# 16-cell honeycomb

16-cell honeycomb :
| Perspective projection: the first layer of adjacent 16-cell facets. ||
| Type | Regular 4-honeycomb Uniform 4-honeycomb |
| Family | Alternated hypercube honeycomb |
| Schläfli symbol | {3,3,4,3} |
| Coxeter diagrams | = = |
| 4-face type | {3,3,4} |
| Cell type | {3,3} |
| Face type | {3} |
| Edge figure | cube |
| Vertex figure | 24-cell |
| Coxeter group | ${\tilde {F}}_{4}$ = [3,3,4,3] |
| Dual | {3,4,3,3} |
| Properties | vertex-transitive, edge-transitive, face-transitive, cell-transitive, 4-face-transitive |
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 B$_4$, D$_4$, or F$_4$ 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.

## D$_4$ lattice

The vertex arrangement of the 16-cell honeycomb is called the D$_4$ lattice or F$_4$ 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 R$^4$, as proved by Oleg Musin in 2003.
The related D$^+$ $_4$ lattice (also called D$^2$ $_4$) can be constructed by the union of two D$_4$ lattices, and is identical to the C$_4$ lattice:
The kissing number for D$^+$ $_4$ is 2$^3$ = 8, (2$^{n–1}$ for n < 8, 240 for n = 8, and 2n(n – 1) for n > 8).
The related D$^*$ $_4$ lattice (also called D$^4$ $_4$ and C$^2$ $_4$) can be constructed by the union of all four D$_4$ lattices, but it is identical to the D$_4$ 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 D$_4$ 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} | | [3,4,3], order 1152 | 24: 16-cell |
| ${\tilde {B}}_{4}$ = [3$^{1,1}$,3,4] | = h{4,3,3,4} | = | [3,3,4], order 384 | 16+8: 16-cell |
| ${\tilde {D}}_{4}$ = [3$^{1,1,1,1}$] | {3,3$^{1,1,1}$} = h{4,3,3$^{1,1}$} | = | [3$^{1,1,1}$], order 192 | 8+8+8: 16-cell |
| 2×½${\tilde {C}}_{4}$ = [[(4,3,3,4,2$^+$)]] | ht$_{0,4}${4,3,3,4} | | | 8+4+4: 4-demicube 8: 16-cell |

## 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 ${\tilde {D}}_{5}$ 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:
D5 honeycombs :
| Extended symmetry | Extended diagram | Extended group | Honeycombs |
| [3$^{1,1}$,3,3$^{1,1}$] | | ${\tilde {D}}_{5}$ | |
| <[3$^{1,1}$,3,3$^{1,1}$]> ↔ [3$^{1,1}$,3,3,4] | ↔ | ${\tilde {D}}_{5}$×2$_1$ = ${\tilde {B}}_{5}$ | , , , , , , |
| [[3$^{1,1}$,3,3$^{1,1}$]] | | ${\tilde {D}}_{5}$×2$_2$ | , |
| <2[3$^{1,1}$,3,3$^{1,1}$]> ↔ [4,3,3,3,4] | ↔ | ${\tilde {D}}_{5}$×4$_1$ = ${\tilde {C}}_{5}$ | , , , , , |
| [<2[3$^{1,1}$,3,3$^{1,1}$]>] ↔ [[4,3,3,3,4]] | ↔ | ${\tilde {D}}_{5}$×8 = ${\tilde {C}}_{5}$×2 | , , |