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.

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 related D$$ lattice (also called D$$) can be constructed by the union of two D4 lattices, and is identical to the C4 lattice:

The kissing number for D$$ is 23 = 8, (2n – 1 for n < 8, 240 for n = 8, and 2n(n – 1) for n > 8).

The related D$$ lattice (also called D$$ and C$$) 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$$ 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.

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.