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.
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, (2"n" – 1 for "n" ; 8, 240 for "n" = 8, and 2"n"("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.
This honeycomb is one of 20 uniform honeycombs constructed by the formula_1 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:
Notes.
 ;templatestyles src="Reflist/styles.css" / ;