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Rectified 24-cell honeycomb

class="wikitable" align="right" style="margin-left:10px" width="320"

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

bgcolor=#ffffff align=center colspan=2|(No image)
bgcolor=#e7dcc3|TypeUniform 4-honeycomb
bgcolor=#e7dcc3|Schläfli symbolr{3,4,3,3}
rr{3,3,4,3}
r2r{4,3,3,4}
r2r{4,3,31,1}
bgcolor=#e7dcc3|Coxeter-Dynkin diagrams{{CDD|node|3|node_1|4|node|3|node|3|node}}

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

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

{{CDD|nodes_11|split2|node|3|node_1|4|node}} = {{CDD|node_h0|4|node_1|3|node|3|node_1|4|node}}

{{CDD|nodes_11|split2|node|split1|nodes_11}} = {{CDD|node|3|node_1|4|node_g|3sg|node_g|3g|node_g}}

{{CDD|nodes_11|split2|node|split1|nodes_11}} = {{CDD|node_h0|4|node_1|3|node|3|node_1|4|node_h0}}

bgcolor=#e7dcc3|4-face typeTesseract 40px
Rectified 24-cell 40px
bgcolor=#e7dcc3|Cell typeCube 20px
Cuboctahedron 20px
bgcolor=#e7dcc3|Face typeSquare
Triangle
bgcolor=#e7dcc3|Vertex figure80px
Tetrahedral prism
bgcolor=#e7dcc3|Coxeter groups{\tilde{F}}_4, [3,4,3,3]
{\tilde{C}}_4, [4,3,3,4]
{\tilde{B}}_4, [4,3,31,1]
{\tilde{D}}_4, [31,1,1,1]
bgcolor=#e7dcc3|PropertiesVertex transitive

In four-dimensional Euclidean geometry, the rectified 24-cell honeycomb is a uniform space-filling honeycomb. It is constructed by a rectification of the regular 24-cell honeycomb, containing tesseract and rectified 24-cell cells.

Alternate names

  • Rectified icositetrachoric tetracomb
  • Rectified icositetrachoric honeycomb
  • Cantellated 16-cell honeycomb
  • Bicantellated tesseractic honeycomb

Symmetry constructions

There are five different symmetry constructions of this tessellation. Each symmetry can be represented by different arrangements of colored rectified 24-cell and tesseract facets. The tetrahedral prism vertex figure contains 4 rectified 24-cells capped by two opposite tesseracts.

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!Coxeter group

!Coxeter
diagram

!Facets

!Vertex figure

!Vertex
figure
symmetry
(order)

align=center

|rowspan=2|{\tilde{F}}_4
= [3,4,3,3]

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

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

|80px

|{{CDD|node|3|node|3|node|2|node}}, [3,3,2]
(48)

align=center

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

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

|80px

|{{CDD|node|3|node|2|node}}, [3,2]
(12)

align=center

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

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

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

|80px

|{{CDD|node|2|node|2|node}}, [2,2]
(8)

align=center

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

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

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

|80px

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

align=center

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

|{{CDD|nodes_11|split2|node|split1|nodes_11}}

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

|80px

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

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 93
  • {{KlitzingPolytopes|flat.htm|4D|Euclidean tesselations}}, o3o3o4x3o, o4x3o3x4o - ricot - O93

{{Honeycombs}}

Category:5-polytopes

Category:Honeycombs (geometry)