Rectified 5-simplexes

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

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 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	{}×{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.

• Rectified hexateron (Acronym: rix) (Jonathan Bowers)

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.

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	fk	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}		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}		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

Stereographic projection

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 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	${\displaystyle {\tilde {E}}_{7}}$ = 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 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}×{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, .

• Birectified hexateron
• dodecateron (Acronym: dot) (For 12-facetted polyteron) (Jonathan Bowers)

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}×{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	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

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]

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.

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

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} ${\displaystyle \left\{{\begin{array}{l}3,3\\3,3\end{array}}\right\}}$	Tetradecapeton 3t{35}	Hexadecaexon 3r{36} = {33,3} ${\displaystyle \left\{{\begin{array}{l}3,3,3\\3,3,3\end{array}}\right\}}$	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	∩	∩	∩	∩	∩	∩	∩

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	t0,1	t0,2	t1,2	t0,3
t1,3	t0,4	t0,1,2	t0,1,3	t0,2,3	t1,2,3	t0,1,4
t0,2,4	t0,1,2,3	t0,1,2,4	t0,1,3,4	t0,1,2,3,4

• 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

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