5-cubic honeycomb

5-cubic honeycomb
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Type	Regular 5-space honeycomb Uniform 5-honeycomb
Family	Hypercube honeycomb
Schläfli symbol	${4,3 3,4} t 0,5 {4,3 3,4} {4,3,3,3 1,1} {4,3,4}\times{\infty} {4,3,4}\times{4,4} {4,3,4}\times{\infty} (2) {4,4} (2) \times{\infty} {\infty} (5)$
Coxeter-Dynkin diagrams
5-face type	${4,3 3}$ (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	$32 {4,3 3}$ (5-orthoplex)
Coxeter group	${\tilde {C}}_{5}$ $[4,3 3,4]$
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

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 {4,3,3,3^1,1}. The lowest symmetry Wythoff construction has 32 types of facets around each vertex and a prismatic product Schläfli symbol {∞}⁽⁵⁾.

Related polytopes and honeycombs

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

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 ${\tilde {C}}_{5}$ ×2, [[4,3³,4]] symmetry, alternately colored from ${\tilde {C}}_{5}$ , [4,3³,4] symmetry, three colors from ${\tilde {B}}_{5}$ , [4,3,3,3^1,1] symmetry, and 4 colors from ${\tilde {D}}_{5}$ , [3^1,1,3,3^1,1] symmetry.

References

^ 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]

v t e Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space	Family	${\tilde {A}}_{n-1}$	${\tilde {C}}_{n-1}$	${\tilde {B}}_{n-1}$	${\tilde {D}}_{n-1}$	${\tilde {G}}_{2}$ / ${\tilde {F}}_{4}$ / ${\tilde {E}}_{n-1}$
E²	Uniform tiling	0_[3]	δ₃	hδ₃	qδ₃	Hexagonal
E³	Uniform convex honeycomb	0_[4]	δ₄	hδ₄	qδ₄
E⁴	Uniform 4-honeycomb	0_[5]	δ₅	hδ₅	qδ₅	24-cell honeycomb
E⁵	Uniform 5-honeycomb	0_[6]	δ₆	hδ₆	qδ₆
E⁶	Uniform 6-honeycomb	0_[7]	δ₇	hδ₇	qδ₇	2₂₂
E⁷	Uniform 7-honeycomb	0_[8]	δ₈	hδ₈	qδ₈	1₃₃ • 3₃₁
E⁸	Uniform 8-honeycomb	0_[9]	δ₉	hδ₉	qδ₉	1₅₂ • 2₅₁ • 5₂₁
E⁹	Uniform 9-honeycomb	0_[10]	δ₁₀	hδ₁₀	qδ₁₀
E¹⁰	Uniform 10-honeycomb	0_[11]	δ₁₁	hδ₁₁	qδ₁₁
E^n-1	Uniform (n-1)-honeycomb	0_[n]	δ_n	hδ_n	qδ_n	1_k2 • 2_k1 • k₂₁

[1] 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.

[1]

Constructions

Related polytopes and honeycombs

Tritruncated 5-cubic honeycomb

See also

References