5-demicubic honeycomb
| Demipenteractic honeycomb | |
|---|---|
| (No image) | |
| Type | uniform honeycomb |
| Family | Alternated hypercubic honeycomb |
| Schläfli symbol | h{4,3,3,3,4} |
| Coxeter diagram |
|
| 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 D+
5 packing (also called D2
5) 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 D*
5[4] lattice (also called D4
5 and C2
5) 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 D*
5 lattice is 10 (2n for n≥5) and it Voronoi tessellation is a tritruncated 5-cubic honeycomb, 
, containing all with bitruncated 5-orthoplex, 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} | =  | [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} | = | [32,1,1] |
16+16: 5-demicube 10: 5-orthoplex |
 = ([[4,3,3,4,2+]]) | ht0,4{4,3,3,4} |  | 16+8+8: 5-demicube 10: 5-orthoplex |
Related honeycombs
This honeycomb is one of 20 uniform honycombs 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:
| Extended symmetry |
Extended diagram |
Order | Honeycombs |
|---|---|---|---|
| [31,1,3,31,1] | |
×1 | |
| [[31,1,3,31,1]] |   |
×2 | , |
| <[31,1,3,31,1]> = [31,1,3,3,4] |
  =  |
×2 | , , ,
|
| <<[31,1,3,31,1]>> = [4,3,3,3,4] |
   =  |
×4 |  , , , , , |
| [<<[31,1,3,1,1]>>] = [[4,3,3,3,4]] |
= |
×8 | , , |
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
- ↑ 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
- ↑ Conway (1998), p. 119
- ↑ http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/Ds5.html
- ↑ Conway (1998), p. 120
- ↑ Conway (1998), p. 466
- 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
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
- Conway JH, Sloane NJH (1998). Sphere Packings, Lattices and Groups (3rd ed.). ISBN 0-387-98585-9.
External links
- Olshevsky, George, Half measure polytope at Glossary for Hyperspace.
| Fundamental convex regular and uniform honeycombs in dimensions 2–11 | |||||
|---|---|---|---|---|---|
| Family |  |
 |
 |
 |
 /  /  |
| Uniform tiling | {3[3]} | δ3 | hδ3 | qδ3 | Hexagonal |
| Uniform convex honeycomb | {3[4]} | δ4 | hδ4 | qδ4 | |
| Uniform 5-honeycomb | {3[5]} | δ5 | hδ5 | qδ5 | 24-cell honeycomb |
| Uniform 6-honeycomb | {3[6]} | δ6 | hδ6 | qδ6 | |
| Uniform 7-honeycomb | {3[7]} | δ7 | hδ7 | qδ7 | 222 |
| Uniform 8-honeycomb | {3[8]} | δ8 | hδ8 | qδ8 | 133 • 331 |
| Uniform 9-honeycomb | {3[9]} | δ9 | hδ9 | qδ9 | 152 • 251 • 521 |
| Uniform n-honeycomb | {3[n]} | δn | hδn | qδn | 1k2 • 2k1 • k21 |