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 the one of three regular space-filling tessellation (or honeycomb) 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

Its vertex arrangement is called the D₄ lattice or F₄ lattice.^[2] The vertices of this lattice are the centers of the 3-spheres in the densest possible packing of equal spheres in 4-space; its kissing number is 24, which is also the highest possible in 4-space.^[3]

=

The D+
4 lattice (also called D2
4) can be constructed by the union of two 4-demicubic lattices, and is identical to the tesseractic honeycomb:

∪ = =

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).^[4]

The D*
4 lattice (also called D4
4 and C2
4) can be constructed by the union of all four 5-demicubic honeycombs, 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.

∪ ∪ ∪ = = ∪ .

The kissing number of the D*
4 lattice (and D₄ lattice) is 24^[5] and its Voronoi tessellation is a 24-cell honeycomb, , containing all rectified 16-cells (24-cell) Voronoi cells, or .^[6]

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	, ,

See also

Regular and uniform honeycombs in 4-space:

• Tesseractic honeycomb
• 24-cell honeycomb
• Truncated 24-cell honeycomb
• Snub 24-cell honeycomb
• 5-cell honeycomb
• Truncated 5-cell honeycomb
• Omnitruncated 5-cell honeycomb

Notes

1. http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/F4.html
2. http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/D4.html
3. O. R. Musin (2003). "The problem of the twenty-five spheres". Russ. Math. Surv. 58: 794–795. doi:10.1070/RM2003v058n04ABEH000651.
4. Conway (1998), p. 119
5. Conway (1998), p. 120
6. Conway (1998), 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)
• Richard Klitzing, 4D, Euclidean tesselations x3o3o4o3o - hext - O104
• Conway JH, Sloane NJH (1998). Sphere Packings, Lattices and Groups (3rd ed.). ISBN 0-387-98585-9.