# Rectified 5-simplexes

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 (15 tetrahedral, and 30 octahedral), and 12 4-faces (6 5-cell and 6 rectified 5-cells). It is also called **0 _{3,1}** for its branching Coxeter-Dynkin diagram, shown as .

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S^{1}_{5}.

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.^{[1]}^{[2]}

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.^{[3]}

The rectified 5-simplex, 0_{31}, is second in a dimensional series of uniform polytopes, expressed by Coxeter as 1_{3k} series. The fifth figure is a Euclidean honeycomb, 3_{31}, and the final is a noncompact hyperbolic honeycomb, 4_{31}. 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^{2}_{5}.

It is also called **0 _{2,2}** for its branching Coxeter-Dynkin diagram, shown as . It is seen in the vertex figure of the 6-dimensional 1

_{22}, .

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.^{[4]}^{[5]}

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.^{[6]}

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,−1,−1,−1).

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_{22}, is second in a dimensional series of uniform polytopes, expressed by Coxeter as k_{22} series. The *birectified 5-simplex* is the vertex figure for the third, the 1_{22}. The fourth figure is a Euclidean honeycomb, 2_{22}, and the final is a noncompact hyperbolic honeycomb, 3_{22}. 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_{31} polytope.

It is also one of 19 uniform polytera based on the [3,3,3,3] Coxeter group, all shown here in A_{5} Coxeter plane orthographic projections. (Vertices are colored by projection overlap order, red, orange, yellow, green, cyan, blue, purple having progressively more vertices)