Rectified 5-simplexes

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5-simplex t0.svg
5-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-simplex t1.svg
Rectified 5-simplex
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-simplex t2.svg
Birectified 5-simplex
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
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.

Contents

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
Coxeter diagram CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
or CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
4-faces126 {3,3,3} Schlegel wireframe 5-cell.png
6 r{3,3,3} Schlegel half-solid rectified 5-cell.png
Cells4515 {3,3} Tetrahedron.png
30 r{3,3} Uniform polyhedron-33-t1.png
Faces8080 {3}
Edges60
Vertices15
Vertex figure Rectified 5-simplex verf.png
{}×{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 03,1 for its branching Coxeter-Dynkin diagram, shown as CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png.

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S1
5
.

Alternate names

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.

As a configuration

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]

A5CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngk-facefkf0f1f2f3f4k-figurenotes
A3A1CDel node.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png( )f01584126842 {3,3}×{ } A5/A3A1 = 6!/4!/2 = 15
A2A1CDel node x.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.png{ }f1260133331 {3}∨( ) A5/A2A1 = 6!/3!/2 = 60
A2A2CDel node.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.png r{3} f23320*3030 {3} A5/A2A2 = 6!/3!/3! =20
A2A1CDel node x.pngCDel 2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node.png {3} 33*601221 { }×( ) A5/A2A1 = 6!/3!/2 = 60
A3A1CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node.png r{3,3} f36124415*20{ }A5/A3A1 = 6!/4!/2 = 15
A3CDel node x.pngCDel 2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node x.png {3,3} 4604*3011A5/A3 = 6!/4! = 30
A4CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node x.png r{3,3,3} f410301020556*( )A5/A4 = 6!/5! = 6
A4CDel node x.pngCDel 2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png {3,3,3} 51001005*6A5/A4 = 6!/5! = 6

Images

Stereographic projection
Rectified Hexateron.png
Stereographic projection of spherical form
orthographic projections
Ak
Coxeter plane
A5A4
Graph 5-simplex t1.svg 5-simplex t1 A4.svg
Dihedral symmetry [6][5]
Ak
Coxeter plane
A3A2
Graph 5-simplex t1 A3.svg 5-simplex t1 A2.svg
Dihedral symmetry [4][3]

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
A3A1A5D6 E7 = E7+=E7++
Coxeter
diagram
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea 1.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png
Symmetry [3−1,3,1][30,3,1][31,3,1][32,3,1][33,3,1][34,3,1]
Order 4872023,0402,903,040
Graph Tetrahedral prism.png 5-simplex t1.svg Demihexeract ortho petrie.svg Up2 2 31 t0 E7.svg --
Name 131 031 131 231 331 431

Birectified 5-simplex

Birectified 5-simplex
Birectified hexateron (dot)
Type uniform 5-polytope
Schläfli symbol 2r{34} = {32,2}
or
Coxeter diagram CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
or CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png
4-faces1212 r{3,3,3} Schlegel half-solid rectified 5-cell.png
Cells6030 {3,3} Tetrahedron.png
30 r{3,3} Uniform polyhedron-33-t1.png
Faces120120 {3}
Edges90
Vertices20
Vertex figure Birectified hexateron verf.png
{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 02,2 for its branching Coxeter-Dynkin diagram, shown as CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png. It is seen in the vertex figure of the 6-dimensional 122, CDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png.

Alternate names

Construction

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]

A5CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngk-facefkf0f1f2f3f4k-figurenotes
A2A2CDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.png( )f02099939333 {3}×{3} A5/A2A2 = 6!/3!/3! = 20
A1A1A1CDel node.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node.png{ }f12902214122 { }∨{ } A5/A1A1A1 = 6!/2/2/2 = 90
A2A1CDel node x.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node.png {3} f23360*12021 { }∨( ) A5/A2A1 = 6!/3!/2 = 60
A2A1CDel node.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node x.png33*6002112
A3A1CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node.png {3,3} f3464015**20{ }A5/A3A1 = 6!/4!/2 = 15
A3CDel node x.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node x.png r{3,3} 61244*30*11A5/A3 = 6!/4! = 30
A3A1CDel node.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png {3,3} 4604**1502A5/A3A1 = 6!/4!/2 = 15
A4CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node x.png r{3,3,3} f4103020105506*( )A5/A4 = 6!/5! = 6
A4CDel node x.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png10301020055*6

Images

The A5 projection has an identical appearance to Metatron's Cube. [7]

orthographic projections
Ak
Coxeter plane
A5A4
Graph 5-simplex t2.svg 5-simplex t2 A4.svg
Dihedral symmetry [6][[5]]=[10]
Ak
Coxeter plane
A3A2
Graph 5-simplex t2 A3.svg 5-simplex t2 A2.svg
Dihedral symmetry [4][[3]]=[6]

Intersection of two 5-simplices

Stereographic projection
Birectified Hexateron.png

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-simplex intersection graphs.png
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.

