16-cell honeycomb
In four-dimensional Euclidean geometry, the 16-cell honeycomb is one of the three regular space-filling tessellations, represented by Schläfli symbol, 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
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 lattice 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,.
The D lattice 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 is 24 and its Voronoi tessellation is a 24-cell honeycomb,, containing all rectified 16-cells 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 |
| = ] | , order 1152 | 24: 16-cell | ||
| = ] | = h | = | , order 384 | 16+8: 16-cell |
| = ] | = h | = | , order 192 | 8+8+8: 16-cell |
| 2×½ = | ht0,4 | 8+4+4: 4-demicube 8: 16-cell |
Related honeycombs
It is related to the regular hyperbolic 5-space 5-orthoplex honeycomb,, with 5-orthoplex facets, the regular 4-polytope 24-cell, with octahedral cell, and cube, with square faces.It has a 2-dimensional analogue,, and as an alternated form it is related to the alternated cubic honeycomb.