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 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	{}x{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 (15 tetrahedral, and 30 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.

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]	[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

Alternate names

• 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.

Images

Stereographic projection

320px Stereographic projection of spherical form

orthographic projections

Ak Coxeter plane	A5	A4
Graph
Dihedral symmetry	[6]	[5]
Ak Coxeter plane	A3	A2
Graph	100px	100px
Dihedral symmetry	[4]	[3]

Birectified 5-simplex

Birectified 5-simplex Birectified hexateron (dot)
Type	uniform 5-polytope
Schläfli symbol	2r{34} = {32,2} 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}x{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)

Images

The A5 projection has an identical appearance to Metatron's Cube.

orthographic projections

Ak Coxeter plane	A5	A4
Graph
Dihedral symmetry	[6]	[[5]]=[10]
Ak Coxeter plane	A3	A2
Graph	100px	100px
Dihedral symmetry	[4]	[[3]]=[6]

Construction

Stereographic projection

320px

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.

320px

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	2A2	A5	E6	${\tilde{E}}_{6}$=E6+	${\bar{T}}_7$=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

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} $\left\{\begin{array}{l}3\\3\end{array}\right\}$	Decachoron 2t{33}	Dodecateron 2r{34} = {32,2} $\left\{\begin{array}{l}3, 3\\3 ,3\end{array}\right\}$	Tetradecapeton 3t{35}	Hexadecaexon 3r{36} = {33,3} $\left\{\begin{array}{l}3, 3, 3\\3, 3, 3\end{array}\right\}$	Octadecazetton 4t{37}
Images					60px	60px60px	60px60px
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} 30px
As intersecting dual simplexes	∩	120px ∩	60px ∩	60px60px ∩	∩	∩	∩

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)

References

• 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.
• Richard Klitzing, 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