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
SpaceFiniteEuclideanHyperbolic
n 4 5 6 7 8
Coxeter
group
A2A2E6=E6+=E6++
Coxeter
diagram
CDel nodes.pngCDel 3ab.pngCDel nodes 11.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node 1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Symmetry [[32,2,-1]][[32,2,0]][[32,2,1]][[32,2,2]][[32,2,3]]
Order 721440103,680
Graph 3-3 duoprism ortho-skew.png 5-simplex t2.svg Up 1 22 t0 E6.svg
Name 122 022 122 222 322

Isotopics polytopes

Isotopic uniform truncated simplices
Dim.2345678
Name
Coxeter
Hexagon
CDel branch 11.png = CDel node 1.pngCDel 6.pngCDel node.png
t{3} = {6}
Octahedron
CDel node 1.pngCDel split1.pngCDel nodes.png = CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
r{3,3} = {31,1} = {3,4}
Decachoron
CDel branch 11.pngCDel 3ab.pngCDel nodes.png
2t{33}
Dodecateron
CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png
2r{34} = {32,2}
Tetradecapeton
CDel branch 11.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png
3t{35}
Hexadecaexon
CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png
3r{36} = {33,3}
Octadecazetton
CDel branch 11.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png
4t{37}
Images Truncated triangle.svg 3-cube t2.svg Uniform polyhedron-33-t1.png 4-simplex t12.svg Schlegel half-solid bitruncated 5-cell.png 5-simplex t2.svg 5-simplex t2 A4.svg 6-simplex t23.svg 6-simplex t23 A5.svg 7-simplex t3.svg 7-simplex t3 A5.svg 8-simplex t34.svg 8-simplex t34 A7.svg
Vertex figure( )∨( ) Octahedron vertfig.svg
{ }×{ }
Bitruncated 5-cell verf.png
{ }∨{ }
Birectified hexateron verf.png
{3}×{3}
Tritruncated 6-simplex verf.png
{3}∨{3}
{3,3}×{3,3} Quadritruncated 8-simplex verf.png
{3,3}∨{3,3}
Facets {3} Regular polygon 3 annotated.svg t{3,3} Uniform polyhedron-33-t01.png r{3,3,3} Schlegel half-solid rectified 5-cell.png 2t{3,3,3,3} 5-simplex t12.svg 2r{3,3,3,3,3} 6-simplex t2.svg 3t{3,3,3,3,3,3} 7-simplex t23.svg
As
intersecting
dual
simplexes
Regular hexagon as intersection of two triangles.png
CDel branch 10.pngCDel branch 01.png
Stellated octahedron A4 A5 skew.png
CDel node.pngCDel split1.pngCDel nodes 10lu.pngCDel node.pngCDel split1.pngCDel nodes 01ld.png
Compound dual 5-cells and bitruncated 5-cell intersection A4 coxeter plane.png
CDel branch.pngCDel 3ab.pngCDel nodes 10l.pngCDel branch.pngCDel 3ab.pngCDel nodes 01l.png
Dual 5-simplex intersection graph a5.png Dual 5-simplex intersection graph a4.png
CDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 10l.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 01l.png
CDel branch.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes 10l.pngCDel branch.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes 01l.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes 10l.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes 01l.pngCDel branch.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes 10l.pngCDel branch.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes 01l.png

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
5-simplex t0.svg
t0
5-simplex t1.svg
t1
5-simplex t2.svg
t2
5-simplex t01.svg
t0,1
5-simplex t02.svg
t0,2
5-simplex t12.svg
t1,2
5-simplex t03.svg
t0,3
5-simplex t13.svg
t1,3
5-simplex t04.svg
t0,4
5-simplex t012.svg
t0,1,2
5-simplex t013.svg
t0,1,3
5-simplex t023.svg
t0,2,3
5-simplex t123.svg
t1,2,3
5-simplex t014.svg
t0,1,4
5-simplex t024.svg
t0,2,4
5-simplex t0123.svg
t0,1,2,3
5-simplex t0124.svg
t0,1,2,4
5-simplex t0134.svg
t0,1,3,4
5-simplex t01234.svg
t0,1,2,3,4

Related Research Articles

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<span class="mw-page-title-main">Rectified 5-cell</span> Uniform polychoron

In four-dimensional geometry, the rectified 5-cell is a uniform 4-polytope composed of 5 regular tetrahedral and 5 regular octahedral cells. Each edge has one tetrahedron and two octahedra. Each vertex has two tetrahedra and three octahedra. In total it has 30 triangle faces, 30 edges, and 10 vertices. Each vertex is surrounded by 3 octahedra and 2 tetrahedra; the vertex figure is a triangular prism.

