Runcinated 16-cell honeycomb

In four-dimensional Euclidean geometry, the runcinated 16-cell honeycomb is a uniform space-filling honeycomb. It can be seen as a runcination of the regular 16-cell honeycomb, containing Rectified 24-cell, runcinated tesseract, cuboctahedral prism, and 3-3 duoprism cells.

Runcinated 16-cell honeycomb
(No image)
TypeUniform 4-honeycomb
Schläfli symbolt03{3,3,4,3}
Coxeter-Dynkin diagrams

4-face typer{3,4,3}
t03{4,3,3}
r{3,4}x{}
{3}x{3}
Cell typer{3,4}
{4,3}
{3,3}
Face type{3}, {4}
Vertex figure
Coxeter groups, [3,4,3,3]
PropertiesVertex transitive

Alternate names

  • Runcinated hexadecachoric tetracomb/honeycomb
  • Small prismated demitesseractic tetracomb (spaht)
  • Small disicositetrachoric tetracomb

The [3,4,3,3], , Coxeter group generates 31 permutations of uniform tessellations, 28 are unique in this family and ten are shared in the [4,3,3,4] and [4,3,31,1] families. The alternation (13) is also repeated in other families.

F4 honeycombs
Extended
symmetry
Extended
diagram
OrderHoneycombs
[3,3,4,3]×1

1, 3, 5, 6, 8,
9, 10, 11, 12

[3,4,3,3]×1

2, 4, 7, 13,
14, 15, 16, 17,
18, 19, 20, 21,
22 23, 24, 25,
26, 27, 28, 29

[(3,3)[3,3,4,3*]]
=[(3,3)[31,1,1,1]]
=[3,4,3,3]

=
=
×4

(2), (4), (7), (13)

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 122
  • Klitzing, Richard. "4D Euclidean tesselations". x3o3o4x3o - spaht - O122
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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