Truncated 5-cell
In geometry, a truncated 5-cell is a uniform 4-polytope formed as the truncation of the regular 5-cell.
There are two degrees of truncations, including a bitruncation.
Truncated 5-cell
The truncated 5-cell, truncated pentachoron or truncated 4-simplex is bounded by 10 cells: 5 tetrahedra, and 5 truncated tetrahedra. Each vertex is surrounded by 3 truncated tetrahedra and one tetrahedron; the vertex figure is an elongated tetrahedron.Construction
The truncated 5-cell may be constructed from the 5-cell by truncating its vertices at 1/3 of its edge length. This transforms the 5 tetrahedral cells into truncated tetrahedra, and introduces 5 new tetrahedral cells positioned near the original vertices.Structure
The truncated tetrahedra are joined to each other at their hexagonal faces, and to the tetrahedra at their triangular faces.Seen in a configuration matrix, all incidence counts between elements are shown. The diagonal f-vector numbers are derived through the Wythoff construction, dividing the full group order of a subgroup order by removing one mirror at a time.
Projections
The tetrahedron-first parallel projection of the truncated 5-cell into 3-dimensional space has the following structure:- The projection envelope is a truncated tetrahedron.
- One of the truncated tetrahedral cells project onto the entire envelope.
- One of the tetrahedral cells project onto a tetrahedron lying at the center of the envelope.
- Four flattened tetrahedra are joined to the triangular faces of the envelope, and connected to the central tetrahedron via 4 radial edges. These are the images of the remaining 4 tetrahedral cells.
- Between the central tetrahedron and the 4 hexagonal faces of the envelope are 4 irregular truncated tetrahedral volumes, which are the images of the 4 remaining truncated tetrahedral cells.
Images
Alternate names
- Truncated pentatope
- Truncated 4-simplex
- Truncated pentachoron
Coordinates
More simply, the vertices of the truncated 5-cell can be constructed on a hyperplane in 5-space as permutations of or of. These coordinates come from positive orthant facets of the truncated pentacross and bitruncated penteract respectively.
Related polytopes
The convex hull of the truncated 5-cell and its dual is a nonuniform polychoron composed of 60 cells: 10 tetrahedra, 20 octahedra, 30 tetrahedra, and 40 vertices. Its vertex figure is a hexakis triangular cupola.Vertex figure'''
Bitruncated 5-cell
The bitruncated 5-cell is a 4-dimensional polytope, or 4-polytope, composed of 10 cells in the shape of truncated tetrahedra.Topologically, under its highest symmetry,
E. L. Elte identified it in 1912 as a semiregular polytope.
Each hexagonal face of the truncated tetrahedra is joined in complementary orientation to the neighboring truncated tetrahedron. Each edge is shared by two hexagons and one triangle. Each vertex is surrounded by 4 truncated tetrahedral cells in a tetragonal disphenoid vertex figure.
The bitruncated 5-cell is the intersection of two pentachora in dual configuration. As such, it is also the intersection of a penteract with the hyperplane that bisects the penteract's long diagonal orthogonally. In this sense it is a 4-dimensional analog of the regular octahedron and the regular hexagon. The 5-dimensional analog is the birectified 5-simplex, and the -dimensional analog is the polytope whose Coxeter–Dynkin diagram is linear with rings on the middle one or two nodes.
The bitruncated 5-cell is one of the two non-regular uniform 4-polytopes which are cell-transitive. The other is the bitruncated 24-cell, which is composed of 48 truncated cubes.
Symmetry
This 4-polytope has a higher extended pentachoric symmetry, doubled to order 240, because the element corresponding to any element of the underlying 5-cell can be exchanged with one of those corresponding to an element of its dual.Alternative names
- Bitruncated 5-cell
- 10-cell as a cell-transitive 4-polytope
- Bitruncated pentachoron
- Bitruncated pentatope
- Bitruncated 4-simplex
- Decachoron
Images
stereographic projection of spherical 4-polytope | Net |
Coordinates
The Cartesian coordinates of an origin-centered bitruncated 5-cell having edge length 2 are:More simply, the vertices of the bitruncated 5-cell can be constructed on a hyperplane in 5-space as permutations of. These represent positive orthant facets of the bitruncated pentacross. Another 5-space construction, centered on the origin are all 20 permutations of.
Related polytopes
The bitruncated 5-cell can be seen as the intersection of two regular 5-cells in dual positions. = ∩.Related regular skew polyhedron
The regular skew polyhedron,, exists in 4-space with 4 hexagonal around each vertex, in a zig-zagging nonplanar vertex figure. These hexagonal faces can be seen on the bitruncated 5-cell, using all 60 edges and 30 vertices. The 20 triangular faces of the bitruncated 5-cell can be seen as removed. The dual regular skew polyhedron,, is similarly related to the square faces of the runcinated 5-cell.Disphenoidal 30-cell
The disphenoidal 30-cell is the dual of the bitruncated 5-cell. It is a 4-dimensional polytope derived from the 5-cell. It is the convex hull of two 5-cells in opposite orientations.Being the dual of a uniform polychoron, it is cell-transitive, consisting of 30 congruent tetragonal disphenoids. In addition, it is vertex-transitive under the group Aut.