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Truncated 24-cell honeycomb

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!bgcolor=#e7dcc3 colspan=2|Truncated 24-cell honeycomb

bgcolor=#ffffff align=center colspan=2|(No image)
bgcolor=#e7dcc3|TypeUniform 4-honeycomb
bgcolor=#e7dcc3|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}
bgcolor=#e7dcc3|Coxeter-Dynkin diagrams{{CDD|node_1|3|node_1|4|node|3|node|3|node}}

{{CDD|node_1|3|node_1|3|node_1|4|node|3|node}}

{{CDD|node|4|node_1|3|node_1|3|node_1|4|node}}

{{CDD|nodes_11|split2|node_1|3|node_1|4|node}}

{{CDD|nodes_11|split2|node_1|split1|nodes_11}}

bgcolor=#e7dcc3|4-face typeTesseract 40px
Truncated 24-cell 40px
bgcolor=#e7dcc3|Cell typeCube 20px
Truncated octahedron 20px
bgcolor=#e7dcc3|Face typeSquare
Triangle
bgcolor=#e7dcc3|Vertex figure80px
Tetrahedral pyramid
bgcolor=#e7dcc3|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]
bgcolor=#e7dcc3|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

  • Truncated icositetrachoric tetracomb
  • Truncated icositetrachoric honeycomb
  • Cantitruncated 16-cell honeycomb
  • Bicantitruncated tesseractic honeycomb

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.

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!Coxeter group

!Coxeter
diagram

!Facets

!Vertex figure

!Vertex
figure
symmetry
(order)

align=center

|{\tilde{F}}_4
= [3,4,3,3]

|{{CDD|node_1|3|node_1|4|node|3|node|3|node}}

|4: {{CDD|node_1|3|node_1|4|node|3|node}}
1: {{CDD|node_1|4|node|3|node|3|node}}

|80px

|{{CDD|node|3|node|3|node}}, [3,3]
(24)

align=center

|{\tilde{F}}_4
= [3,3,4,3]

|{{CDD|node_1|3|node_1|3|node_1|4|node|3|node}}

|3: {{CDD|node|3|node|4|node_1|3|node_1}}
1: {{CDD|node|4|node_1|3|node_1|3|node_1}}
1: {{CDD|node|4|node|3|node_1|2|node_1}}

|80px

|{{CDD|node|3|node}}, [3]
(6)

align=center

|{\tilde{C}}_4
= [4,3,3,4]

|{{CDD|node|4|node_1|3|node_1|3|node_1|4|node}}

|2,2: {{CDD|node_1|3|node_1|3|node_1|4|node}}
1: {{CDD|node|4|node_1|2|node_1|4|node}}

|80px

|{{CDD|node|2|node}}, [2]
(4)

align=center

|{\tilde{B}}_4
= [31,1,3,4]

|{{CDD|nodes_11|split2|node_1|3|node_1|4|node}}

|1,1: {{CDD|node_1|3|node_1|3|node_1|4|node}}
2: {{CDD|nodes_11|split2|node_1|3|node_1}}
1: {{CDD|node_1|2|node_1|2|node_1|4|node}}

|80px

|{{CDD|node}}, [ ]
(2)

align=center

|{\tilde{D}}_4
= [31,1,1,1]

|{{CDD|nodes_11|split2|node_1|split1|nodes_11}}

|1,1,1,1:
{{CDD|nodes_11|split2|node_1|3|node_1}}
1: {{CDD|node_1|2|node_1|2|node_1|2|node_1}}

|80px

|[ ]+
(1)

See also

Regular and uniform honeycombs in 4-space:

References

  • Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, {{isbn|0-486-61480-8}} p. 296, Table II: Regular honeycombs
  • Kaleidoscopes: Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, {{isbn|978-0-471-01003-6}} [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html]
  • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
  • George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs) Model 99
  • {{KlitzingPolytopes|flat.htm|4D|Euclidean tesselations}} o4x3x3x4o, x3x3x *b3x4o, x3x3x *b3x *b3x, o3o3o4x3x, x3x3x4o3o - ticot - O99

{{Honeycombs}}

Category:5-polytopes

Category:Honeycombs (geometry)

Category:Truncated tilings