| bgcolor=#e7dcc3 colspan=2 | 4-simplex honeycomb | |
|---|---|---|
| bgcolor=#ffffff align=center colspan=2 | (No image) | |
| Type | Uniform 4-honeycomb | |
| Family | Simplectic honeycomb | |
| Schläfli symbol | = 0[5] | |
| Coxeter diagram | ||
| 4-face types | ||
| Cell types | ||
| Face types | ||
| Vertex figure | t0,3 | |
| Symmetry | {\tilde{A}}4 [3<sup>[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
{\tilde{A}}4
The A 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
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]
| bgcolor=#e7dcc3 colspan=2 | Rectified 5-cell honeycomb | |
|---|---|---|
| bgcolor=#ffffff align=center colspan=2 | (No image) | |
| Type | Uniform 4-honeycomb | |
| Schläfli symbol | t0,2 or r | |
| Coxeter diagram | ||
| 4-face types | ||
| Cell types | ||
| Vertex figure | triangular elongated-antiprismatic prism | |
| Symmetry | {\tilde{A}}4 [3<sup>[5]] | |
| Properties | vertex-transitive |
| bgcolor=#e7dcc3 colspan=2 | Cyclotruncated 5-cell honeycomb | |
|---|---|---|
| bgcolor=#ffffff align=center colspan=2 | (No image) | |
| Type | Uniform 4-honeycomb | |
| Family | Truncated simplectic honeycomb | |
| Schläfli symbol | t0,1 | |
| Coxeter diagram | ||
| 4-face types | ||
| Cell types | ||
| Face types | ||
| Vertex figure | Tetrahedral antiprism [3,4,2<sup>+</sup>], order 48 | |
| Symmetry | {\tilde{A}}4 [3<sup>[5]] | |
| Properties | vertex-transitive |
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.[7]
| bgcolor=#e7dcc3 colspan=2 | Truncated 4-simplex honeycomb | |
|---|---|---|
| bgcolor=#ffffff align=center colspan=2 | (No image) | |
| Type | Uniform 4-honeycomb | |
| Schläfli symbol | t0,1,2 or t | |
| Coxeter diagram | ||
| 4-face types | ||
| Cell types | ||
| Vertex figure | triangular elongated-antiprismatic pyramid | |
| Symmetry | {\tilde{A}}4 [3<sup>[5]] | |
| Properties | vertex-transitive |
| bgcolor=#e7dcc3 colspan=2 | Cantellated 5-cell honeycomb | |
|---|---|---|
| bgcolor=#ffffff align=center colspan=2 | (No image) | |
| Type | Uniform 4-honeycomb | |
| Schläfli symbol | t0,1,3 or rr | |
| Coxeter diagram | ||
| 4-face types | ||
| Cell types | ||
| Vertex figure | Bidiminished rectified pentachoron | |
| Symmetry | {\tilde{A}}4 [3<sup>[5]] | |
| Properties | vertex-transitive |
| bgcolor=#e7dcc3 colspan=2 | Bitruncated 5-cell honeycomb | |
|---|---|---|
| bgcolor=#ffffff align=center colspan=2 | (No image) | |
| Type | Uniform 4-honeycomb | |
| Schläfli symbol | t0,1,2,3 or 2t | |
| Coxeter diagram | ||
| 4-face types | ||
| Cell types | ||
| Vertex figure | tilted rectangular duopyramid | |
| Symmetry | {\tilde{A}}4 [3<sup>[5]] | |
| Properties | vertex-transitive |
| bgcolor=#e7dcc3 colspan=2 | Omnitruncated 4-simplex honeycomb |
|---|---|
| bgcolor=#ffffff align=center colspan=2 | (No image) |
| Type | Uniform 4-honeycomb |
| Family | Omnitruncated simplectic honeycomb |
| Schläfli symbol | t0,1,2,3,4 or tr |
| Coxeter diagram | |
| 4-face types | |
| Cell types | |
| Face types | |
| Vertex figure | Irr. 5-cell |
| Symmetry | {\tilde{A}}4 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.[8] 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
A4* latticeThe A 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.[9] ∪ ∪ ∪ ∪ = dual of Alternated formThis 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 alsoRegular and uniform honeycombs in 4-space:
References
] |