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.
Contents
- Alternate names
- Projection by folding
- A4 lattice
- Related polytopes and honeycombs
- Rectified 5 cell honeycomb
- Cyclotruncated 5 cell honeycomb
- Truncated 5 cell honeycomb
- Alaternate names
- Cantellated 5 cell honeycomb
- Bitruncated 5 cell honeycomb
- Omnitruncated 5 cell honeycomb
- References
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.
Alternate names
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:
A4 lattice
This vertex arrangement 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
The A*
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
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.
This honeycomb is one of seven unique uniform honeycombs constructed by the
Rectified 5-cell honeycomb
The rectified 4-simplex honeycomb or rectified 5-cell honeycomb is a space-filling tessellation honeycomb.
Alternate names
Cyclotruncated 5-cell honeycomb
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.
Alternate names
Truncated 5-cell honeycomb
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
Cantellated 5-cell honeycomb
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
Bitruncated 5-cell honeycomb
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
Omnitruncated 5-cell honeycomb
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.
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
A4* lattice
The A*
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.