16-cell honeycomb

16-cell honeycomb
Perspective projection: the first layer of adjacent 16-cell facets.
Type	Regular 4-space honeycomb Uniform 4-honeycomb
Family	Alternated hypercube honeycomb
Schläfli symbol	{3,3,4,3}
Coxeter-Dynkin diagram
4-face type	{3,3,4}
Cell type	{3,3}
Face type	{3}
Edge figure	cube
Vertex figure	24-cell
Coxeter group	${\tilde {F}}_{4}$ = [3,3,4,3]
Dual	{3,4,3,3}
Properties	vertex-transitive, edge-transitive, face-transitive, cell-transitive, 4-face-transitive

In four-dimensional Euclidean geometry, the 16-cell honeycomb is one of the three regular space-filling tessellations (or honeycombs) in Euclidean 4-space. The other two are its dual the 24-cell honeycomb, and the tesseractic honeycomb. This honeycomb is constructed from 16-cell facets, three around every face. It has a 24-cell vertex figure.

This vertex arrangement or lattice is called the B₄, D₄, or F₄ lattice.^[1]^[2]

Alternate names

Hexadecachoric tetracomb/honeycomb
Demitesseractic tetracomb/honeycomb

Coordinates

As a regular honeycomb, {3,3,4,3}, it has a 2-dimensional analogue, {3,6}, and as an alternated form (the demitesseractic honeycomb, h{4,3,3,4}) it is related to the alternated cubic honeycomb.

Vertices can be placed at all integer coordinates (i,j,k,l), such that the sum of the coordinates is even.

D₄ lattice

The vertex arrangement of the 16-cell honeycomb is called the D₄ lattice or F₄ lattice.^[2] The vertices of this lattice are the centers of the 3-spheres in the densest known packing of equal spheres in 4-space;^[3] its kissing number is 24, which is also the same as the kissing number in R⁴, as proved by Oleg Musin in 2003.^[4]^[5]

The D+
4 lattice (also called D2
4) can be constructed by the union of two D₄ lattices, and is identical to the tesseractic honeycomb:^[6]

∪

This packing is only a lattice for even dimensions. The kissing number is 2³ = 8, (2^{n – 1} for n < 8, 240 for n = 8, and 2n(n – 1) for n > 8).^[7]

The D*
4 lattice (also called D4
4 and C2
4) can be constructed by the union of all four D₄ lattices, but it is identical to the D₄ lattice: It is also the 4-dimensional body centered cubic, the union of two 4-cube honeycombs in dual positions.^[8]

∪

The kissing number of the D*
4 lattice (and D₄ lattice) is 24^[9] and its Voronoi tessellation is a 24-cell honeycomb, , containing all rectified 16-cells (24-cell) Voronoi cells, or .^[10]

Symmetry constructions

There are three different symmetry constructions of this tessellation. Each symmetry can be represented by different arrangements of colored 16-cell facets.

Name	Coxeter group	Schläfli symbol	Coxeter diagram	Vertex figure Symmetry	Facets/verf
16-cell honeycomb	${\tilde {F}}_{4}$ = [3,3,4,3]	{3,3,4,3}		[3,4,3], order 1152	24: 16-cell
4-demicube honeycomb	${\tilde {B}}_{4}$ = [3^1,1,3,4]	= h{4,3,3,4}	=	[3,3,4], order 384	16+8: 16-cell
	${\tilde {D}}_{4}$ = [3^1,1,1,1]	{3,3^1,1,1} = h{4,3,3^1,1}	=	[3^1,1,1], order 192	8+8+8: 16-cell

Related honeycombs

It is related to the regular hyperbolic 5-space 5-orthoplex honeycomb, {3,3,3,4,3}, with 5-orthoplex facets, the regular 4-polytope 24-cell, {3,4,3} with octahedral (3-orthoplex) cell, and cube {4,3}, with (2-orthoplex) square faces.

This honeycomb is one of 20 uniform honeycombs constructed by the ${\tilde {D}}_{5}$ 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]		${\tilde {D}}_{5}$
<[3^1,1,3,3^1,1]> ↔ [3^1,1,3,3,4]	↔	${\tilde {D}}_{5}$ ×2₁ = ${{\tilde {B}}}_{5}$	, , , , , ,
[[3^1,1,3,3^1,1]]		${\tilde {D}}_{5}$ ×2₂	,
<2[3^1,1,3,3^1,1]> ↔ [4,3,3,3,4]	↔	${\tilde {D}}_{5}$ ×4₁ = ${{\tilde {C}}}_{5}$	, , , , ,
[<2[3^1,1,3,3^1,1]>] ↔ [[4,3,3,3,4]]	↔	${\tilde {D}}_{5}$ ×8 = ${{\tilde {C}}}_{5}$ ×2	, ,

Notes

↑ http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/F4.html
1 2 http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/D4.html
↑ Conway and Sloane, Sphere packings, lattices, and groups, 1.4 n-dimensional packings, p.9
↑ Conway and Sloane, Sphere packings, lattices, and groups, 1.5 Sphere packing problem summary of results. , p.12
↑ O. R. Musin (2003). "The problem of the twenty-five spheres". Russ. Math. Surv. 58: 794–795. doi:10.1070/RM2003v058n04ABEH000651.
↑ Conway and Sloane, Sphere packings, lattices, and groups, 7.3 The packing D₃⁺, p.119
↑ Conway and Sloane, Sphere packings, lattices, and groups, p. 119
↑ Conway and Sloane, Sphere packings, lattices, and groups, 7.4 The dual lattice D₃^*, p.120
↑ Conway and Sloane, Sphere packings, lattices, and groups, p. 120
↑ Conway and Sloane, Sphere packings, lattices, and groups, p. 466

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]
George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)
Klitzing, Richard. "4D Euclidean tesselations". x3o3o4o3o - hext - O104
Conway JH, Sloane NJH (1998). Sphere Packings, Lattices and Groups (3rd ed.). ISBN 0-387-98585-9.

Fundamental convex regular and uniform honeycombs in dimensions 3–10 (or 2-9)
Family	${\tilde {A}}_{n-1}$	${\tilde {C}}_{n-1}$	${\tilde {B}}_{n-1}$	${\tilde {D}}_{n-1}$	${\tilde {G}}_{2}$ / ${\tilde {F}}_{4}$ / ${\tilde {E}}_{n-1}$
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]}	δ_n	hδ_n	qδ_n	1_k2 • 2_k1 • k₂₁

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