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 related D⁺
₄ lattice (also called D²
₄) can be constructed by the union of two D₄ lattices, and is identical to the C₄ lattice:^[6]

∪ = =

The kissing number for D⁺
₄ is 2³ = 8, (2^{n – 1} for n < 8, 240 for n = 8, and 2n(n – 1) for n > 8).^[7]

The related 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
${\displaystyle {\tilde {B}}_{4}}$ = [31,1,3,4]	= h{4,3,3,4}	=	[3,3,4], order 384	16+8: 16-cell
${\displaystyle {\tilde {D}}_{4}}$ = [31,1,1,1]	{3,31,1,1} = h{4,3,31,1}	=	[31,1,1], order 192	8+8+8: 16-cell
2×½${\displaystyle {\tilde {C}}_{4}}$ = [[(4,3,3,4,2+)]]	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
[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

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