16-cell honeycomb

In four-dimensional Euclidean geometry, the 16-cell honeycomb is one of the three regular space-filling tessellations (or honeycombs), represented by Schläfli symbol {3,3,4,3}, and constructed by a 4-dimensional packing of 16-cell facets, three around every face.

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

Its dual is the 24-cell honeycomb. Its vertex figure is a 24-cell. The vertex arrangement is called the B₄, D₄, or F₄ lattice.[1][2]

Alternate names

• Hexadecachoric tetracomb/honeycomb
• Demitesseractic tetracomb/honeycomb

Coordinates

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.

Coxeter group	Schläfli symbol	Coxeter diagram	Vertex figure Symmetry	Facets/verf
${\displaystyle {\tilde {F}}_{4}}$ = [3,3,4,3]	{3,3,4,3}		[3,4,3], order 1152	24: 16-cell
${\displaystyle {\tilde {B}}_{4}}$ = [3^1,1,3,4]	= h{4,3,3,4}	=	[3,3,4], order 384	16+8: 16-cell
${\displaystyle {\tilde {D}}_{4}}$ = [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
2×½${\displaystyle {\tilde {C}}_{4}}$ = [[(4,3,3,4,2<sup>+</sup>)]]	ht0,4{4,3,3,4}			8+4+4: 4-demicube 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.

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.

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

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. Conway and Sloane, Sphere packings, lattices, and groups, 1.4 n-dimensional packings, p.9
4. Conway and Sloane, Sphere packings, lattices, and groups, 1.5 Sphere packing problem summary of results. , p.12
5. O. R. Musin (2003). "The problem of the twenty-five spheres". Russ. Math. Surv. 58 (4): 794–795. Bibcode:2003RuMaS..58..794M. doi:10.1070/RM2003v058n04ABEH000651.
6. Conway and Sloane, Sphere packings, lattices, and groups, 7.3 The packing D₃⁺, p.119
7. Conway and Sloane, Sphere packings, lattices, and groups, p. 119
8. Conway and Sloane, Sphere packings, lattices, and groups, 7.4 The dual lattice D₃^*, p.120
9. Conway and Sloane, Sphere packings, lattices, and groups, p. 120
10. 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 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