Truncated 6-simplex honeycomb

From Wikipedia, the free encyclopedia
Truncated 6-simplex honeycomb
(No image)
TypeUniform honeycomb
FamilyTruncated simplectic honeycomb
Schläfli symbolt0,1{3[7]}
Coxeter–Dynkin diagrams
6-face types{35}
t{35}
2t{35}
3t{35}
Vertex figureElongated 5-simplex antiprism
Symmetry{{\tilde {A}}}_{6}×2, [[3[7]]]
Propertiesvertex-transitive

In six-dimensional Euclidean geometry, the truncated 6-simplex honeycomb is a space-filling tessellation (or honeycomb). The tessellation fills space by 6-simplex, truncated 6-simplex, bitruncated 6-simplex, and tritruncated 6-simplex facets. These facet types occur in proportions of 2:2:2:1 respectively in the whole honeycomb.

It can be constructed by seven sets of parallel hyperplanes that divide space. The hyperplane intersections generate truncated 5-simplex honeycomb divisions on each hyperplane.

Related polytopes and honeycombs

This honeycomb is one of 17 unique uniform honeycombs[1] constructed by the {{\tilde {A}}}_{6} Coxeter group, grouped by their extended symmetry of the Coxeter–Dynkin diagrams:

Heptagon
symmetry		 Extended
symmetry		 Extended
diagram		 Extended
order		 Honeycombs
a1 [3[7]] ×1

i2 [[3[7]]] ×2

1

2

r14 [7[3[7]]] ×14

3

See also

Regular and uniform honeycombs in 6-space:

Notes

  1. ↑

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]
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
This article is issued from Wikipedia. The text is available under the Creative Commons Attribution/Share Alike; additional terms may apply for the media files.