| bgcolor=#e7dcc3 colspan=2 | 16-cell honeycomb | |
|---|---|---|
| bgcolor=#ffffff align=center colspan=2 | Perspective projection the first layer of adjacent 16-cell facets. | |
| Type | Regular 4-honeycomb Uniform 4-honeycomb | |
| Family | Alternated hypercube honeycomb | |
| Schläfli symbol | ||
| Coxeter diagrams | = = | |
| 4-face type | ||
| Cell type | ||
| Face type | ||
| Edge figure | cube | |
| Vertex figure | 24-cell | |
| Coxeter group | {\tilde{F}}4 | |
| Dual | ||
| 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]
Vertices can be placed at all integer coordinates (i,j,k,l), such that the sum of the coordinates is even.
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 lattice (also called D) can be constructed by the union of two D4 lattices, and is identical to the C4 lattice:[6]
∪ = =
The kissing number for D is 23 = 8, (2n – 1 for n < 8, 240 for n = 8, and 2n(n – 1) for n > 8).[7]
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.[8]
∪ ∪ ∪ = = ∪ .
The kissing number of the D 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]
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 | [3,4,3], order 1152 | 24: 16-cell | |||
{\tilde{B}}4 | = h | = | [3,3,4], order 384 | 16+8: 16-cell | |
{\tilde{D}}4 | = h | = | [3<sup>1,1,1</sup>], order 192 | 8+8+8: 16-cell | |
| 2×½ {\tilde{C}}4 | ht0,4 | 8+4+4: 4-demicube 8: 16-cell |
It is related to the regular hyperbolic 5-space 5-orthoplex honeycomb,, with 5-orthoplex facets, the regular 4-polytope 24-cell, with octahedral (3-orthoplex) cell, and cube, with (2-orthoplex) square faces.
It has a 2-dimensional analogue, , and as an alternated form (the demitesseractic honeycomb, h) it is related to the alternated cubic honeycomb.
Regular and uniform honeycombs in 4-space: