Truncated 24-cell honeycomb

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.

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, [3,4,3,3]
, [4,3,31,1]
, [4,3,3,4]
, [31,1,1,1]
PropertiesVertex transitive

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)
gollark: Wait, can't a lot of "alive" stuff only replicate if it has a suitable environment, too?
gollark: Also, it would consider sterile humans not alive.
gollark: The "and another member of your species" bit does have the interesting implication that you can't really call something alive or not if you just have one of it, then.
gollark: That is true, except I think some cells can't because of DNA damage or something.
gollark: I mean, individual animals can't reproduce on their own, except the weird ones which can.

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, [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 {3[3]} δ3 hδ3 qδ3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4
E4 Uniform 4-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb
E5 Uniform 5-honeycomb {3[6]} δ6 hδ6 qδ6
E6 Uniform 6-honeycomb {3[7]} δ7 hδ7 qδ7 222
E7 Uniform 7-honeycomb {3[8]} δ8 hδ8 qδ8 133331
E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152251521
E9 Uniform 9-honeycomb {3[10]} δ10 hδ10 qδ10
En-1 Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k22k1k21
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