5-cell honeycomb

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.Olshevsky (2006), Model 134

• Cyclopentachoric tetracomb
• Pentachoric-dispentachoric tetracomb

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:

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.{{cite journal |last1=Baake |first1=M. |last2=Kramer |first2=P. |last3=Schlottmann |first3=M. |last4=Zeidler |first4=D. |title=Planar Patterns with Fivefold Symmetry as Sections of Periodic Structures in 4-Space |journal=International Journal of Modern Physics B |date=December 1990 |volume=04 |issue=15n16 |pages=2217–2268 |doi=10.1142/S0217979290001054}}

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 $\{\backslash tilde\{A\}\}\_4$ Coxeter group.{{Cite web|url=http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A4.html|title = The Lattice A4}}{{Cite web|url=https://m.wolframalpha.com/input/?i=A4+root+lattice&lk=3|title = A4 root lattice - Wolfram|Alpha}} It is the 4-dimensional case of a simplectic honeycomb.

The A{{sup sub|*|4}} lattice{{Cite web|url=http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/As4.html|title=The Lattice A4}} 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

: {{CDD|node_1|split1|nodes|3ab|branch}} ∪ {{CDD|node|split1|nodes_10lur|3ab|branch}} ∪ {{CDD|node|split1|nodes_01lr|3ab|branch}} ∪ {{CDD|node|split1|nodes|3ab|branch_10l}} ∪ {{CDD|node|split1|nodes|3ab|branch_01l}} = dual of {{CDD|node_1|split1|nodes_11|3ab|branch_11}}

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.Olshevsky (2006), Klitzing, elong( x3o3o3o3o3*a ) - ecypit - O141, schmo( x3o3o3o3o3*a ) - zucypit - O142, elongschmo( x3o3o3o3o3*a ) - ezucypit - O143

{{4-simplex honeycomb family}}

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

• small cyclorhombated pentachoric tetracomb
• small prismatodispentachoric tetracomb

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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.Olshevsky, (2006) Model 135

• Cyclotruncated pentachoric tetracomb
• Small truncated-pentachoric tetracomb

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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.

• Great cyclorhombated pentachoric tetracomb
• Great truncated-pentachoric tetracomb

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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.

• Cycloprismatorhombated pentachoric tetracomb
• Great prismatodispentachoric tetracomb

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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.

• Great cycloprismated pentachoric tetracomb
• Grand prismatodispentachoric tetracomb

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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.{{cite book|title=The Beauty of Geometry: Twelve Essays|year= 1999|publisher= Dover Publications|lccn=99035678|isbn= 0-486-40919-8 }} (The classification of Zonohededra, page 73)

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).

• Omnitruncated cyclopentachoric tetracomb
• Great-prismatodecachoric tetracomb

The A{{sup sub|*|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.[http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/As4.html The Lattice A4*]

: {{CDD|node_1|split1|nodes|3ab|branch}} ∪ {{CDD|node|split1|nodes_10lur|3ab|branch}} ∪ {{CDD|node|split1|nodes_01lr|3ab|branch}} ∪ {{CDD|node|split1|nodes|3ab|branch_10l}} ∪ {{CDD|node|split1|nodes|3ab|branch_01l}} = dual of {{CDD|node_1|split1|nodes_11|3ab|branch_11}}

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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.

Regular and uniform honeycombs in 4-space:

• Tesseractic honeycomb
• 16-cell honeycomb
• 24-cell honeycomb
• Truncated 24-cell honeycomb
• Snub 24-cell honeycomb

{{reflist}}

• 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}} [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html]
• (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
• {{KlitzingPolytopes|flat.htm|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}}

{{Honeycombs}}

Category:Honeycombs (geometry)

Category:5-polytopes