Rectified 5-simplexes

• 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

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 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