# 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.
TypeRegular 4-honeycomb
Uniform 4-honeycomb
FamilyAlternated hypercube honeycomb
Schläfli symbol{3,3,4,3}
Coxeter diagrams
=
=
4-face type{3,3,4}
Cell type{3,3}
Face type{3}
Edge figurecube
Vertex figure
24-cell
Coxeter group${\tilde {F}}_{4}$ = [3,3,4,3]
Dual{3,4,3,3}
Propertiesvertex-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.

## 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. The vertices of this lattice are the centers of the 3-spheres in the densest known packing of equal spheres in 4-space; its kissing number is 24, which is also the same as the kissing number in R4, as proved by Oleg Musin in 2003.

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:

= =

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

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.

= = .

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

## 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
${\tilde {F}}_{4}$ = [3,3,4,3]{3,3,4,3}
[3,4,3], order 1152
24: 16-cell
${\tilde {B}}_{4}$ = [31,1,3,4]= h{4,3,3,4} =
[3,3,4], order 384
16+8: 16-cell
${\tilde {D}}_{4}$ = [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${\tilde {C}}_{4}$ = [[(4,3,3,4,2<sup>+</sup>)]]ht0,4{4,3,3,4} 8+4+4: 4-demicube
8: 16-cell

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 ${\tilde {D}}_{5}$ 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:

## See also

Regular and uniform honeycombs in 4-space:

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