5-cubic honeycomb

Summary

5-cubic honeycomb
(no image)
Type Regular 5-space honeycomb
Uniform 5-honeycomb
Family Hypercube honeycomb
Schläfli symbol {4,33,4}
t0,5{4,33,4}
{4,3,3,31,1}
{4,3,4}×{∞}
{4,3,4}×{4,4}
{4,3,4}×{∞}(2)
{4,4}(2)×{∞}
{∞}(5)
Coxeter-Dynkin diagrams















5-face type {4,33} (5-cube)
4-face type {4,3,3} (tesseract)
Cell type {4,3} (cube)
Face type {4} (square)
Face figure {4,3} (octahedron)
Edge figure 8 {4,3,3} (16-cell)
Vertex figure 32 {4,33} (5-orthoplex)
Coxeter group
[4,33,4]
Dual self-dual
Properties vertex-transitive, edge-transitive, face-transitive, cell-transitive

In geometry, the 5-cubic honeycomb or penteractic honeycomb is the only regular space-filling tessellation (or honeycomb) in Euclidean 5-space. Four 5-cubes meet at each cubic cell, and it is more explicitly called an order-4 penteractic honeycomb.

It is analogous to the square tiling of the plane and to the cubic honeycomb of 3-space, and the tesseractic honeycomb of 4-space.

Constructions

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There are many different Wythoff constructions of this honeycomb. The most symmetric form is regular, with Schläfli symbol {4,33,4}. Another form has two alternating 5-cube facets (like a checkerboard) with Schläfli symbol {4,3,3,31,1}. The lowest symmetry Wythoff construction has 32 types of facets around each vertex and a prismatic product Schläfli symbol {∞}(5).

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The [4,33,4],            , Coxeter group generates 63 permutations of uniform tessellations, 35 with unique symmetry and 34 with unique geometry. The expanded 5-cubic honeycomb is geometrically identical to the 5-cubic honeycomb.

The 5-cubic honeycomb can be alternated into the 5-demicubic honeycomb, replacing the 5-cubes with 5-demicubes, and the alternated gaps are filled by 5-orthoplex facets.

It is also related to the regular 6-cube which exists in 6-space with three 5-cubes on each cell. This could be considered as a tessellation on the 5-sphere, an order-3 penteractic honeycomb, {4,34}.

The Penrose tilings are 2-dimensional aperiodic tilings that can be obtained as a projection of the 5-cubic honeycomb along a 5-fold rotational axis of symmetry. The vertices correspond to points in the 5-dimensional cubic lattice, and the tiles are formed by connecting points in a predefined manner.[1]

Tritruncated 5-cubic honeycomb

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A tritruncated 5-cubic honeycomb,      , contains all bitruncated 5-orthoplex facets and is the Voronoi tessellation of the D5* lattice. Facets can be identically colored from a doubled  ×2, [[4,33,4]] symmetry, alternately colored from  , [4,33,4] symmetry, three colors from  , [4,3,3,31,1] symmetry, and 4 colors from  , [31,1,3,31,1] symmetry.

See also

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Regular and uniform honeycombs in 5-space:

References

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  1. ^ de Bruijn, N. G. (1981). "Algebraic theory of Penrose's non-periodic tilings of the plane, I, II" (PDF). Indagationes Mathematicae. 43 (1): 39–66. doi:10.1016/1385-7258(81)90017-2.
  • 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 [1]
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
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