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{4,3,31,1}
t{31,1,1,1}
Coxeter-Dynkin diagrams

CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
CDel nodes 11.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
CDel nodes 11.pngCDel split2.pngCDel node 1.pngCDel split1.pngCDel nodes 11.png

4-face type Tesseract Schlegel wireframe 8-cell.png
Truncated 24-cell Schlegel half-solid truncated 24-cell.png
Cell type Cube Hexahedron.png
Truncated octahedron Truncated octahedron.png
Face type Square
Triangle
Vertex figure Truncated 24-cell honeycomb verf.png
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[edit]

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

Symmetry constructions[edit]

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]
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png 4: CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
1: CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Truncated 24-cell honeycomb verf.png CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png, [3,3]
(24)

= [3,3,4,3]
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png 3: CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
1: CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
1: CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png
Truncated 24-cell honeycomb F4b verf.png CDel node.pngCDel 3.pngCDel node.png, [3]
(6)

= [4,3,3,4]
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png 2,2: CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
1: CDel node.pngCDel 4.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.png
Truncated 24-cell honeycomb C4 verf.png CDel node.pngCDel 2.pngCDel node.png, [2]
(4)

= [31,1,3,4]
CDel nodes 11.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png 1,1: CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
2: CDel nodes 11.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.png
1: CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.png
Truncated 24-cell honeycomb B4 verf.png CDel node.png, [ ]
(2)

= [31,1,1,1]
CDel nodes 11.pngCDel split2.pngCDel node 1.pngCDel split1.pngCDel nodes 11.png 1,1,1,1:
CDel nodes 11.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.png
1: CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png
Truncated 24-cell honeycomb D4 verf.png [ ]+
(1)

See also[edit]

Regular and uniform honeycombs in 4-space:

References[edit]

  • 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 {3[3]} δ3 3 3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 4 4
E4 Uniform 4-honeycomb {3[5]} δ5 5 5 24-cell honeycomb
E5 Uniform 5-honeycomb {3[6]} δ6 6 6
E6 Uniform 6-honeycomb {3[7]} δ7 7 7 222
E7 Uniform 7-honeycomb {3[8]} δ8 8 8 133331
E8 Uniform 8-honeycomb {3[9]} δ9 9 9 152251521
E9 Uniform 9-honeycomb {3[10]} δ10 10 10
En-1 Uniform (n-1)-honeycomb {3[n]} δn n n 1k22k1k21