Steric tesseractic honeycomb

Steric tesseractic honeycomb
(No image)
Type Uniform honeycomb
Schläfli symbol h4{4,3,3,4}
Coxeter-Dynkin diagram =
4-face type {4,3,3}
t0,3{4,3,3}
{3,3,4}
{3,3}×{}
Cell type {4,3}
{3,3}
{3}×{}
Face type {4}
{3}
Vertex figure
Coxeter group = [4,3,31,1]
Dual ?
Properties vertex-transitive

In four-dimensional Euclidean geometry, the steric tesseractic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 4-space.

Alternate names

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  • Small diprismatodemitesseractic tetracomb (siphatit)
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The [4,3,31,1],        , Coxeter group generates 31 permutations of uniform tessellations, 23 with distinct symmetry and 4 with distinct geometry. There are two alternated forms: the alternations (19) and (24) have the same geometry as the 16-cell honeycomb and snub 24-cell honeycomb respectively.

B4 honeycombs
Extended
symmetry
Extended
diagram
Order Honeycombs
[4,3,31,1]:         ×1

        5,         6,         7,         8

<[4,3,31,1]>:
↔[4,3,3,4]
       
         
×2

        9,         10,         11,         12,         13,         14,

        (10),         15,         16,         (13),         17,         18,         19

[3[1+,4,3,31,1]]
↔ [3[3,31,1,1]]
↔ [3,3,4,3]
       
      
         
×3

        1,         2,         3,         4

[(3,3)[1+,4,3,31,1]]
↔ [(3,3)[31,1,1,1]]
↔ [3,4,3,3]
       
     
         
×12

        20,         21,         22,         23

See also

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Regular and uniform honeycombs in 4-space:

Notes

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References

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  • 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)
  • Klitzing, Richard. "4D Euclidean tesselations". x3o3o *b3o4x - siphatit - O108
Space Family           /   /  
E2 Uniform tiling 0[3] δ3 3 3 Hexagonal
E3 Uniform convex honeycomb 0[4] δ4 4 4
E4 Uniform 4-honeycomb 0[5] δ5 5 5 24-cell honeycomb
E5 Uniform 5-honeycomb 0[6] δ6 6 6
E6 Uniform 6-honeycomb 0[7] δ7 7 7 222
E7 Uniform 7-honeycomb 0[8] δ8 8 8 133331
E8 Uniform 8-honeycomb 0[9] δ9 9 9 152251521
E9 Uniform 9-honeycomb 0[10] δ10 10 10
E10 Uniform 10-honeycomb 0[11] δ11 11 11
En-1 Uniform (n-1)-honeycomb 0[n] δn n n 1k22k1k21