5-cell honeycomb

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4-simplex honeycomb
(No image)
Type	Uniform 4-honeycomb
Family	Simplectic honeycomb
Schläfli symbol	{3^[5]}
Coxeter diagram
4-face types	{3,3,3} t₁{3,3,3}
Cell types	{3,3} t₁{3,3}
Face types	{3}
Vertex figure	t_0,3{3,3,3}
Symmetry	${{\tilde {A}}}_{4}$ ×2, [[3^[5]]]
Properties	vertex-transitive

In four-dimensional Euclidean geometry, the 4-simplex honeycomb, 5-cell honeycomb or pentachoric-dispentachoric honeycomb is a space-filling tessellation honeycomb. It is composed of 5-cells and rectified 5-cells facets in a ratio of 1:1.

Cells of the vertex figure are ten tetrahedrons and 20 triangular prisms, corresponding to the ten 5-cells and 20 rectified 5-cells that meet at each vertex. All the vertices lie in parallel realms in which they form alternated cubic honeycombs, the tetrahedra being either tops of the rectified 5-cell or the bases of the 5-cell, and the octahedra being the bottoms of the rectified 5-cell.^[1]

A4 lattice

This vertex arrangement is called the A4 lattice, or 4-simplex lattice. The 20 vertices of its vertex figure, the runcinated 5-cell represent the 20 roots of the ${{\tilde {A}}}_{4}$ Coxeter group.^[2] It is the 4-dimensional case of a simplectic honeycomb.

The A*
4 lattice is the union of five A₄ lattices, and is the dual to the omnitruncated 5-simplex honeycomb, and therefore the Voronoi cell of this lattice is an omnitruncated 5-cell

= dual of

Alternate names

Cyclopentachoric tetracomb
Pentachoric-dispentachoric tetracomb

Related polytopes and honeycombs

The tops of the 5-cells in this honeycomb adjoin the bases of the 5-cells, and vice versa, in adjacent laminae; but alternating laminae may be inverted so that the tops of the rectified 5-cells adjoin the tops of the rectified 5-cells and the bases of the 5-cells adjoin the bases of other 5-cells. This inversion results in another non-Wythoffian uniform convex honeycomb. Octahedral prisms and tetrahedral prisms may be inserted in between alternated laminae as well, resulting in two more non-Wythoffian elongated uniform honeycombs.^[3]

This honeycomb is one of seven unique uniform honeycombs^[4] 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 order	Honeycomb diagrams
a1	[3^[5]]	×1	(None)
i2	[[3^[5]]]	×2	₁, ₂, ₃, ₄, ₅, ₆
r10	[5[3^[5]]]	×10	₇

Projection by folding

The 5-cell honeycomb can be projected into the 2-dimensional square tiling by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same vertex arrangement:

${{\tilde {A}}}_{3}$
${{\tilde {C}}}_{2}$

Notes

↑ Olshevsky (2006), Model 134
↑ http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A4.html
↑ Olshevsky (2006), Klitzing, elong( x3o3o3o3o3*a ) - ecypit - O141, schmo( x3o3o3o3o3*a ) - zucypit - O142, elongschmo( x3o3o3o3o3*a ) - ezucypit - O143
↑ , 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 134
Richard Klitzing, 4D, Euclidean tesselations, x3o3o3o3o3*a - cypit - O134
Affine Coxeter group Wa(A4), Quaternions, and Decagonal Quasicrystals Mehmet Koca, Nazife O. Koca, Ramazan Koc (2013)

v t e 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]}	δ_n	hδ_n	qδ_n	1_k2 • 2_k1 • k₂₁

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