# 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.

5-simplex |
Rectified 5-simplex |
Birectified 5-simplex |

Orthogonal projections in A_{5} Coxeter plane |
---|

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 | |

Schläfli symbol | r{3^{4}} or | |

Coxeter diagram | 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 | |

Vertices | 15 | |

Vertex figure | {}x{3,3} | |

Coxeter group | A_{5}, [3^{4}], order 720 | |

Dual | ||

Base point | (0,0,0,0,1,1) | |

Circumradius | 0.645497 | |

Properties | 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 (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}.

### 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.[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]

A_{5} | k-face | f_{k} | f_{0} | f_{1} | f_{2} | f_{3} | f_{4} | k-figure | notes | ||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

A_{3}A_{1} | ( ) | f_{0} |
15 | 8 | 4 | 12 | 6 | 8 | 4 | 2 | {3,3}x{ } | 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}V( ) | 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} | 3 | 3 | * | 60 | 1 | 2 | 2 | 1 | { }x( ) | 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} | 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} | 5 | 10 | 0 | 10 | 0 | 5 | * | 6 | A_{5}/A_{4} = 6!/5! = 6 |

### Images

Stereographic projection of spherical form |

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.

n | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|

Coxeter group |
A_{3}A_{1} |
A_{5} |
D_{6} |
E_{7} |
= E_{7}^{+} |
=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 | |

Schläfli symbol | 2r{3^{4}} = {3^{2,2}}or | |

Coxeter diagram | or | |

4-faces | 12 | 12 r{3,3,3} |

Cells | 60 | 30 {3,3} 30 r{3,3} |

Faces | 120 | 120 {3} |

Edges | 90 | |

Vertices | 20 | |

Vertex figure | {3}x{3} | |

Coxeter group | A_{5}×2, [[3^{4}]], order 1440 | |

Dual | ||

Base point | (0,0,0,1,1,1) | |

Circumradius | 0.866025 | |

Properties | 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

_{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.[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]

A_{5} | k-face | f_{k} | f_{0} | f_{1} | f_{2} | f_{3} | f_{4} | k-figure | notes | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

A_{2}A_{2} | ( ) | f_{0} |
20 | 9 | 9 | 9 | 3 | 9 | 3 | 3 | 3 | {3}x{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 | { }v( ) | A_{5}/A_{2}A_{1} = 6!/3!/2 = 60 | |

A_{2}A_{1} | 3 | 3 | * | 60 | 0 | 2 | 1 | 1 | 2 | |||||

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} | 6 | 12 | 4 | 4 | * | 30 | * | 1 | 1 | A_{5}/A_{3} = 6!/4! = 30 | |||

A_{3}A_{1} | {3,3} | 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} | 10 | 30 | 10 | 20 | 0 | 5 | 5 | * | 6 |

### Images

The A5 projection has an identical appearance to *Metatron's Cube*.[7]

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.

Space | Finite | Euclidean | Hyperbolic | ||
---|---|---|---|---|---|

n | 4 | 5 | 6 | 7 | 8 |

Coxeter group |
A_{2}A_{2} |
E_{6} |
=E_{6}^{+} |
=E_{6}^{++} | |

Coxeter diagram |
|||||

Symmetry | [[3<sup>2,2,-1</sup>]] | [[3<sup>2,2,0</sup>]] | [[3<sup>2,2,1</sup>]] | [[3<sup>2,2,2</sup>]] | [[3<sup>2,2,3</sup>]] |

Order | 72 | 1440 | 103,680 | ∞ | |

Graph | ∞ | ∞ | |||

Name | −1_{22} |
0_{22} |
1_{22} |
2_{22} |
3_{22} |

#### Isotopics polytopes

Dim. | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|

Name Coxeter |
Hexagon t{3} = {6} |
Octahedron r{3,3} = {3 ^{1,1}} = {3,4} |
Decachoron 2t{3 ^{3}} |
Dodecateron 2r{3 ^{4}} = {3^{2,2}} |
Tetradecapeton 3t{3 ^{5}} |
Hexadecaexon 3r{3 ^{6}} = {3^{3,3}} |
Octadecazetton 4t{3 ^{7}} |

Images | |||||||

Vertex figure | ( )v( ) | { }×{ } |
{ }v{ } |
{3}×{3} |
{3}v{3} |
{3,3}x{3,3} | {3,3}v{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} |

## References

- Coxeter, Regular Polytopes, sec 1.8 Configurations
- Coxeter, Complex Regular Polytopes, p.117
- Klitzing, Richard. "o3x3o3o3o - rix".
- Coxeter, Regular Polytopes, sec 1.8 Configurations
- Coxeter, Complex Regular Polytopes, p.117
- Klitzing, Richard. "o3o3x3o3o - dot".
- Melchizedek, Drunvalo (1999).
*The Ancient Secret of the Flower of Life*.**1**. Light Technology Publishing. p.160 Figure 6-12

- H.S.M. Coxeter:
- H.S.M. Coxeter,
*Regular Polytopes*, 3rd Edition, Dover New York, 1973 **Kaleidoscopes: Selected Writings of H.S.M. Coxeter**, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6- (Paper 22) H.S.M. Coxeter,
*Regular and Semi Regular Polytopes I*, [Math. Zeit. 46 (1940) 380-407, MR 2,10] - (Paper 23) H.S.M. Coxeter,
*Regular and Semi-Regular Polytopes II*, [Math. Zeit. 188 (1985) 559-591] - (Paper 24) H.S.M. Coxeter,
*Regular and Semi-Regular Polytopes III*, [Math. Zeit. 200 (1988) 3-45]

- (Paper 22) H.S.M. Coxeter,

- H.S.M. Coxeter,
- Norman Johnson
*Uniform Polytopes*, Manuscript (1991)- N.W. Johnson:
*The Theory of Uniform Polytopes and Honeycombs*, Ph.D.

- N.W. Johnson:
- Klitzing, Richard. "5D uniform polytopes (polytera)". o3x3o3o3o - rix, o3o3x3o3o - dot

## External links

- Glossary for hyperspace, George Olshevsky.
- Polytopes of Various Dimensions, Jonathan Bowers
- Rectified uniform polytera (Rix), Jonathan Bowers

- Multi-dimensional Glossary