5-demicubic honeycomb

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.

Demipenteractic honeycomb
(No image)
Type	Uniform 5-honeycomb
Family	Alternated hypercubic honeycomb
Schläfli symbols	h{4,3,3,3,4} h{4,3,3,3^1,1} ht0,5{4,3,3,3,4} h{4,3,3,4}h{∞} h{4,3,3^1,1}h{∞} ht0,4{4,3,3,4}h{∞} h{4,3,4}h{∞}h{∞} h{4,3^1,1}h{∞}h{∞}
Coxeter diagrams	= =
Facets	{3,3,3,4} h{4,3,3,3}
Vertex figure	t1{3,3,3,4}
Coxeter group	${\displaystyle {\tilde {B}}_{5}}$ [4,3,3,3^1,1] ${\displaystyle {\tilde {D}}_{5}}$ [3^1,1,3,3^1,1]

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 different colors on the 32 5-demicube facets around each vertex.

Coxeter group	Schläfli symbol	Coxeter-Dynkin diagram	Vertex figure Symmetry	Facets/verf
${\displaystyle {\tilde {B}}_{5}}$ = [3^1,1,3,3,4] = [1^+,4,3,3,4]	h{4,3,3,3,4}	=	[3,3,3,4]	32: 5-demicube 10: 5-orthoplex
${\displaystyle {\tilde {D}}_{5}}$ = [3^1,1,3,3^1,1] = [1^+,4,3,3^1,1]	h{4,3,3,3^1,1}	=	[3^2,1,1]	16+16: 5-demicube 10: 5-orthoplex
2×½${\displaystyle {\tilde {C}}_{5}}$ = [[(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 ${\displaystyle {\tilde {D}}_{5}}$ 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
[3^1,1,3,3^1,1]		${\displaystyle {\tilde {D}}_{5}}$
<[3^1,1,3,3^1,1]> ↔ [3^1,1,3,3,4]	↔	${\displaystyle {\tilde {D}}_{5}}$×21 = ${\displaystyle {\tilde {B}}_{5}}$	, , , , , ,
[[3^1,1,3,3^1,1]]		${\displaystyle {\tilde {D}}_{5}}$×22	,
<2[3^1,1,3,3^1,1]> ↔ [4,3,3,3,4]	↔	${\displaystyle {\tilde {D}}_{5}}$×41 = ${\displaystyle {\tilde {C}}_{5}}$	, , , , ,
[<2[3^1,1,3,3^1,1]>] ↔ [[4,3,3,3,4]]	↔	${\displaystyle {\tilde {D}}_{5}}$×8 = ${\displaystyle {\tilde {C}}_{5}}$×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

1. http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/D5.html
2. Sphere packings, lattices, and groups, by John Horton Conway, Neil James Alexander Sloane, Eiichi Bannai
3. Conway (1998), p. 119
4. http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/Ds5.html
5. Conway (1998), p. 120
6. 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}={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

Fundamental convex regular and uniform honeycombs in dimensions 2-9
Space	Family	${\displaystyle {\tilde {A}}_{n-1}}$	${\displaystyle {\tilde {C}}_{n-1}}$	${\displaystyle {\tilde {B}}_{n-1}}$	${\displaystyle {\tilde {D}}_{n-1}}$	${\displaystyle {\tilde {G}}_{2}}$ / ${\displaystyle {\tilde {F}}_{4}}$ / ${\displaystyle {\tilde {E}}_{n-1}}$
E^2	Uniform tiling	{3^[3]}	δ3	hδ3	qδ3	Hexagonal
E^3	Uniform convex honeycomb	{3^[4]}	δ4	hδ4	qδ4
E^4	Uniform 4-honeycomb	{3^[5]}	δ5	hδ5	qδ5	24-cell honeycomb
E^5	Uniform 5-honeycomb	{3^[6]}	δ6	hδ6	qδ6
E^6	Uniform 6-honeycomb	{3^[7]}	δ7	hδ7	qδ7	222
E^7	Uniform 7-honeycomb	{3^[8]}	δ8	hδ8	qδ8	133 • 331
E^8	Uniform 8-honeycomb	{3^[9]}	δ9	hδ9	qδ9	152 • 251 • 521
E^9	Uniform 9-honeycomb	{3^[10]}	δ10	hδ10	qδ10
E^n-1	Uniform (n-1)-honeycomb	{3^[n]}	δn	hδn	qδn	1k2 • 2k1 • k21