- 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
- See also
- Notes
- References
| 4-simplex honeycomb | |
|---|---|
| (No image) | |
| Type | Uniform 4-honeycomb |
| Family | Simplectic honeycomb |
| Schläfli symbol | {3[5]} |
| Coxeter diagram | node_1|split1|nodes|3ab|branch}} |
| 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, [5]">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:
 | node_1|split1|nodes|3ab|branch}} |
|---|---|
 | node_1|4|node|4|node}} |
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.[2][3] It is the 4-dimensional case of a simplectic honeycomb.
The A{{sup sub|*|4}} lattice[4] 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}}
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.[5]
{{4-simplex honeycomb family}}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 | node|split1|nodes_11|3ab|branch}} |
| 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, [5]">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 | branch_11|3ab|nodes|split2|node}} |
| 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 | ×2, [5]">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.[6]
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 | node_1|split1|nodes_11|3ab|branch}} |
| 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, [5]">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 | node_1|split1|nodes|3ab|branch_11}} |
| 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 | triangular-prismatic antifastigium |
| Symmetry | ×2, [5]">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 | node|split1|nodes_11|3ab|branch_11}} |
| 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 | ×2, [5]">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 | node_1|split1|nodes_11|3ab|branch_11}} |
| 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 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.[7]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{{sup sub|*|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.[8]
{{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}}{{-}}
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
2. ^http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A4.html
3. ^https://m.wolframalpha.com/input/?i=A4+root+lattice&lk=3
4. ^http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/As4.html
5. ^Olshevsky (2006), Klitzing, elong( x3o3o3o3o3*a ) - ecypit - O141, schmo( x3o3o3o3o3*a ) - zucypit - O142, elongschmo( x3o3o3o3o3*a ) - ezucypit - O143
6. ^Olshevsky, (2006) Model 135
7. ^{{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)
8. ^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
- {{KlitzingPolytopes|flat.htm|4D|Euclidean tesselations}}, x3o3o3o3o3a - cypit - O134, x3x3x3x3x3a - otcypit - 135, x3x3x3o3o3a - gocyropit - O137, x3x3o3x3o3a - cypropit - O138, x3x3x3x3o3a - gocypapit - O139, x3x3x3x3x3a - otcypit - 140
- Affine Coxeter group Wa(A4), Quaternions, and Decagonal Quasicrystals Mehmet Koca, Nazife O. Koca, Ramazan Koc (2013) [https://arxiv.org/ftp/arxiv/papers/1209/1209.1878.pdf]
- Edward Baxter
- Edward B. Butler
- Edward B. Cole
- Edward B. Dudley
- Edward Beale McLean
- Edward Beetham
- Edward Belcher
- Edward Bellamy
- Edward Bellew
- Edward B. Ellington
- Edward Bell (singer/songwriter)
- Edward Benes
- Edward Benjamin Britten
- Edward Benjamin Britten, Baron Britten
- Edward Benjamin Cushing
- Edward Benlowes
- Edward Bennett Williams
- Edward Bernard Raczynski
- Edward Bernard Raczyński
- Edward Bernays
- Edward Bernds
- Edward Bernstein
- Edward Berry
- Edward Bertram Johnston
- Edward Betham Beetham