4-simplex honeycomb | |
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(No image) | |
Type | Uniform 4-honeycomb |
Family | Simplectic honeycomb |
Schläfli symbol | {3^{[5]}} |
Coxeter diagram | |
4-face types | {3,3,3} t_{1}{3,3,3} |
Cell types | {3,3} t_{1}{3,3} |
Face types | {3} |
Vertex figure | t_{0,3}{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.
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]}
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.^{[2]}
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 A_{4} lattices, and is the dual to the omnitruncated 5-simplex honeycomb, and therefore the Voronoi cell of this lattice is an omnitruncated 5-cell
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 | _{1}, _{2}, _{3},
_{4}, _{5}, _{6} | |
r10 | [5[3^{[5]}]] | ×10 | _{7} |
Rectified 5-cell honeycomb | |
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(No image) | |
Type | Uniform 4-honeycomb |
Schläfli symbol | t_{0,2}{3^{[5]}} or r{3^{[5]}} |
Coxeter diagram | |
4-face types | t_{1}{3^{3}} t_{0,2}{3^{3}} t_{0,3}{3^{3}} |
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.
Cyclotruncated 5-cell honeycomb | |
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(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 | 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]}
Truncated 4-simplex honeycomb | |
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(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}{3^{3}} t_{0,1,2}{3^{3}} t_{0,3}{3^{3}} |
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.
Cantellated 5-cell honeycomb | |
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(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}{3^{3}} t_{1,2}{3^{3}} t_{0,1,3}{3^{3}} |
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.
Bitruncated 5-cell honeycomb | |
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(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}{3^{3}} t_{0,1,2}{3^{3}} t_{0,1,2,3}{3^{3}} |
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.
Omnitruncated 4-simplex honeycomb | |
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(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 | ×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).
The A^{*}
_{4} lattice is the union of five A_{4} 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]}
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:
Space | Family | / / | ||||
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E^{2} | Uniform tiling | {3^{[3]}} | δ_{3} | hδ_{3} | qδ_{3} | Hexagonal |
E^{3} | Uniform convex honeycomb | {3^{[4]}} | δ_{4} | hδ_{4} | qδ_{4} | |
E^{4} | Uniform 4-honeycomb | {3^{[5]}} | δ_{5} | hδ_{5} | qδ_{5} | 24-cell honeycomb |
E^{5} | Uniform 5-honeycomb | {3^{[6]}} | δ_{6} | hδ_{6} | qδ_{6} | |
E^{6} | Uniform 6-honeycomb | {3^{[7]}} | δ_{7} | hδ_{7} | qδ_{7} | 2_{22} |
E^{7} | Uniform 7-honeycomb | {3^{[8]}} | δ_{8} | hδ_{8} | qδ_{8} | 1_{33} • 3_{31} |
E^{8} | Uniform 8-honeycomb | {3^{[9]}} | δ_{9} | hδ_{9} | qδ_{9} | 1_{52} • 2_{51} • 5_{21} |
E^{9} | Uniform 9-honeycomb | {3^{[10]}} | δ_{10} | hδ_{10} | qδ_{10} | |
E^{10} | Uniform 10-honeycomb | {3^{[11]}} | δ_{11} | hδ_{11} | qδ_{11} | |
E^{n-1} | Uniform (n-1)-honeycomb | {3^{[n]}} | δ_{n} | hδ_{n} | qδ_{n} | 1_{k2} • 2_{k1} • k_{21} |