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Rectified 5-simplexes

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Source updated May 19, 2026

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.

01Rectified 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.

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.

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

02Birectified 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. 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.

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.

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

k₂₂ 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	A₅	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

Isotopic 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	∩	∩	∩	∩	∩	∩	∩

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Sources and credits

This article is adapted from the Wikipedia article “Rectified 5-simplexes”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

Images, from Wikimedia Commons:

5-simplex t0.svg by Tomruen, Public domain

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Rectified 5-simplexes

01Rectified 5-simplex

Alternate names

Coordinates

As a configuration

Images

Related polytopes

02Birectified 5-simplex

Alternate names

Construction

Images

Intersection of two 5-simplices

Related polytopes

k22 polytopes

Isotopic polytopes

03Related uniform 5-polytopes

Sources and credits

k₂₂ polytopes