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 $\left\{{\begin{array}{l}3,3,3\\3\end{array}}\right\}$
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	{}×{3,3}
Coxeter group	A₅, [3⁴], 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 (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 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]

A₅	k-face	f_k	f₀	f₁	f₂		f₃		f₄		k-figure	notes
A₃A₁	( )	f₀	15	8	4	12	6	8	4	2	{3,3}×{ }	A₅/A₃A₁ = 6!/4!/2 = 15
A₂A₁	{ }	f₁	2	60	1	3	3	3	3	1	{3}∨( )	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}	f₂	3	3	*	60	1	2	2	1	{ }×( )	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}	f₃	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}	f₄	5	10	0	10	0	5	*	6	( )	A₅/A₄ = 6!/5! = 6

Images

Stereographic projection
Stereographic projection of spherical form

orthographic projections
A_k Coxeter plane	A₅	A₄
Graph
Dihedral symmetry	[6]	[5]
A_k 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 1_3k 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	${\tilde {E}}_{7}$ = E₇⁺	${\bar {T}}_{8}$ =E₇⁺⁺
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	−1₃₁	0₃₁	1₃₁	2₃₁	3₃₁	4₃₁

Birectified 5-simplex

Birectified 5-simplex Birectified hexateron (dot)
Type	uniform 5-polytope
Schläfli symbol	2r{3⁴} = {3^2,2} or $\left\{{\begin{array}{l}3,3\\3,3\end{array}}\right\}$
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}×{3}
Coxeter group	A₅×2, [[3⁴]], 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 S²
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 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	f_k	f₀	f₁	f₂		f₃			f₄		k-figure	notes
A₂A₂	( )	f₀	20	9	9	9	3	9	3	3	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	{3}	f₂	3	3	60	*	1	2	0	2	1	{ }∨( )	A₅/A₂A₁ = 6!/3!/2 = 60
A₂A₁	{3}	f₂	3	3	*	60	0	2	1	1	2	{ }∨( )	A₅/A₂A₁ = 6!/3!/2 = 60
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	r{3,3,3}	f₄	10	30	10	20	0	5	5	*	6	( )	A₅/A₄ = 6!/5! = 6

Images

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

orthographic projections
A_k Coxeter plane	A₅	A₄
Graph
Dihedral symmetry	[6]	[[5]]=[10]
A_k Coxeter plane	A₃	A₂
Graph
Dihedral symmetry	[4]	[[3]]=[6]

Intersection of two 5-simplices

Stereographic projection

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	${\tilde {E}}_{6}$ =E₆⁺	${\bar {T}}_{7}$ =E₆⁺⁺
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	−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} = {3^1,1} = {3,4} $\left\{{\begin{array}{l}3\\3\end{array}}\right\}$	Decachoron 2t{3³}	Dodecateron 2r{3⁴} = {3^2,2} $\left\{{\begin{array}{l}3,3\\3,3\end{array}}\right\}$	Tetradecapeton 3t{3⁵}	Hexadecaexon 3r{3⁶} = {3^3,3} $\left\{{\begin{array}{l}3,3,3\\3,3,3\end{array}}\right\}$	Octadecazetton 4t{3⁷}
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 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)

A5 polytopes
t₀	t₁	t₂	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. Vol. 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

External links

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

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	A_n	B_n	I₂(p) / D_n	E₆ / E₇ / E₈ / F₄ / G₂	H_n
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	Pentachoron	16-cell • Tesseract	Demitesseract	24-cell	120-cell • 600-cell
Uniform 5-polytope	5-simplex	5-orthoplex • 5-cube	5-demicube
Uniform 6-polytope	6-simplex	6-orthoplex • 6-cube	6-demicube	1₂₂ • 2₂₁
Uniform 7-polytope	7-simplex	7-orthoplex • 7-cube	7-demicube	1₃₂ • 2₃₁ • 3₂₁
Uniform 8-polytope	8-simplex	8-orthoplex • 8-cube	8-demicube	1₄₂ • 2₄₁ • 4₂₁
Uniform 9-polytope	9-simplex	9-orthoplex • 9-cube	9-demicube
Uniform 10-polytope	10-simplex	10-orthoplex • 10-cube	10-demicube
Uniform n-polytope	n-simplex	n-orthoplex • n-cube	n-demicube	1_k2 • 2_k1 • k₂₁	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds

[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. Vol. 1. Light Technology Publishing. p.160 Figure 6-12

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[2]

[3]

[4]

[5]

[6]

[7]