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	= =
Facets	{3,3,3,4} h{4,3,3,3}
Vertex figure	t1{3,3,3,4}
Coxeter group	${\displaystyle {\tilde {B}}_{5}}$ [4,3,3,31,1] ${\displaystyle {\tilde {D}}_{5}}$ [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 its Voronoi tessellation is a tritruncated 5-cubic honeycomb, , containing all 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
${\displaystyle {\tilde {B}}_{5}}$ = [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
${\displaystyle {\tilde {D}}_{5}}$ = [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
2×½${\displaystyle {\tilde {C}}_{5}}$ = [[(4,3,3,3,4,2+)]]	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
[31,1,3,31,1]		${\displaystyle {\tilde {D}}_{5}}$
<[31,1,3,31,1]> ↔ [31,1,3,3,4]	↔	${\displaystyle {\tilde {D}}_{5}}$×21 = ${\displaystyle {\tilde {B}}_{5}}$	, , , , , ,
[[31,1,3,31,1]]		${\displaystyle {\tilde {D}}_{5}}$×22	,
<2[31,1,3,31,1]> ↔ [4,3,3,3,4]	↔	${\displaystyle {\tilde {D}}_{5}}$×41 = ${\displaystyle {\tilde {C}}_{5}}$	, , , , ,
[<2[31,1,3,31,1]>] ↔ [[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

• 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	${\displaystyle {\tilde {A}}_{n-1}}$	${\displaystyle {\tilde {C}}_{n-1}}$	${\displaystyle {\tilde {B}}_{n-1}}$	${\displaystyle {\tilde {D}}_{n-1}}$	${\displaystyle {\tilde {G}}_{2}}$ / ${\displaystyle {\tilde {F}}_{4}}$ / ${\displaystyle {\tilde {E}}_{n-1}}$
E2	Uniform tiling	{3[3]}	δ3	hδ3	qδ3	Hexagonal
E3	Uniform convex honeycomb	{3[4]}	δ4	hδ4	qδ4
E4	Uniform 4-honeycomb	{3[5]}	δ5	hδ5	qδ5	24-cell honeycomb
E5	Uniform 5-honeycomb	{3[6]}	δ6	hδ6	qδ6
E6	Uniform 6-honeycomb	{3[7]}	δ7	hδ7	qδ7	222
E7	Uniform 7-honeycomb	{3[8]}	δ8	hδ8	qδ8	133 • 331
E8	Uniform 8-honeycomb	{3[9]}	δ9	hδ9	qδ9	152 • 251 • 521
E9	Uniform 9-honeycomb	{3[10]}	δ10	hδ10	qδ10
E10	Uniform 10-honeycomb	{3[11]}	δ11	hδ11	qδ11
En-1	Uniform (n-1)-honeycomb	{3[n]}	δn	hδn	qδn	1k2 • 2k1 • k21