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}
Coxeter-Dynkin 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 .

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, 4_{31. 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	A3×A1	A5	D6	E7	= E7+	E7++
Coxeter diagram
Symmetry (order)	[3-1,3,1] (48)	[30,3,1] (720)	[31,3,1] (23,040)	[32,3,1] (2,903,040)	[33,3,1] (∞)	[34,3,1] (∞)
Graph					∞	∞
Name	−131	031	131	231	331	431

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.

Images

orthographic projections

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

Stereographic projection

Stereographic projection of spherical form

Birectified 5-simplex

Birectified 5-simplex Birectified hexateron (dot)
Type	uniform 5-polytope
Schläfli symbol	2r{34}
Coxeter-Dynkin 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	{{{dot-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). It is also called 0_2,2 for its branching Coxeter-Dynkin diagram, shown as .

Alternate names

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

Construction

The birectified 5-simplex is the intersection of two regular 5-simplices in dual configuration. As such, it is also the intersection of a 6-cube with the hyperplane that bisects the hexeract'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-cell 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.

Images

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

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, 3_{22. Each progressive uniform polytope is constructed from the previous as its vertex figure.}

k₂₂ figures in n dimensions

n	4	5	6	7	8
Coxeter group	A22	A5	E6	=E6+	E6++
Coxeter diagram
Symmetry (order)	[[32,2,-1]] (72)	[[32,2,0]] (1440)	[[32,2,1]] (103,680)	[[32,2,2]] (∞)	[[32,2,3]] (∞)
Graph				∞	∞
Name	−122	022	122	222	322

Isotopics polytopes

Isotopic uniform truncated simplices

Dim.	2	3	4	5	6	7	8
Name	t{3} Hexagon	r{3,3} Octahedron	2t{3,3,3} Decachoron	2r{3,3,3,3} Dodecateron	3t{3,3,3,3,3} Tetradecapeton	3r{3,3,3,3,3,3} Hexadecaexon	4t{3,3,3,3,3,3,3} Octadecazetton
Coxeter diagram
Images
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}

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)

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

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, editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
    • (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