Snub 24-cell honeycomb

Snub 24-cell honeycomb
(No image)
TypeUniform 4-honeycomb
Schläfli symbolss{3,4,3,3}
sr{3,3,4,3}
2sr{4,3,3,4}
2sr{4,3,31,1}
s{31,1,1,1}
Coxeter diagrams





=

4-face typesnub 24-cell
16-cell
5-cell
Cell type{3,3}
{3,5}
Face typetriangle {3}
Vertex figure
Irregular decachoron
Symmetries[3+,4,3,3]
[3,4,(3,3)+]
[4,(3,3)+,4]
[4,(3,31,1)+]
[31,1,1,1]+
PropertiesVertex transitive, nonWythoffian

In four-dimensional Euclidean geometry, the snub 24-cell honeycomb, or snub icositetrachoric honeycomb is a uniform space-filling tessellation (or honeycomb) by snub 24-cells, 16-cells, and 5-cells. It was discovered by Thorold Gosset with his 1900 paper of semiregular polytopes. It is not semiregular by Gosset's definition of regular facets, but all of its cells (ridges) are regular, either tetrahedra or icosahedra.

It can be seen as an alternation of a truncated 24-cell honeycomb, and can be represented by Schläfli symbol s{3,4,3,3}, s{31,1,1,1}, and 3 other snub constructions.

It is defined by an irregular decachoron vertex figure (10-celled 4-polytope), faceted by four snub 24-cells, one 16-cell, and five 5-cells. The vertex figure can be seen topologically as a modified tetrahedral prism, where one of the tetrahedra is subdivided at mid-edges into a central octahedron and four corner tetrahedra. Then the four side-facets of the prism, the triangular prisms become tridiminished icosahedra.

Symmetry constructions

There are five different symmetry constructions of this tessellation. Each symmetry can be represented by different arrangements of colored snub 24-cell, 16-cell, and 5-cell facets. In all cases, four snub 24-cells, five 5-cells, and one 16-cell meet at each vertex, but the vertex figures have different symmetry generators.

Symmetry Coxeter
Schläfli		 Facets (on vertex figure)
Snub 24-cell
(4)		 16-cell
(1)		 5-cell
(5)
[3+,4,3,3]
s{3,4,3,3}		 4:
[3,4,(3,3)+]
sr{3,3,4,3}		 3:
1:
[[4,(3,3)+,4]]
2sr{4,3,3,4}		 2,2:
[(31,1,3)+,4]
2sr{4,3,31,1}		 1,1:
2:
[31,1,1,1]+
s{31,1,1,1}		 1,1,1,1:

See also

Regular and uniform honeycombs in 4-space:

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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