Steriruncic tesseractic honeycomb

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

Steriruncic tesseractic honeycomb
(No image)
TypeUniform honeycomb
Schläfli symbolh3,4{4,3,3,4}
Coxeter-Dynkin diagram =
4-face typer{4,3,4}
t{4,3,4}
t0,1,3{4,3,4}
{3,3}×{}
Cell typet{4,3}
{3,3}
r{4,3}
{3}×{}
t{4}×{}
Face type{8}
{4}
{3}
Vertex figure
Coxeter group = [4,3,31,1]
Dual?
Propertiesvertex-transitive

Alternate names

  • Prismatorhombated demitesseractic tetracomb (pirhatit)
  • Great prismatodemitesseractic tetracomb

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

Regular and uniform honeycombs in 4-space:

Notes

    References

    • 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)
    • Klitzing, Richard. "4D Euclidean tesselations". x3o3o *b3x4x - pirhatit - O110
    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
    E10 Uniform 10-honeycomb {3[11]} δ11 hδ11 qδ11
    En-1 Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k22k1k21
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