# Birectified 16-cell honeycomb

Birectified 16-cell honeycomb :
| (No image) ||
| Type | Uniform honeycomb |
| Schläfli symbol | t$_2${3,3,4,3} |
| Coxeter-Dynkin diagram | = |
| 4-face type | Rectified tesseract Rectified 24-cell |
| Cell type | Cube Cuboctahedron Tetrahedron |
| Face type | {3}, {4} |
| Vertex figure | {3}×{3} duoprism |
| Coxeter group | ${\tilde {F}}_{4}$ = [3,3,4,3] ${\tilde {B}}_{4}$ = [4,3,3$^{1,1}$] ${\tilde {D}}_{4}$ = [3$^{1,1,1,1}$] |
| Dual | ? |
| Properties | vertex-transitive |
In four-dimensional Euclidean geometry, the birectified 16-cell honeycomb (or runcic tesseractic honeycomb) is a uniform space-filling tessellation (or honeycomb) in Euclidean 4-space.

## Symmetry constructions

There are 3 different symmetry constructions, all with 3-3 duoprism vertex figures. The ${\tilde {B}}_{4}$ symmetry doubles on ${\tilde {D}}_{4}$ in three possible ways, while ${\tilde {F}}_{4}$ contains the highest symmetry.
| Affine Coxeter group | ${\tilde {F}}_{4}$ [3,3,4,3] | ${\tilde {B}}_{4}$ [4,3,3$^{1,1}$] | ${\tilde {D}}_{4}$ [3$^{1,1,1,1}$] |
| Coxeter diagram | | | |
| Vertex figure | | | |
| Vertex figure symmetry | [3,2,3] (order 36) | [3,2] (order 12) | [3] (order 6) |
| 4-faces | | | |
| Cells | | | |

## Related honeycombs

The [3,4,3,3], , Coxeter group generates 31 permutations of uniform tessellations, 28 are unique in this family and ten are shared in the [4,3,3,4] and [4,3,3$^{1,1}$] families. The alternation (13) is also repeated in other families.
F4 honeycombs :
| Extended symmetry | Extended diagram | Order | Honeycombs |
| [3,3,4,3] | | ×1 | $_1$, $_3$, $_5$, $_6$, $_8$, $_9$, $_{10}$, $_{11}$, $_{12}$ |
| [3,4,3,3] | | ×1 | $_2$, $_4$, $_7$, $_{13}$, $_{14}$, $_{15}$, $_{16}$, $_{17}$, $_{18}$, $_{19}$, $_{20}$, $_{21}$, $_{22}$ $_{23}$, $_{24}$, $_{25}$, $_{26}$, $_{27}$, $_{28}$, $_{29}$ |
| [(3,3)[3,3,4,3$^*$]] =[(3,3)[3$^{1,1,1,1}$]] =[3,4,3,3] | = = | ×4 | $_{(2)}$, $_{(4)}$, $_{(7)}$, $_{(13)}$ |
The [4,3,3$^{1,1}$], , Coxeter group generates 31 permutations of uniform tessellations, 23 with distinct symmetry and 4 with distinct geometry. There are two alternated forms: the alternations (19) and (24) have the same geometry as the 16-cell honeycomb and snub 24-cell honeycomb respectively.
B4 honeycombs :
| Extended symmetry | Extended diagram | Order | Honeycombs |
| [4,3,3$^{1,1}$]: | | ×1 | $_5$, $_6$, $_7$, $_8$ |
| <[4,3,3$^{1,1}$]>: ↔[4,3,3,4] | ↔ | ×2 | $_9$, $_{10}$, $_{11}$, $_{12}$, $_{13}$, $_{14}$, $_{(10)}$, $_{15}$, $_{16}$, $_{(13)}$, $_{17}$, $_{18}$, $_{19}$ |
| [3[1$^+$,4,3,3$^{1,1}$]] ↔ [3[3,3$^{1,1,1}$]] ↔ [3,3,4,3] | ↔ ↔ | ×3 | $_1$, $_2$, $_3$, $_4$ |
| [(3,3)[1$^+$,4,3,3$^{1,1}$]] ↔ [(3,3)[3$^{1,1,1,1}$]] ↔ [3,4,3,3] | ↔ ↔ | ×12 | $_{20}$, $_{21}$, $_{22}$, $_{23}$ |
There are ten uniform honeycombs constructed by the ${\tilde {D}}_{4}$ Coxeter group, all repeated in other families by extended symmetry, seen in the graph symmetry of rings in the Coxeter–Dynkin diagrams. The 10th is constructed as an alternation. As subgroups in Coxeter notation: [3,4,(3,3)$^*$] (index 24), [3,3,4,3$^*$] (index 6), [1$^+$,4,3,3,4,1$^+$] (index 4), [3$^{1,1}$,3,4,1$^+$] (index 2) are all isomorphic to [3$^{1,1,1,1}$].
The ten permutations are listed with its highest extended symmetry relation:
D4 honeycombs :
| Extended symmetry | Extended diagram | Extended group | Honeycombs |
| [3$^{1,1,1,1}$] | | ${\tilde {D}}_{4}$ | (none) |
| <[3$^{1,1,1,1}$]> ↔ [3$^{1,1}$,3,4] | ↔ | ${\tilde {D}}_{4}$×2 = ${\tilde {B}}_{4}$ | (none) |
| <2[$^{1,1}$3$^{1,1}$]> ↔ [4,3,3,4] | ↔ | ${\tilde {D}}_{4}$×4 = ${\tilde {C}}_{4}$ | $_1$, $_2$ |
| [3[3,3$^{1,1,1}$]] ↔ [3,3,4,3] | ↔ | ${\tilde {D}}_{4}$×6 = ${\tilde {F}}_{4}$ | $_3$, $_4$, $_5$, $_6$ |
| [4[$^{1,1}$3$^{1,1}$]] ↔ [[4,3,3,4]] | ↔ | ${\tilde {D}}_{4}$×8 = ${\tilde {C}}_{4}$×2 | $_7$, $_8$, $_9$ |
| [(3,3)[3$^{1,1,1,1}$]] ↔ [3,4,3,3] | ↔ | ${\tilde {D}}_{4}$×24 = ${\tilde {F}}_{4}$ | $_7$, $_8$, $_9$ |
| [(3,3)[3$^{1,1,1,1}$]]$^+$ ↔ [3$^+$,4,3,3] | ↔ | ½${\tilde {D}}_{4}$×24 = ½${\tilde {F}}_{4}$ | $_{10}$ |