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.

 Orthogonal projections in A5 Coxeter plane 5-simplex Rectified 5-simplex Birectified 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-faces126 {3,3,3}
6 r{3,3,3}
Cells4515 {3,3}
30 r{3,3}
Faces8080 {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.

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]

A5k-facefkf0f1f2f3f4k-figurenotes
A3A1( ) f0 1584126842{3,3}x{ }A5/A3A1 = 6!/4!/2 = 15
A2A1{ } f1 260133331{3}V( )A5/A2A1 = 6!/3!/2 = 60
A2A2r{3} f2 3320*3030{3}A5/A2A2 = 6!/3!/3! =20
A2A1{3} 33*601221{ }x( )A5/A2A1 = 6!/3!/2 = 60
A3A1r{3,3} f3 6124415*20{ }A5/A3A1 = 6!/4!/2 = 15
A3{3,3} 4604*3011A5/A3 = 6!/4! = 30
A4r{3,3,3} f4 10301020556*( )A5/A4 = 6!/5! = 6
A4{3,3,3} 51001005*6A5/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]

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-faces1212 r{3,3,3}
Cells6030 {3,3}
30 r{3,3}
Faces120120 {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]

A5k-facefkf0f1f2f3f4k-figurenotes
A2A2( ) f0 2099939333{3}x{3}A5/A2A2 = 6!/3!/3! = 20
A1A1A1{ } f1 2902214122{ }∨{ }A5/A1A1A1 = 6!/2/2/2 = 90
A2A1{3} f2 3360*12021{ }v( )A5/A2A1 = 6!/3!/2 = 60
A2A1 33*6002112
A3A1{3,3} f3 464015**20{ }A5/A3A1 = 6!/4!/2 = 15
A3r{3,3} 61244*30*11A5/A3 = 6!/4! = 30
A3A1{3,3} 4604**1502A5/A3A1 = 6!/4!/2 = 15
A4r{3,3,3} f4 103020105506*( )A5/A4 = 6!/5! = 6
A4 10301020055*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.

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 [[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 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

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
• (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