Truncated 24-cell honeycomb

Truncated 24-cell honeycomb
(No image)
TypeUniform 4-honeycomb
Schläfli symbolt{3,4,3,3}
tr{3,3,4,3}
t2r{4,3,3,4}
t2r{4,3,31,1}
t{31,1,1,1}
Coxeter-Dynkin diagrams





4-face typeTesseract
Truncated 24-cell
Cell typeCube
Truncated octahedron
Face typeSquare
Triangle
Vertex figure
Tetrahedral pyramid
Coxeter groups{\tilde{F}}_4, [3,4,3,3]
{\tilde{B}}_4, [4,3,31,1]
{\tilde{C}}_4, [4,3,3,4]
{\tilde{D}}_4, [31,1,1,1]
PropertiesVertex transitive

In four-dimensional Euclidean geometry, the truncated 24-cell honeycomb is a uniform space-filling honeycomb. It can be seen as a truncation of the regular 24-cell honeycomb, containing tesseract and truncated 24-cell cells.

It has a uniform alternation, called the snub 24-cell honeycomb. It is a snub from the {\tilde{D}}_4 construction. This truncated 24-cell has Schläfli symbol t{31,1,1,1}, and its snub is represented as s{31,1,1,1}.

Alternate names

Symmetry constructions

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

Coxeter group Coxeter
diagram		 Facets Vertex figure Vertex
figure
symmetry
(order)
{\tilde{F}}_4
= [3,4,3,3]		 4:
1: 		, [3,3]
(24)
{\tilde{F}}_4
= [3,3,4,3]		 3:
1:
1: 		, [3]
(6)
{\tilde{C}}_4
= [4,3,3,4]		 2,2:
1: 		, [2]
(4)
{\tilde{B}}_4
= [31,1,3,4]		 1,1:
2:
1: 		, [ ]
(2)
{\tilde{D}}_4
= [31,1,1,1]		 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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