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Truncated 24-cell honeycomb

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Truncated 24-cell honeycomb
(No image)
Type Uniform 4-honeycomb
Schläfli symbol t{3,4,3,3}
tr{3,3,4,3}
t2r{4,3,3,4}
t2r{4,3,3}
t{3}
Coxeter-Dynkin diagrams





4-face type Tesseract
Truncated 24-cell
Cell type Cube
Truncated octahedron
Face type Square
Triangle
Vertex figure
Tetrahedral pyramid
Coxeter groups F ~ 4 {\displaystyle {\tilde {F}}_{4}} ,
B ~ 4 {\displaystyle {\tilde {B}}_{4}} ,
C ~ 4 {\displaystyle {\tilde {C}}_{4}} ,
D ~ 4 {\displaystyle {\tilde {D}}_{4}} ,
Properties Vertex 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 D ~ 4 {\displaystyle {\tilde {D}}_{4}} construction. This truncated 24-cell has Schläfli symbol t{3}, and its snub is represented as s{3}.

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.

Coxeter group Coxeter
diagram
Facets Vertex figure Vertex
figure
symmetry
(order)
F ~ 4 {\displaystyle {\tilde {F}}_{4}}
=
4:
1:
,
(24)
F ~ 4 {\displaystyle {\tilde {F}}_{4}}
=
3:
1:
1:
,
(6)
C ~ 4 {\displaystyle {\tilde {C}}_{4}}
=
2,2:
1:
,
(4)
B ~ 4 {\displaystyle {\tilde {B}}_{4}}
=
1,1:
2:
1:
,
(2)
D ~ 4 {\displaystyle {\tilde {D}}_{4}}
=
1,1,1,1:

1:

(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
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III,
  • 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
  • Klitzing, Richard. "4D Euclidean tesselations". o4x3x3x4o, x3x3x *b3x4o, x3x3x *b3x *b3x, o3o3o4x3x, x3x3x4o3o - ticot - O99
Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space Family A ~ n 1 {\displaystyle {\tilde {A}}_{n-1}} C ~ n 1 {\displaystyle {\tilde {C}}_{n-1}} B ~ n 1 {\displaystyle {\tilde {B}}_{n-1}} D ~ n 1 {\displaystyle {\tilde {D}}_{n-1}} G ~ 2 {\displaystyle {\tilde {G}}_{2}} / F ~ 4 {\displaystyle {\tilde {F}}_{4}} / E ~ n 1 {\displaystyle {\tilde {E}}_{n-1}}
E Uniform tiling 0 δ3 3 3 Hexagonal
E Uniform convex honeycomb 0 δ4 4 4
E Uniform 4-honeycomb 0 δ5 5 5 24-cell honeycomb
E Uniform 5-honeycomb 0 δ6 6 6
E Uniform 6-honeycomb 0 δ7 7 7 222
E Uniform 7-honeycomb 0 δ8 8 8 133331
E Uniform 8-honeycomb 0 δ9 9 9 152251521
E Uniform 9-honeycomb 0 δ10 10 10
E Uniform 10-honeycomb 0 δ11 11 11
E Uniform (n-1)-honeycomb 0 δn n n 1k22k1k21
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