# Rectified 5-simplexes

| 5-simplex | Rectified 5-simplex | Birectified 5-simplex |
| Orthogonal projections in A$_5$ Coxeter plane |||
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.

## Rectified 5-simplex

Rectified 5-simplex Rectified hexateron (rix) :
| Type | uniform 5-polytope | uniform 5-polytope |
| Schläfli symbol | r{3$^4$} or $\left\{{\begin{array}{l}3,3,3\\3\end{array}}\right\}$ | r{3$^4$} or $\left\{{\begin{array}{l}3,3,3\\3\end{array}}\right\}$ |
| Coxeter diagram | or | or |
| 4-faces | 12 | 6 {3,3,3} 6 r{3,3,3} |
| Cells | 45 | 15 {3,3} 30 r{3,3} |
| Faces | 80 | 80 {3} |
| Edges | 60 | 60 |
| Vertices | 15 | 15 |
| Vertex figure |×{3,3} |×{3,3} |
| Coxeter group | A$_5$, [3$^4$], order 720 | A$_5$, [3$^4$], order 720 |
| Dual | | |
| Base point | (0,0,0,0,1,1) | (0,0,0,0,1,1) |
| Circumradius | 0.645497 | 0.645497 |
| Properties | convex, isogonal isotoxal | convex, isogonal isotoxal |
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$_{3,1}$ for its branching Coxeter-Dynkin diagram, shown as .
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S$^1$ $_5$.

### Alternate names

* Rectified hexateron (Acronym: rix) (Jonathan Bowers)

### Coordinates

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.

### As a configuration

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.
| A$_5$ | k-face | f$_k$ | f$_0$ | f$_1$ | f$_2$ | f$_2$ | f$_3$ | f$_3$ | f$_4$ | f$_4$ | k-figure | notes |
| A$_3$A$_1$ | ( ) | f$_0$ | 15 | 8 | 4 | 12 | 6 | 8 | 4 | 2 | {3,3}×{ } | A$_5$/A$_3$A$_1$ = 6!/4!/2 = 15 |
| A$_2$A$_1$ | { } | f$_1$ | 2 | 60 | 1 | 3 | 3 | 3 | 3 | 1 | {3}∨( ) | A$_5$/A$_2$A$_1$ = 6!/3!/2 = 60 |
| A$_2$A$_2$ | r{3} | f$_2$ | 3 | 3 | 20 | * | 3 | 0 | 3 | 0 | {3} | A$_5$/A$_2$A$_2$ = 6!/3!/3! =20 |
| A$_2$A$_1$ | {3} | f$_2$ | 3 | 3 | * | 60 | 1 | 2 | 2 | 1 | { }×( ) | A$_5$/A$_2$A$_1$ = 6!/3!/2 = 60 |
| A$_3$A$_1$ | r{3,3} | f$_3$ | 6 | 12 | 4 | 4 | 15 | * | 2 | 0 | { } | A$_5$/A$_3$A$_1$ = 6!/4!/2 = 15 |
| A$_3$ | {3,3} | f$_3$ | 4 | 6 | 0 | 4 | * | 30 | 1 | 1 | { } | A$_5$/A$_3$ = 6!/4! = 30 |
| A$_4$ | r{3,3,3} | f$_4$ | 10 | 30 | 10 | 20 | 5 | 5 | 6 | * | ( ) | A$_5$/A$_4$ = 6!/5! = 6 |
| A$_4$ | {3,3,3} | f$_4$ | 5 | 10 | 0 | 10 | 0 | 5 | * | 6 | ( ) | A$_5$/A$_4$ = 6!/5! = 6 |

### Images

Stereographic projection :
* Stereographic projection of spherical form
orthographic projections :
| A$_k$ Coxeter plane | A$_5$ | A$_4$ |
| Graph | | |
| Dihedral symmetry | [6] | [5] |
| A$_k$ Coxeter plane | A$_3$ | A$_2$ |
| Graph | | |
| Dihedral symmetry | [4] | [3] |

