5-cell honeycomb

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]

Alternate names

Cyclopentachoric tetracomb
Pentachoric-dispentachoric tetracomb

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:

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

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

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 (or layers); 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:

A4 honeycombs
Pentagon symmetry	Extended symmetry	Extended diagram	Extended group	Honeycomb diagrams
a1	[3^[5]]		${{\tilde {A}}}_{4}$	(None)
i2	[[3^[5]]]		${{\tilde {A}}}_{4}$ ×2	₁, ₂, ₃, ₄, ₅, ₆
r10	[5[3^[5]]]		${{\tilde {A}}}_{4}$ ×10	₇

Rectified 5-cell honeycomb

Rectified 5-cell honeycomb
(No image)
Type	Uniform 4-honeycomb
Schläfli symbol	t_0,2{3^[5]} or r{3^[5]}
Coxeter diagram
4-face types	t₁{3³} t_0,2{3³} t_0,3{3³}
Cell types	Tetrahedron Octahedron Cuboctahedron Triangular prism
Vertex figure	triangular elongated-antiprismatic prism
Symmetry	${{\tilde {A}}}_{4}$ ×2, [[3^[5]]]
Properties	vertex-transitive

The rectified 4-simplex honeycomb or rectified 5-cell honeycomb is a space-filling tessellation honeycomb.

Alternate names

small cyclorhombated pentachoric tetracomb
small prismatodispentachoric tetracomb

Cyclotruncated 5-cell honeycomb

Cyclotruncated 5-cell honeycomb
(No image)
Type	Uniform 4-honeycomb
Family	Truncated simplectic honeycomb
Schläfli symbol	t_0,1{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 types	Triangle {3} Hexagon {6}
Vertex figure	Elongated tetrahedral antiprism [3,4,2⁺], order 48
Symmetry	${{\tilde {A}}}_{4}$ ×2, [[3^[5]]]
Properties	vertex-transitive

The cyclotruncated 4-simplex honeycomb or cyclotruncated 5-cell honeycomb is a space-filling tessellation honeycomb. It can also be seen as a birectified 5-cell 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.^[5]

Alternate names

Cyclotruncated pentachoric tetracomb
Small truncated-pentachoric tetracomb

Truncated 5-cell honeycomb

Truncated 4-simplex honeycomb
(No image)
Type	Uniform 4-honeycomb
Schläfli symbol	t_0,1,2{3^[5]} or t{3^[5]}
Coxeter diagram
4-face types	t_0,1{3³} t_0,1,2{3³} t_0,3{3³}
Cell types	Tetrahedron Truncated tetrahedron Truncated octahedron Triangular prism
Vertex figure	triangular elongated-antiprismatic pyramid
Symmetry	${{\tilde {A}}}_{4}$ ×2, [[3^[5]]]
Properties	vertex-transitive

The truncated 4-simplex honeycomb or truncated 5-cell honeycomb is a space-filling tessellation honeycomb. It can also be called a cyclocantitruncated 5-cell honeycomb.

Alaternate names

Great cyclorhombated pentachoric tetracomb
Great truncated-pentachoric tetracomb

Cantellated 5-cell honeycomb

Cantellated 5-cell honeycomb
(No image)
Type	Uniform 4-honeycomb
Schläfli symbol	t_0,1,3{3^[5]} or rr{3^[5]}
Coxeter diagram
4-face types	t_0,2{3³} t_1,2{3³} t_0,1,3{3³}
Cell types	Truncated tetrahedron Octahedron Cuboctahedron Triangular prism Hexagonal prism
Vertex figure	triangular-prismatic antifastigium
Symmetry	${{\tilde {A}}}_{4}$ ×2, [[3^[5]]]
Properties	vertex-transitive

The cantellated 4-simplex honeycomb or cantellated 5-cell honeycomb is a space-filling tessellation honeycomb. It can also be called a cycloruncitruncated 5-cell honeycomb.

