Rectified 5-cell
In four-dimensional geometry, the rectified 5-cell is a uniform 4-polytope composed of 5 regular tetrahedral and 5 regular octahedral cells. Each edge has one tetrahedron and two octahedra. Each vertex has two tetrahedra and three octahedra. In total it has 30 triangle faces, 30 edges, and 10 vertices. Each vertex is surrounded by 3 octahedra and 2 tetrahedra; the vertex figure is a triangular prism.
Topologically, under its highest symmetry, , there is only one geometrical form, containing 5 regular tetrahedra and 5 rectified tetrahedra. It is also topologically identical to a tetrahedron-octahedron segmentochoron.
The vertex figure of the rectified 5-cell is a uniform triangular prism, formed by three octahedra around the sides, and two tetrahedra on the opposite ends.
Despite having the same number of vertices as cells and the same number of edges as faces, the rectified 5-cell is not self-dual because the vertex figure is not a dual of the polychoron's cells.
Wythoff construction
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.Structure
Together with the simplex and 24-cell, this shape and its dual was one of the first 2-simple 2-simplicial 4-polytopes known. This means that all of its two-dimensional faces, and all of the two-dimensional faces of its dual, are triangles. In 1997, Tom Braden found another dual pair of examples, by gluing two rectified 5-cells together; since then, infinitely many 2-simple 2-simplicial polytopes have been constructed.Semiregular polytope
It is one of three semiregular 4-polytope made of two or more cells which are Platonic solids, discovered by Thorold Gosset in his 1900 paper. He called it a tetroctahedric for being made of tetrahedron and octahedron cells.E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as tC5.
Alternate names
- Tetroctahedric
- Dispentachoron
- Rectified 5-cell
- Rectified 4-simplex
- Fully truncated 4-simplex
- Rectified pentachoron
- Ambopentachoron
- -hypersimplex
Images
stereographic projection | Net |
| Tetrahedron-centered perspective projection into 3D space, with nearest tetrahedron to the 4D viewpoint rendered in red, and the 4 surrounding octahedra in green. Cells lying on the far side of the polytope have been culled for clarity. The rotation is only of the 3D projection image, in order to show its structure, not a rotation in 4D space. |
Coordinates
The Cartesian coordinates of the vertices of an origin-centered rectified 5-cell having edge length 2 are:More simply, the vertices of the rectified 5-cell can be positioned on a hyperplane in 5-space as permutations of or. These construction can be seen as positive orthant facets of the rectified pentacross or birectified penteract respectively.
Related polytopes
The convex hull of the rectified 5-cell and its dual is a nonuniform polychoron composed of 30 cells: 10 tetrahedra, 20 octahedra, and 20 vertices. Its vertex figure is a triangular bifrustum.Related 4-polytopes
This polytope is the vertex figure of the 5-demicube, and the edge figure of the uniform 221 polytope.It is also one of 9 Uniform 4-polytopes constructed from the Coxeter group.