5-cell honeycomb
4-simplex honeycomb | |
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(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 | ×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.
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:
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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 Coxeter group.[3][4] It is the 4-dimensional case of a simplectic honeycomb.
The A*
4 lattice[5] 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.[6]
This honeycomb is one of seven unique uniform honeycombs[7] constructed by the Coxeter group. The symmetry can be multiplied by the symmetry of rings in the Coxeter–Dynkin diagrams:
A4 honeycombs | ||||
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Pentagon symmetry |
Extended symmetry |
Extended diagram |
Extended group |
Honeycomb diagrams |
a1 | [3[5]] | ![]() ![]() ![]() ![]() ![]() |
(None) | |
i2 | [[3[5]]] | ![]() ![]() ![]() ![]() ![]() |
×2 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
r10 | [5[3[5]]] | ![]() ![]() ![]() ![]() ![]() |
×10 | ![]() ![]() ![]() ![]() ![]() |
Rectified 5-cell honeycomb
Rectified 5-cell honeycomb | |
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(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 | ×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 | |
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(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 | ×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 | |
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(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 | ×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 | |
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(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 | ×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 | |
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(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 |
Vertex figure | tilted rectangular duopyramid |
Symmetry | ×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 | |
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(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 | ×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.
See also
Regular and uniform honeycombs in 4-space:
- Tesseractic honeycomb
- 16-cell honeycomb
- 24-cell honeycomb
- Truncated 24-cell honeycomb
- Snub 24-cell honeycomb
Notes
- ^ 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.
- ^ "The Lattice A4".
- ^ "A4 root lattice - Wolfram|Alpha".
- ^ "The Lattice A4".
- ^ 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
- ^ 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 [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
Space | Family | / / | ||||
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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 |