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 03,1 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.[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.
| A5 | k-face | fk | f0 | f1 | f2 | f3 | f4 | k-figure | notes | |||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| A3A1 | f0 | 15 | 8 | 4 | 12 | 6 | 8 | 4 | 2 | A5/A3A1 = 6!/4 | /2 = 15 | |||
| A2A1 | f1 | 2 | 60 | 1 | 3 | 3 | 3 | 3 | 1 | A5/A2A1 = 6!/3 | /2 = 60 | |||
| A2A2 | r | f2 | 3 | 3 | 20 | 3 | 0 | 3 | 0 | A5/A2A2 = 6!/3 | /3! =20 | |||
| A2A1 | 3 | 3 | 60 | 1 | 2 | 2 | 1 | A5/A2A1 = 6!/3 | /2 = 60 | |||||
| A3A1 | r | f3 | 6 | 12 | 4 | 4 | 15 | 2 | 0 | A5/A3A1 = 6!/4 | /2 = 15 | |||
| A3 | 4 | 6 | 0 | 4 | 30 | 1 | 1 | A5/A3 = 6!/4 | = 30 | |||||
| A4 | r | f4 | 10 | 30 | 10 | 20 | 5 | 5 | 6 | A5/A4 = 6!/5 | = 6 | |||
| A4 | 5 | 10 | 0 | 10 | 0 | 5 | 6 | A5/A4 = 6!/5 | = 6 | |||||
The rectified 5-simplex, 031, is second in a dimensional series of uniform polytopes, expressed by Coxeter as 13k series. The fifth figure is a Euclidean honeycomb, 331, and the final is a noncompact hyperbolic honeycomb, 431. 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 02,2 for its branching Coxeter-Dynkin diagram, shown as . It is seen in the vertex figure of the 6-dimensional 122, .
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.[3] [4]
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.
| A5 | k-face | fk | f0 | f1 | f2 | f3 | f4 | k-figure | notes | ||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| A2A2 | f0 | 20 | 9 | 9 | 9 | 3 | 9 | 3 | 3 | 3 | A5/A2A2 = 6!/3 | /3! = 20 | |||
| A1A1A1 | f1 | 2 | 90 | 2 | 2 | 1 | 4 | 1 | 2 | 2 | A5/A1A1A1 = 6!/2/2/2 = 90 | ||||
| A2A1 | f2 | 3 | 3 | 60 | 1 | 2 | 0 | 2 | 1 | A5/A2A1 = 6 | /3!/2 = 60 | ||||
| A2A1 | 3 | 3 | 60 | 0 | 2 | 1 | 1 | 2 | |||||||
| A3A1 | f3 | 4 | 6 | 4 | 0 | 15 | 2 | 0 | A5/A3A1 = 6!/4 | /2 = 15 | |||||
| A3 | r | 6 | 12 | 4 | 4 | 30 | 1 | 1 | A5/A3 = 6!/4 | = 30 | |||||
| A3A1 | 4 | 6 | 0 | 4 | 15 | 0 | 2 | A5/A3A1 = 6!/4 | /2 = 15 | ||||||
| A4 | r | f4 | 10 | 30 | 20 | 10 | 5 | 5 | 0 | 6 | A5/A4 = 6 | /5! = 6 | |||
| A4 | 10 | 30 | 10 | 20 | 0 | 5 | 5 | 6 | |||||||
The A5 projection has an identical appearance to Metatron's Cube.[5]
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, 022, is second in a dimensional series of uniform polytopes, expressed by Coxeter as k22 series. The birectified 5-simplex is the vertex figure for the third, the 122. The fourth figure is a Euclidean honeycomb, 222, and the final is a noncompact hyperbolic honeycomb, 322. 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 231 polytope.
It is also one of 19 uniform polytera based on the [3,3,3,3] Coxeter group, all shown here in A5 Coxeter plane orthographic projections. (Vertices are colored by projection overlap order, red, orange, yellow, green, cyan, blue, purple having progressively more vertices)