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    Truncated 24-cell honeycomb

    From Handwiki - Reading time: 2 min
    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 [math]\displaystyle{ {\tilde{F}}_4 }[/math], [3,4,3,3]
    [math]\displaystyle{ {\tilde{B}}_4 }[/math], [4,3,31,1]
    [math]\displaystyle{ {\tilde{C}}_4 }[/math], [4,3,3,4]
    [math]\displaystyle{ {\tilde{D}}_4 }[/math], [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 [math]\displaystyle{ {\tilde{D}}_4 }[/math] 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)
    [math]\displaystyle{ {\tilde{F}}_4 }[/math]
    = [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)
    [math]\displaystyle{ {\tilde{F}}_4 }[/math]
    = [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)
    [math]\displaystyle{ {\tilde{C}}_4 }[/math]
    = [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)
    [math]\displaystyle{ {\tilde{B}}_4 }[/math]
    = [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)
    [math]\displaystyle{ {\tilde{D}}_4 }[/math]
    = [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

    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". https://bendwavy.org/klitzing/dimensions/flat.htm.  o4x3x3x4o, x3x3x *b3x4o, x3x3x *b3x *b3x, o3o3o4x3x, x3x3x4o3o - ticot - O99
    Fundamental convex regular and uniform honeycombs in dimensions 2-9
    Space Family [math]\displaystyle{ {\tilde{A}}_{n-1} }[/math] [math]\displaystyle{ {\tilde{C}}_{n-1} }[/math] [math]\displaystyle{ {\tilde{B}}_{n-1} }[/math] [math]\displaystyle{ {\tilde{D}}_{n-1} }[/math] [math]\displaystyle{ {\tilde{G}}_2 }[/math] / [math]\displaystyle{ {\tilde{F}}_4 }[/math] / [math]\displaystyle{ {\tilde{E}}_{n-1} }[/math]
    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 133 • 331
    E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152 • 251 • 521
    E9 Uniform 9-honeycomb {3[10]} δ10 hδ10 qδ10
    En-1 Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k2 • 2k1 • k21



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