### Related polytopes

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.
k$_{31}$ dimensional figures :
| n | 4 | 5 | 6 | 7 | 8 | 9 |
| Coxeter group | A$_3$A$_1$ | A$_5$ | D$_6$ | E$_7$ | ${\tilde {E}}_{7}$ = E$_7$$^+$ | ${\bar {T}}_{8}$=E$_7$$^{++}$ |
| Coxeter diagram | | | | | | |
| Symmetry | [3$^{−1,3,1}$] | [3$^{0,3,1}$] | [3$^{1,3,1}$] | [3$^{2,3,1}$] | [3$^{3,3,1}$] | [3$^{4,3,1}$] |
| Order | 48 | 720 | 23,040 | 2,903,040 | ∞ | ∞ |
| Graph | | | | | - | - |
| Name | −1$_{31}$ | 0$_{31}$ | 1$_{31}$ | 2$_{31}$ | 3$_{31}$ | 4$_{31}$ |

## Birectified 5-simplex

Birectified 5-simplex Birectified hexateron (dot) :
| Type | uniform 5-polytope | uniform 5-polytope |
| Schläfli symbol | 2r{3$^4$} = {3$^{2,2}$} or $\left\{{\begin{array}{l}3,3\\3,3\end{array}}\right\}$ | 2r{3$^4$} = {3$^{2,2}$} or $\left\{{\begin{array}{l}3,3\\3,3\end{array}}\right\}$ |
| Coxeter diagram | or | or |
| 4-faces | 12 | 12 r{3,3,3} |
| Cells | 60 | 30 {3,3} 30 r{3,3} |
| Faces | 120 | 120 {3} |
| Edges | 90 | 90 |
| Vertices | 20 | 20 |
| Vertex figure | {3}×{3} | {3}×{3} |
| Coxeter group | A$_5$×2, [[3$^4$]], order 1440 | A$_5$×2, [[3$^4$]], order 1440 |
| Dual | | |
| Base point | (0,0,0,1,1,1) | (0,0,0,1,1,1) |
| Circumradius | 0.866025 | 0.866025 |
| Properties | convex, isogonal isotoxal | convex, isogonal isotoxal |
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$_{22}$, .

### Alternate names

* Birectified hexateron
* dodecateron (Acronym: dot) (For 12-facetted polyteron) (Jonathan Bowers)

### Construction

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.
| A$_5$ | k-face | f$_k$ | f$_0$ | f$_1$ | f$_2$ | f$_2$ | f$_3$ | f$_3$ | f$_3$ | f$_4$ | f$_4$ | k-figure | notes |
| A$_2$A$_2$ | ( ) | f$_0$ | 20 | 9 | 9 | 9 | 3 | 9 | 3 | 3 | 3 | {3}×{3} | A$_5$/A$_2$A$_2$ = 6!/3!/3! = 20 |
| A$_1$A$_1$A$_1$ | { } | f$_1$ | 2 | 90 | 2 | 2 | 1 | 4 | 1 | 2 | 2 | { }∨{ } | A$_5$/A$_1$A$_1$A$_1$ = 6!/2/2/2 = 90 |
| A$_2$A$_1$ | {3} | f$_2$ | 3 | 3 | 60 | * | 1 | 2 | 0 | 2 | 1 | { }∨( ) | A$_5$/A$_2$A$_1$ = 6!/3!/2 = 60 |
| A$_2$A$_1$ | {3} | f$_2$ | 3 | 3 | * | 60 | 0 | 2 | 1 | 1 | 2 | { }∨( ) | A$_5$/A$_2$A$_1$ = 6!/3!/2 = 60 |
| A$_3$A$_1$ | {3,3} | f$_3$ | 4 | 6 | 4 | 0 | 15 | * | * | 2 | 0 | { } | A$_5$/A$_3$A$_1$ = 6!/4!/2 = 15 |
| A$_3$ | r{3,3} | f$_3$ | 6 | 12 | 4 | 4 | * | 30 | * | 1 | 1 | { } | A$_5$/A$_3$ = 6!/4! = 30 |
| A$_3$A$_1$ | {3,3} | f$_3$ | 4 | 6 | 0 | 4 | * | * | 15 | 0 | 2 | { } | A$_5$/A$_3$A$_1$ = 6!/4!/2 = 15 |
| A$_4$ | r{3,3,3} | f$_4$ | 10 | 30 | 20 | 10 | 5 | 5 | 0 | 6 | * | ( ) | A$_5$/A$_4$ = 6!/5! = 6 |
| A$_4$ | r{3,3,3} | f$_4$ | 10 | 30 | 10 | 20 | 0 | 5 | 5 | * | 6 | ( ) | A$_5$/A$_4$ = 6!/5! = 6 |