Alternate names

Cycloprismatorhombated pentachoric tetracomb
Great prismatodispentachoric tetracomb

Bitruncated 5-cell honeycomb

Bitruncated 5-cell honeycomb
(No image)
Type	Uniform 4-honeycomb
Schläfli symbol	t_0,1,2,3{3^[5]} or 2t{3^[5]}
Coxeter diagram
4-face types	t_0,1,3{3³} t_0,1,2{3³} t_0,1,2,3{3³}
Cell types	Cuboctahedron Truncated octahedron Truncated tetrahedron Hexagonal prism Triangular prism
Vertex figure	tilted rectangular duopyramid
Symmetry	${{\tilde {A}}}_{4}$ ×2, [[3^[5]]]
Properties	vertex-transitive

The bitruncated 4-simplex honeycomb or bitruncated 5-cell honeycomb is a space-filling tessellation honeycomb. It can also be called a cycloruncicantitruncated 5-cell honeycomb.

Alternate names

Great cycloprismated pentachoric tetracomb
Grand prismatodispentachoric tetracomb

Omnitruncated 5-cell honeycomb

Omnitruncated 4-simplex honeycomb
(No image)
Type	Uniform 4-honeycomb
Family	Omnitruncated simplectic honeycomb
Schläfli symbol	t_0,1,2,3,4{3^[5]} or tr{3^[5]}
Coxeter diagram
4-face types	t_0,1,2,3{3,3,3}
Cell types	t_0,1,2{3,3} {6}x{}
Face types	{4} {6}
Vertex figure	Irr. 5-cell
Symmetry	${{\tilde {A}}}_{4}$ ×10, [5[3^[5]]]
Properties	vertex-transitive, cell-transitive

The omnitruncated 4-simplex honeycomb or omnitruncated 5-cell honeycomb is a space-filling tessellation honeycomb. It can also be seen as a cantitruncated 5-cell honeycomb and also a cyclosteriruncicantitruncated 5-cell honeycomb. .

It is composed entirely of omnitruncated 5-cell (omnitruncated 4-simplex) facets.

Coxeter calls this Hinton's honeycomb after C. H. Hinton, who described it in his book The Fourth Dimension in 1906.^[6]

The facets of all omnitruncated simplectic honeycombs are called permutahedra and can be positioned in n+1 space with integral coordinates, permutations of the whole numbers (0,1,..,n).

Alternate names

Omnitruncated cyclopentachoric tetracomb
Great-prismatodecachoric tetracomb

A₄^* lattice

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.^[7]

∪

= dual of

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
↑ Olshevsky, (2006) Model 135
↑ The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. LCCN 99035678. (The classification of Zonohededra, page 73)
↑ The Lattice A4*

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
Klitzing, Richard. "4D Euclidean tesselations". , x3o3o3o3o3*a - cypit - O134, x3x3x3x3x3*a - otcypit - 135, x3x3x3o3o3*a - gocyropit - O137, x3x3o3x3o3*a - cypropit - O138, x3x3x3x3o3*a - gocypapit - O139, x3x3x3x3x3*a - otcypit - 140
Affine Coxeter group Wa(A4), Quaternions, and Decagonal Quasicrystals Mehmet Koca, Nazife O. Koca, Ramazan Koc (2013)

Fundamental convex regular and uniform honeycombs in dimensions 3–10 (or 2-9)
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 10-honeycomb	{3^[10]}	δ₁₀	hδ₁₀	qδ₁₀
Uniform n-honeycomb	{3^[n]}	δ_n	hδ_n	qδ_n	1_k2 • 2_k1 • k₂₁

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5-cell honeycomb

Alternate names

Projection by folding

A4 lattice

Related polytopes and honeycombs

Rectified 5-cell honeycomb

Alternate names

Cyclotruncated 5-cell honeycomb

Alternate names

Truncated 5-cell honeycomb

Alaternate names

Cantellated 5-cell honeycomb

Alternate names

Bitruncated 5-cell honeycomb

Alternate names

Omnitruncated 5-cell honeycomb

Alternate names

A4* lattice

See also

Notes

References

A₄^* lattice