Truncated 5-cell honeycomb

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Truncated 4-simplex honeycomb
(No image)
TypeUniform 4-honeycomb
FamilyTruncated simplectic honeycomb
Schläfli symbolt0,1{3[5]}
Coxeter diagram
4-face types{3,3,3}
t{3,3,3}
2t{3,3,3}
Cell types{3,3}
t{3,3}
Face typesTriangle {3}
Hexagon {6}
Vertex figure
Elongated tetrahedral antiprism
[3,4,2+], order 48
Symmetry{{\tilde {A}}}_{4}×2, [[3[5]]]
Propertiesvertex-transitive

In four-dimensional Euclidean geometry, the truncated 4-simplex honeycomb, truncated 5-cell honeycomb is a space-filling tessellation honeycomb. It is composed of 5-cells, truncated 5-cells, and bitruncated 5-cells facets in a ratio of 2:2:1. Its vertex figure is an Elongated tetrahedral antiprism, with 8 equilateral triangle and 24 isosceles triangle faces, defining 8 5-cell and 24 truncated 5-cell facets around a vertex.

It can be constructed as five sets of parallel hyperplanes that divide space into two half-spaces. The 3-space hyperplanes contain quarter cubic honeycombs as a collection facets.[1]

Alternate names

  • Cyclotruncated pentachoric tetracomb
  • Small truncated-pentachoric tetracomb

Related polytopes and honeycombs

This honeycomb is one of seven unique uniform honeycombs[2] constructed by the {{\tilde {A}}}_{4} Coxeter group. The symmetry can be multiplied by the symmetry of rings in the Coxeter–Dynkin diagrams:

Pentagon
symmetry		 Extended
symmetry		 Extended
diagram		 Extended
order		 Honeycomb diagrams
a1 [3[5]] ×1 (None)
i2 [[3[5]]] ×2  1, 2, 3,

 4, 5, 6

r10 [5[3[5]]] ×10  7

See also

Regular and uniform honeycombs in 4-space:

Notes

  1. ↑ Olshevsky, 2006 (Model 135)
  2. ↑ , A000029 8-1 cases, skipping one with zero marks

References

  • Norman Johnson Uniform Polytopes, Manuscript (1991)
  • 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 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10] (1.9 Uniform space-fillings)
    • (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 135
  • Richard Klitzing, 4D, Euclidean tesselations, x3x3x3x3x3*a - otcypit - 135
Fundamental convex regular and uniform honeycombs in dimensions 2–11
Family {{\tilde {A}}}_{{n-1}} {{\tilde {C}}}_{{n-1}} {{\tilde {B}}}_{{n-1}} {{\tilde {D}}}_{{n-1}} {{\tilde {G}}}_{2} / {{\tilde {F}}}_{4} / {{\tilde {E}}}_{{n-1}}
Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal
Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4
Uniform 5-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb
Uniform 6-honeycomb {3[6]} δ6 hδ6 qδ6
Uniform 7-honeycomb {3[7]} δ7 hδ7 qδ7 222
Uniform 8-honeycomb {3[8]} δ8 hδ8 qδ8 133 • 331
Uniform 9-honeycomb {3[9]} δ9 hδ9 qδ9 152 • 251 • 521
Uniform n-honeycomb {3[n]} δn hδn qδn 1k2 • 2k1 • k21
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