In five-dimensional geometry, a rectified 5-simplex is a convex uniform 5-polytope, being a rectification of the regular 5-simplex.
There are three unique degrees of rectifications, including the zeroth, the 5-simplex itself. Vertices of the rectified 5-simplex are located at the edge-centers of the 5-simplex. Vertices of the birectified 5-simplex are located in the triangular face centers of the 5-simplex.
In five-dimensional geometry, a rectified 5-simplex is a uniform 5-polytope with 15 vertices, 60 edges, 80 triangular faces, 45 cells (30 tetrahedral, and 15 octahedral), and 12 4-faces (6 5-cell and 6 rectified 5-cells). It is also called 0<sub>3,1</sub> for its branching Coxeter-Dynkin diagram, shown as .
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S.
The vertices of the rectified 5-simplex can be more simply positioned on a hyperplane in 6-space as permutations of (0,0,0,0,1,1) or (0,0,1,1,1,1). These construction can be seen as facets of the rectified 6-orthoplex or birectified 6-cube respectively.
This configuration matrix represents the rectified 5-simplex. The rows and columns correspond to vertices, edges, faces, cells and 4-faces. The diagonal numbers say how many of each element occur in the whole rectified 5-simplex. The nondiagonal numbers say how many of the column's element occur in or at the row's element.
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.
The rectified 5-simplex, 0<sub>31</sub>, is second in a dimensional series of uniform polytopes, expressed by Coxeter as 1<sub>3k</sub> series. The fifth figure is a Euclidean honeycomb, 3<sub>31</sub>, and the final is a noncompact hyperbolic honeycomb, 4<sub>31</sub>. Each progressive uniform polytope is constructed from the previous as its vertex figure.
The birectified 5-simplex is isotopic, with all 12 of its facets as rectified 5-cells. It has 20 vertices, 90 edges, 120 triangular faces, 60 cells (30 tetrahedral, and 30 octahedral).
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S.
It is also called 0<sub>2,2</sub> for its branching Coxeter-Dynkin diagram, shown as . It is seen in the vertex figure of the 6-dimensional 1<sub>22</sub>, .
The elements of the regular polytopes can be expressed in a configuration matrix. Rows and columns reference vertices, edges, faces, and cells, with diagonal element their counts (f-vectors). The nondiagonal elements represent the number of row elements are incident to the column element.
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.
The A5 projection has an identical appearance to Metatron's Cube.
The birectified 5-simplex is the intersection of two regular 5-simplexes in dual configuration. The vertices of a birectification exist at the center of the faces of the original polytope(s). This intersection is analogous to the 3D stellated octahedron, seen as a compound of two regular tetrahedra and intersected in a central octahedron, while that is a first rectification where vertices are at the center of the original edges.
It is also the intersection of a 6-cube with the hyperplane that bisects the 6-cube's long diagonal orthogonally. In this sense it is the 5-dimensional analog of the regular hexagon, octahedron, and bitruncated 5-cell. This characterization yields simple coordinates for the vertices of a birectified 5-simplex in 6-space: the 20 distinct permutations of (1,1,1,âÂÂ1,âÂÂ1,âÂÂ1).
The vertices of the birectified 5-simplex can also be positioned on a hyperplane in 6-space as permutations of (0,0,0,1,1,1). This construction can be seen as facets of the birectified 6-orthoplex.
The birectified 5-simplex, 0<sub>22</sub>, is second in a dimensional series of uniform polytopes, expressed by Coxeter as k<sub>22</sub> series. The birectified 5-simplex is the vertex figure for the third, the 1<sub>22</sub>. The fourth figure is a Euclidean honeycomb, 2<sub>22</sub>, and the final is a noncompact hyperbolic honeycomb, 3<sub>22</sub>. Each progressive uniform polytope is constructed from the previous as its vertex figure.
This polytope is the vertex figure of the 6-demicube, and the edge figure of the uniform 2<sub>31</sub> polytope.
It is also one of 19 uniform polytera based on the [3,3,3,3] Coxeter group, all shown here in A<sub>5</sub> Coxeter plane orthographic projections. (Vertices are colored by projection overlap order, red, orange, yellow, green, cyan, blue, purple having progressively more vertices)