### Images

The A5 projection has an identical appearance to Metatron's Cube.
orthographic projections :
| A$_k$ Coxeter plane | A$_5$ | A$_4$ |
| Graph | | |
| Dihedral symmetry | [6] | [[5]]=[10] |
| A$_k$ Coxeter plane | A$_3$ | A$_2$ |
| Graph | | |
| Dihedral symmetry | [4] | [[3]]=[6] |

### Intersection of two 5-simplices

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.
* Dual 5-simplexes (red and blue), and their birectified 5-simplex intersection in green, viewed in A5 and A4 Coxeter planes. The simplexes overlap in the A5 projection and are drawn in magenta.
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.

### Related polytopes
#### k_22 polytopes

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.
k$_{22}$ figures in n dimensions :
| Space | Finite | Finite | Finite | Euclidean | Hyperbolic |
| n | 4 | 5 | 6 | 7 | 8 |
| Coxeter group | A$_2$A$_2$ | E$_6$ | ${\tilde {E}}_{6}$=E$_6$$^+$ | ${\bar {T}}_{7}$=E$_6$$^{++}$ |
| Coxeter diagram | | | | | |
| Symmetry | [[3$^{2,2,-1}$]] | [[3$^{2,2,0}$]] | [[3$^{2,2,1}$]] | [[3$^{2,2,2}$]] | [[3$^{2,2,3}$]] |
| Order | 72 | 1440 | 103,680 | ∞ | ∞ |
| Graph | | | | ∞ | ∞ |
| Name | −1$_{22}$ | 0$_{22}$ | 1$_{22}$ | 2$_{22}$ | 3$_{22}$ |

#### Isotopics polytopes

Isotopic uniform truncated simplices :
| Dim. | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| Name Coxeter | Hexagon = t{3} = {6} | Octahedron = r{3,3} = {3$^{1,1}$} = {3,4} $\left\{{\begin{array}{l}3\\3\end{array}}\right\}$ | Decachoron 2t{3$^3$} | Dodecateron 2r{3$^4$} = {3$^{2,2}$} $\left\{{\begin{array}{l}3,3\\3,3\end{array}}\right\}$ | Tetradecapeton 3t{3$^5$} | Hexadecaexon 3r{3$^6$} = {3$^{3,3}$} $\left\{{\begin{array}{l}3,3,3\\3,3,3\end{array}}\right\}$ | Octadecazetton 4t{3$^7$} |
| Images | | | | | | | |
| Vertex figure | ( )∨( ) | { }×{ } | { }∨{ } | {3}×{3} | {3}∨{3} | {3,3}×{3,3} | {3,3}∨{3,3} |
| Facets | | {3} | t{3,3} | r{3,3,3} | 2t{3,3,3,3} | 2r{3,3,3,3,3} | 3t{3,3,3,3,3,3} |
| As intersecting dual simplexes | ∩ | ∩ | ∩ | ∩ | ∩ | ∩ | ∩ |

## Related uniform 5-polytopes

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)
A5 polytopes :
| t$_0$ | t$_1$ | t$_2$ | t$_{0,1}$ | t$_{0,2}$ | t$_{1,2}$ | t$_{0,3}$ |
| t$_{1,3}$ | t$_{0,4}$ | t$_{0,1,2}$ | t$_{0,1,3}$ | t$_{0,2,3}$ | t$_{1,2,3}$ | t$_{0,1,4}$ |
| t$_{0,2,4}$ | t$_{0,1,2,3}$ | t$_{0,1,2,4}$ | t$_{0,1,3,4}$ | t$_{0,1,2,3,4}$ |