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Truncated 24-cell honeycomb

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{4,3,31,1}
t{31,1,1,1}
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 , [3,4,3,3]
, [4,3,31,1]
, [4,3,3,4]
, [31,1,1,1]
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 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.

Coxeter group Coxeter
diagram 		Facets Vertex figure Vertex
figure
symmetry
(order)

= [3,4,3,3] 		4:
1: 		, [3,3]
(24)

= [3,3,4,3] 		3:
1:
1: 		, [3]
(6)

= [4,3,3,4] 		2,2:
1: 		, [2]
(4)

= [31,1,3,4] 		1,1:
2:
1: 		, [ ]
(2)

= [31,1,1,1] 		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 [1]
    • (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
  • 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 / /
E2 Uniform tiling 0[3] δ3 hδ3 qδ3 Hexagonal
E3 Uniform convex honeycomb 0[4] δ4 hδ4 qδ4
E4 Uniform 4-honeycomb 0[5] δ5 hδ5 qδ5 24-cell honeycomb
E5 Uniform 5-honeycomb 0[6] δ6 hδ6 qδ6
E6 Uniform 6-honeycomb 0[7] δ7 hδ7 qδ7 222
E7 Uniform 7-honeycomb 0[8] δ8 hδ8 qδ8 133 • 331
E8 Uniform 8-honeycomb 0[9] δ9 hδ9 qδ9 152 • 251 • 521
E9 Uniform 9-honeycomb 0[10] δ10 hδ10 qδ10
E10 Uniform 10-honeycomb 0[11] δ11 hδ11 qδ11
En-1 Uniform (n-1)-honeycomb 0[n] δn hδn qδn 1k2 • 2k1 • k21
Quelle: https://en.wikipedia.org/wiki/Truncated_24-cell_honeycomb