# Rectified 5-simplexes

(Redirected from Birectified 5-simplex)
 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.

## Rectified 5-simplex

Rectified 5-simplex
Rectified hexateron (rix)
Type uniform 5-polytope
Schläfli symbol r{34}
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)
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 03,1 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 13k 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)

## 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 of spherical form

## Birectified 5-simplex

Birectified 5-simplex
Birectified hexateron (dot)
Type uniform 5-polytope
Schläfli symbol 2r{34}
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 {{{dot-dual}}}
Base point (0,0,0,1,1,1)
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 02,2 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-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.

### 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]

### 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 t{3}
Hexagon
r{3,3}
Octahedron
2t{3,3,3}
Decachoron
2r{3,3,3,3}
Dodecateron
3t{3,3,3,3,3}
3r{3,3,3,3,3,3}
4t{3,3,3,3,3,3,3}
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, 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