5-demicubic honeycomb

Demipenteractic honeycomb
(No image)
Type	uniform honeycomb
Family	Alternated hypercubic honeycomb
Schläfli symbol	h{4,3,3,3,4}
Coxeter diagram	or or
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 (2^n-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]	= 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]	= h{4,3,3,31,1}	=	[32,1,1]	16+16: 5-demicube 10: 5-orthoplex
= [[(4,3,3,3,4,2<sup>+</sup>)]]	ht0,5{4,3,3,3,4}			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]
<[31,1,3,31,1]> ↔ [31,1,3,3,4]	↔	×21 =	, , , , , ,
[[31,1,3,31,1]]		×22	,
<2[31,1,3,31,1]> ↔ [4,3,3,3,4]	↔	×41 =	, , , , ,
[<2[31,1,3,31,1]>] ↔ [[4,3,3,3,4]]	↔	×8 = ×2	, ,

See also

• Uniform polytope

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, 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}={3^1,1,4}, 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". Glossary for Hyperspace. Archived from the original on 4 February 2007. 

Fundamental convex regular and uniform honeycombs in dimensions 3–10 (or 2-9)
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 10-honeycomb	{3[10]}	δ10	hδ10	qδ10
Uniform n-honeycomb	{3[n]}	δn	hδn	qδn	1k2 • 2k1 • k21