# Rectified 5-simplexes

 Orthogonal projections in A5 Coxeter plane 5-simplex Rectified 5-simplex Birectified 5-simplex

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
Schläfli symbol r{34} or ${\displaystyle \left\{{\begin{array}{l}3,3,3\\3\end{array}}\right\}}$
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 A5, [34], order 720
Dual
Base point (0,0,0,0,1,1)
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 03,1 for its branching Coxeter-Dynkin diagram, shown as .

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S1
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.

### 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.[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]

A5 k-face fk f0 f1 f2 f3 f4 k-figure notes
A3A1 ( ) f0 15 8 4 12 6 8 4 2 {3,3}x{ } A5/A3A1 = 6!/4!/2 = 15
A2A1 { } f1 2 60 1 3 3 3 3 1 {3}V( ) A5/A2A1 = 6!/3!/2 = 60
A2A2 r{3} f2 3 3 20 * 3 0 3 0 {3} A5/A2A2 = 6!/3!/3! =20
A2A1 {3} 3 3 * 60 1 2 2 1 { }x( ) A5/A2A1 = 6!/3!/2 = 60
A3A1 r{3,3} f3 6 12 4 4 15 * 2 0 { } A5/A3A1 = 6!/4!/2 = 15
A3 {3,3} 4 6 0 4 * 30 1 1 A5/A3 = 6!/4! = 30
A4 r{3,3,3} f4 10 30 10 20 5 5 6 * ( ) A5/A4 = 6!/5! = 6
A4 {3,3,3} 5 10 0 10 0 5 * 6 A5/A4 = 6!/5! = 6

### Images

 Stereographic projection of spherical form
orthographic projections
Ak
Coxeter plane
A5 A4
Graph
Dihedral symmetry [6] [5]
Ak
Coxeter plane
A3 A2
Graph
Dihedral symmetry [4] [3]

### Related polytopes

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.

k31 dimensional figures
n 4 5 6 7 8 9
Coxeter
group
A3A1 A5 D6 E7 ${\displaystyle {\tilde {E}}_{7}}$ = E7+ ${\displaystyle {\bar {T}}_{8}}$=E7++
Coxeter
diagram
Symmetry [3−1,3,1] [30,3,1] [31,3,1] [32,3,1] [33,3,1] [34,3,1]
Order 48 720 23,040 2,903,040
Graph - -
Name −131 031 131 231 331 431

## Birectified 5-simplex

Birectified 5-simplex
Birectified hexateron (dot)
Type uniform 5-polytope
Schläfli symbol 2r{34} = {32,2}
or ${\displaystyle \left\{{\begin{array}{l}3,3\\3,3\end{array}}\right\}}$
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 A5×2, [[34]], order 1440
Dual
Base point (0,0,0,1,1,1)
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 S2
5
.

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, .

### 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]

A5 k-face fk f0 f1 f2 f3 f4 k-figure notes
A2A2 ( ) f0 20 9 9 9 3 9 3 3 3 {3}x{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 {3} f2 3 3 60 * 1 2 0 2 1 { }v( ) A5/A2A1 = 6!/3!/2 = 60
A2A1 3 3 * 60 0 2 1 1 2
A3A1 {3,3} f3 4 6 4 0 15 * * 2 0 { } A5/A3A1 = 6!/4!/2 = 15
A3 r{3,3} 6 12 4 4 * 30 * 1 1 A5/A3 = 6!/4! = 30
A3A1 {3,3} 4 6 0 4 * * 15 0 2 A5/A3A1 = 6!/4!/2 = 15
A4 r{3,3,3} f4 10 30 20 10 5 5 0 6 * ( ) A5/A4 = 6!/5! = 6
A4 10 30 10 20 0 5 5 * 6

### Images

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

orthographic projections
Ak
Coxeter plane
A5 A4
Graph
Dihedral symmetry [6] [[5]]=[10]
Ak
Coxeter plane
A3 A2
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, 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.

k22 figures in n dimensions
Space Finite Euclidean Hyperbolic
n 4 5 6 7 8
Coxeter
group
A2A2 E6 ${\displaystyle {\tilde {E}}_{6}}$=E6+ ${\displaystyle {\bar {T}}_{7}}$=E6++
Coxeter
diagram
Symmetry [[32,2,-1]] [[32,2,0]] [[32,2,1]] [[32,2,2]] [[32,2,3]]
Order 72 1440 103,680
Graph
Name −122 022 122 222 322

#### Isotopics polytopes

Isotopic uniform truncated simplices
Dim. 2 3 4 5 6 7 8
Name
Coxeter
Hexagon
=
t{3} = {6}
Octahedron
=
r{3,3} = {31,1} = {3,4}
${\displaystyle \left\{{\begin{array}{l}3\\3\end{array}}\right\}}$
Decachoron

2t{33}
Dodecateron

2r{34} = {32,2}
${\displaystyle \left\{{\begin{array}{l}3,3\\3,3\end{array}}\right\}}$

3t{35}

3r{36} = {33,3}
${\displaystyle \left\{{\begin{array}{l}3,3,3\\3,3,3\end{array}}\right\}}$

4t{37}
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 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)

## References

1. ^ Coxeter, Regular Polytopes, sec 1.8 Configurations
2. ^ Coxeter, Complex Regular Polytopes, p.117
3. ^ Klitzing, Richard. "o3x3o3o3o - rix".
4. ^ Coxeter, Regular Polytopes, sec 1.8 Configurations
5. ^ Coxeter, Complex Regular Polytopes, p.117
6. ^ Klitzing, Richard. "o3o3x3o3o - dot".
7. ^ 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 [1]
• (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]
• Norman Johnson Uniform Polytopes, Manuscript (1991)
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
• Klitzing, Richard. "5D uniform polytopes (polytera)". o3x3o3o3o - rix, o3o3x3o3o - dot