Birectified 16-cell honeycomb

Birectified 16-cell honeycomb
(No image)
TypeUniform honeycomb
Schläfli symbolt2{3,3,4,3}
Coxeter-Dynkin diagram
=
4-face typeRectified tesseract
Rectified 24-cell
Cell typeCube
Cuboctahedron
Tetrahedron
Face type{3}, {4}
Vertex figure
{3}×{3} duoprism
Coxeter group{\tilde{F}}_4 = [3,3,4,3]
{\tilde{B}}_4 = [4,3,31,1]
{\tilde{D}}_4 = [31,1,1,1]
Dual?
Propertiesvertex-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
[4,3,31,1]		 {\tilde{D}}_4
[31,1,1,1]
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 [4,3,31,1] families. The alternation (13) is also repeated in other families.

The [4,3,31,1], , 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.

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), [31,1,3,4,1+] (index 2) are all isomorphic to [31,1,1,1].

See also

Regular and uniform honeycombs in 4-space:

Notes

    References

    Fundamental convex regular and uniform honeycombs in dimensions 2–10
    Family {\tilde{A}}_{n-1} {\tilde{C}}_{n-1} {\tilde{B}}_{n-1} {\tilde{D}}_{n-1} {\tilde{G}}_2 / {\tilde{F}}_4 / {\tilde{E}}_{n-1}
    Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal
    Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4
    Uniform 5-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb
    Uniform 6-honeycomb {3[6]} δ6 hδ6 qδ6
    Uniform 7-honeycomb {3[7]} δ7 hδ7 qδ7 222
    Uniform 8-honeycomb {3[8]} δ8 hδ8 qδ8 133 • 331
    Uniform 9-honeycomb {3[9]} δ9 hδ9 qδ9 152 • 251 • 521
    Uniform n-honeycomb {3[n]} δn hδn qδn 1k2 • 2k1 • k21
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