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5-cell honeycomb

Geometric figure

4-simplex honeycomb
(No image)
Type	Uniform 4-honeycomb
Family	Simplectic honeycomb
Schläfli symbol	{3[5]} = 0[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	t0,3{3,3,3}
Symmetry	A ~ 4 {\displaystyle {\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.

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Structure

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:

A ~ 3 {\displaystyle {\tilde {A}}_{3}}
C ~ 2 {\displaystyle {\tilde {C}}_{2}}

Two different aperiodic tilings with 5-fold symmetry can be obtained by projecting two-dimensional slices of the honeycomb: the Penrose tiling composed of rhombi, and the Tübingen triangle tiling composed of isosceles triangles.²

A4 lattice

The vertex arrangement of the 5-cell honeycomb 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 A ~ 4 {\displaystyle {\tilde {A}}_{4}} Coxeter group.³ ⁴ It is the 4-dimensional case of a simplectic honeycomb.

The A*4 lattice⁵ 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

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.⁶

This honeycomb is one of seven unique uniform honeycombs ⁷ constructed by the A ~ 4 {\displaystyle {\tilde {A}}_{4}} Coxeter group. The symmetry can be multiplied by the symmetry of rings in the Coxeter–Dynkin diagrams:

A4 honeycombs
Pentagonsymmetry	Extendedsymmetry	Extendeddiagram	Extendedgroup	Honeycomb diagrams
a1	[3[5]]		A ~ 4 {\displaystyle {\tilde {A}}_{4}}	(None)
i2	[[3[5]]]		A ~ 4 {\displaystyle {\tilde {A}}_{4}} ×2	1, 2, 3, 4, 5, 6
r10	[5[3[5]]]		A ~ 4 {\displaystyle {\tilde {A}}_{4}} ×10	7

Rectified 5-cell honeycomb

Rectified 5-cell honeycomb
(No image)
Type	Uniform 4-honeycomb
Schläfli symbol	t0,2{3[5]} or r{3[5]}
Coxeter diagram
4-face types	t1{33} t0,2{33} t0,3{33}
Cell types	Tetrahedron Octahedron Cuboctahedron Triangular prism
Vertex figure	triangular elongated-antiprismatic prism
Symmetry	A ~ 4 {\displaystyle {\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	t0,1{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	Tetrahedral antiprism[3,4,2+], order 48
Symmetry	A ~ 4 {\displaystyle {\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 a tetrahedral antiprism, with 2 regular tetrahedron, 8 triangular pyramid, and 6 tetragonal disphenoid cells, defining 2 5-cell, 8 truncated 5-cell, and 6 bitruncated 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.⁸

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	t0,1,2{3[5]} or t{3[5]}
Coxeter diagram
4-face types	t0,1{33} t0,1,2{33} t0,3{33}
Cell types	Tetrahedron Truncated tetrahedron Truncated octahedron Triangular prism
Vertex figure	triangular elongated-antiprismatic pyramid
Symmetry	A ~ 4 {\displaystyle {\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	t0,1,3{3[5]} or rr{3[5]}
Coxeter diagram
4-face types	t0,2{33} t1,2{33} t0,1,3{33}
Cell types	Truncated tetrahedron Octahedron Cuboctahedron Triangular prism Hexagonal prism
Vertex figure	Bidiminished rectified pentachoron
Symmetry	A ~ 4 {\displaystyle {\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	t0,1,2,3{3[5]} or 2t{3[5]}
Coxeter diagram
4-face types	t0,1,3{33} t0,1,2{33} t0,1,2,3{33}
Cell types	Cuboctahedron Truncated octahedron Truncated tetrahedron Hexagonal prism Triangular prism
Vertex figure	tilted rectangular duopyramid
Symmetry	A ~ 4 {\displaystyle {\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	t0,1,2,3,4{3[5]} or tr{3[5]}
Coxeter diagram
4-face types	t0,1,2,3{3,3,3}
Cell types	t0,1,2{3,3} {6}x{}
Face types	{4}{6}
Vertex figure	Irr. 5-cell
Symmetry	A ~ 4 {\displaystyle {\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 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 1906.⁹

The facets of all omnitruncated simplectic honeycombs are called permutohedra 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-cell honeycomb, and therefore the Voronoi cell of this lattice is an omnitruncated 5-cell.¹⁰

∪ ∪ ∪ ∪ = dual of

Alternated form

This honeycomb can be alternated, creating omnisnub 5-cells with irregular 5-cells created at the deleted vertices. Although it is not uniform, the 5-cells have a symmetry of order 10.

Notes

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 [1]
- (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) arXiv:1209.1878

v t e Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space	Family	A ~ n − 1 {\displaystyle {\tilde {A}}_{n-1}}	C ~ n − 1 {\displaystyle {\tilde {C}}_{n-1}}	B ~ n − 1 {\displaystyle {\tilde {B}}_{n-1}}	D ~ n − 1 {\displaystyle {\tilde {D}}_{n-1}}	G ~ 2 {\displaystyle {\tilde {G}}_{2}} / F ~ 4 {\displaystyle {\tilde {F}}_{4}} / E ~ n − 1 {\displaystyle {\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
En-1	Uniform (n-1)-honeycomb	0[n]	δn	hδn	qδn	1k2 • 2k1 • k21

References

Olshevsky (2006), Model 134 ↩
Baake, M.; Kramer, P.; Schlottmann, M.; Zeidler, D. (December 1990). "Planar Patterns with Fivefold Symmetry as Sections of Periodic Structures in 4-Space". International Journal of Modern Physics B. 04 (15n16): 2217–2268. doi:10.1142/S0217979290001054. /wiki/Doi_(identifier) ↩
"The Lattice A4". http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A4.html ↩
"A4 root lattice - Wolfram|Alpha". https://m.wolframalpha.com/input/?i=A4+root+lattice&lk=3 ↩
"The Lattice A4". http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/As4.html ↩
Olshevsky (2006), Klitzing, elong( x3o3o3o3o3*a ) - ecypit - O141, schmo( x3o3o3o3o3*a ) - zucypit - O142, elongschmo( x3o3o3o3o3*a ) - ezucypit - O143 ↩
mathworld: Necklace, OEIS sequence A000029 8-1 cases, skipping one with zero marks http://mathworld.wolfram.com/Necklace.html ↩
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) 0-486-40919-8 ↩
The Lattice A4* http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/As4.html ↩

Structure

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

Alternated form

See also

Notes

References