<span class="mw-page-title-main">Truncated 5-cell</span>

In geometry, a truncated 5-cell is a uniform 4-polytope formed as the truncation of the regular 5-cell.

<span class="mw-page-title-main">Cantellated 5-cell</span>

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<span class="mw-page-title-main">5-polytope</span> 5-dimensional geometric object

In geometry, a five-dimensional polytope is a polytope in five-dimensional space, bounded by (4-polytope) facets, pairs of which share a polyhedral cell.

In five-dimensional geometry, a 5-simplex is a self-dual regular 5-polytope. It has six vertices, 15 edges, 20 triangle faces, 15 tetrahedral cells, and 6 5-cell facets. It has a dihedral angle of cos−1(1/5), or approximately 78.46°.

2<sub> 31</sub> polytope

In 7-dimensional geometry, 231 is a uniform polytope, constructed from the E7 group.

1<sub> 22</sub> polytope Uniform 6-polytope

In 6-dimensional geometry, the 122 polytope is a uniform polytope, constructed from the E6 group. It was first published in E. L. Elte's 1912 listing of semiregular polytopes, named as V72 (for its 72 vertices).

1<sub> 32</sub> polytope

In 7-dimensional geometry, 132 is a uniform polytope, constructed from the E7 group.

1 <sub>42</sub> polytope

In 8-dimensional geometry, the 142 is a uniform 8-polytope, constructed within the symmetry of the E8 group.

3<sub> 21</sub> polytope

In 7-dimensional geometry, the 321 polytope is a uniform 7-polytope, constructed within the symmetry of the E7 group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 7-ic semi-regular figure.

In geometry, the 222 honeycomb is a uniform tessellation of the six-dimensional Euclidean space. It can be represented by the Schläfli symbol {3,3,32,2}. It is constructed from 221 facets and has a 122 vertex figure, with 54 221 polytopes around every vertex.

<span class="mw-page-title-main">Rectified 5-orthoplexes</span>

In five-dimensional geometry, a rectified 5-orthoplex is a convex uniform 5-polytope, being a rectification of the regular 5-orthoplex.

In seven-dimensional geometry, a rectified 7-orthoplex is a convex uniform 7-polytope, being a rectification of the regular 7-orthoplex.

In eight-dimensional geometry, a rectified 8-orthoplex is a convex uniform 8-polytope, being a rectification of the regular 8-orthoplex.

<span class="mw-page-title-main">Rectified 6-orthoplexes</span>

In six-dimensional geometry, a rectified 6-orthoplex is a convex uniform 6-polytope, being a rectification of the regular 6-orthoplex.

<span class="mw-page-title-main">6-polytope</span>

In six-dimensional geometry, a six-dimensional polytope or 6-polytope is a polytope, bounded by 5-polytope facets.

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In seven-dimensional geometry, a rectified 7-simplex is a convex uniform 7-polytope, being a rectification of the regular 7-simplex.

In nine-dimensional geometry, a rectified 9-simplex is a convex uniform 9-polytope, being a rectification of the regular 9-orthoplex.

<span class="mw-page-title-main">Rectified 9-simplexes</span>

In nine-dimensional geometry, a rectified 9-simplex is a convex uniform 9-polytope, being a rectification of the regular 9-simplex.

References

  1. Coxeter, Regular Polytopes, sec 1.8 Configurations
  2. Coxeter, Complex Regular Polytopes, p.117
  3. Klitzing, Richard. "o3x3o3o3o - rix".
  4. Coxeter, Regular Polytopes, sec 1.8 Configurations
  5. Coxeter, Complex Regular Polytopes, p.117
  6. Klitzing, Richard. "o3o3x3o3o - dot".
  7. Melchizedek, Drunvalo (1999). The Ancient Secret of the Flower of Life. Vol. 1. Light Technology Publishing. p.160 Figure 6-12
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds