5-demicubic honeycomb
| Demipenteractic honeycomb | |
|---|---|
| (No image) | |
| Type | Uniform 5-honeycomb |
| Family | Alternated hypercubic honeycomb |
| Schläfli symbols | h{4,3,3,3,4} h{4,3,3,31,1} ht0,5{4,3,3,3,4} h{4,3,3,4}h{∞} h{4,3,31,1}h{∞} ht0,4{4,3,3,4}h{∞} h{4,3,4}h{∞}h{∞} h{4,31,1}h{∞}h{∞} |
| Coxeter diagrams |
Template:CDD = 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 its Voronoi tessellation is a tritruncated 5-cubic honeycomb, Template:CDD, containing all 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 different 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] |
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] |
h{4,3,3,31,1} | Template:CDD = Template:CDD | Template:CDD [32,1,1] |
16+16: 5-demicube 10: 5-orthoplex |
| 2×½ = [[(4,3,3,3,4,2+)]] | ht0,5{4,3,3,3,4} | Template:CDD | 16+8+8: 5-demicube 10: 5-orthoplex |
Related honeycombs
This honeycomb is one of 20 uniform honeycombs constructed by the Coxeter group, all but 3 repeated in other families by extended symmetry, seen in the graph symmetry of rings in the Coxeter–Dynkin diagrams. The 20 permutations are listed with its highest extended symmetry relation:
| D5 honeycombs | |||
|---|---|---|---|
| Extended symmetry |
Extended diagram |
Extended group |
Honeycombs |
| [31,1,3,31,1] | Template:CDD | Template:CDD | |
| <[31,1,3,31,1]> ↔ [31,1,3,3,4] |
Template:CDD ↔ Template:CDD |
×21 = | Template:CDD, Template:CDD, Template:CDD, Template:CDD |
| [[31,1,3,31,1]] | Template:CDD | ×22 | Template:CDD, Template:CDD |
| <2[31,1,3,31,1]> ↔ [4,3,3,3,4] |
Template:CDD ↔ Template:CDD |
×41 = | Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD |
| [<2[31,1,3,31,1]>] ↔ [[4,3,3,3,4]] |
Template:CDD ↔ Template:CDD |
×8 = ×2 | Template:CDD, Template:CDD, Template:CDD |
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
- Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, Template:ISBN
- 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, Template:ISBN [1]
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
- Template:Cite book
External links
| Template:Navbar-collapsible | ||||||
|---|---|---|---|---|---|---|
| Space | Family | Template:Dark mode invert | Template:Dark mode invert | Template:Dark mode invert | Template:Dark mode invert | Template:Dark mode invert / Template:Dark mode invert / Template:Dark mode invert |
| E2 | Uniform tiling | 0[3] | δ3 | hδ3 | qδ3 | Hexagonal |
| E3 | Uniform convex honeycomb | 0[4] | δ4 | hδ4 | qδ4 | |
| E4 | Uniform 4-honeycomb | 0[5] | δ5 | hδ5 | qδ5 | 24-cell honeycomb |
| E5 | Uniform 5-honeycomb | 0[6] | δ6 | hδ6 | qδ6 | |
| E6 | Uniform 6-honeycomb | 0[7] | δ7 | hδ7 | qδ7 | 222 |
| E7 | Uniform 7-honeycomb | 0[8] | δ8 | hδ8 | qδ8 | 133 • 331 |
| E8 | Uniform 8-honeycomb | 0[9] | δ9 | hδ9 | qδ9 | 152 • 251 • 521 |
| E9 | Uniform 9-honeycomb | 0[10] | δ10 | hδ10 | qδ10 | |
| E10 | Uniform 10-honeycomb | 0[11] | δ11 | hδ11 | qδ11 | |
| En−1 | Uniform (n−1)-honeycomb | 0[n] | δn | hδn | qδn | 1k2 • 2k1 • k21 |
- ↑ Template:Cite web
- ↑ Sphere packings, lattices, and groups, by John Horton Conway, Neil James Alexander Sloane, Eiichi Bannai [2]
- ↑ Conway (1998), p. 119
- ↑ Template:Cite web
- ↑ Conway (1998), p. 120
- ↑ Conway (1998), p. 466