncated 5-simplex honeycomb
References
43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.
- Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8
- pp. 154–156: Partial truncation or alternation, represented by h prefix: h{4,4}={4,4}; h{4,3,4}={31,1,4}, h{4,3,3,4}={3,3,4,3}, ...
- 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 [2]
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
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External links
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- ↑ http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/D5.html
- ↑ Sphere packings, lattices, and groups, by John Horton Conway, Neil James Alexander Sloane, Eiichi Bannai [3]
- ↑ Conway (1998), p. 119
- ↑ http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/Ds5.html
- ↑ Conway (1998), p. 120
- ↑ Conway (1998), p. 466