7-simplex
Hexicated 7-simplex
Hexitruncated 7-simplex
Hexicantellated 7-simplex

Hexiruncinated 7-simplex
Hexicantitruncated 7-simplex
Hexiruncitruncated 7-simplex
Hexiruncicantellated 7-simplex

Hexisteritruncated 7-simplex
Hexistericantellated 7-simplex
Hexipentitruncated 7-simplex
Hexiruncicantitruncated 7-simplex

Hexistericantitruncated 7-simplex
Hexisteriruncitruncated 7-simplex
Hexisteriruncicantellated 7-simplex
Hexipenticantitruncated 7-simplex

Hexipentiruncitruncated 7-simplex
Hexisteriruncicantitruncated 7-simplex
Hexipentiruncicantitruncated 7-simplex
Hexipentistericantitruncated 7-simplex

Hexipentisteriruncicantitruncated 7-simplex
(Omnitruncated 7-simplex)
Orthogonal projections in A7 Coxeter plane

In seven-dimensional geometry, a hexicated 7-simplex is a convex uniform 7-polytope, including 6th-order truncations (hexication) from the regular 7-simplex.

There are 20 unique hexications for the 7-simplex, including all permutations of truncations, cantellations, runcinations, sterications, and pentellations.

The simple hexicated 7-simplex is also called an expanded 7-simplex, with only the first and last nodes ringed, is constructed by an expansion operation applied to the regular 7-simplex. The highest form, the hexipentisteriruncicantitruncated 7-simplex is more simply called a omnitruncated 7-simplex with all of the nodes ringed.

Hexicated 7-simplex

Hexicated 7-simplex
Typeuniform 7-polytope
Schläfli symbolt0,6{36}
Coxeter-Dynkin diagrams
6-faces254:
8+8 {35}
28+28 {}x{34}
56+56 {3}x{3,3,3}
70 {3,3}x{3,3}
5-faces
4-faces
Cells
Faces
Edges336
Vertices56
Vertex figure5-simplex antiprism
Coxeter groupA7×2, [[36]], order 80640
Propertiesconvex

In seven-dimensional geometry, a hexicated 7-simplex is a convex uniform 7-polytope, a hexication (6th order truncation) of the regular 7-simplex, or alternately can be seen as an expansion operation.

The vertices of the A7 2D orthogonal projection are seen in the Ammann–Beenker tiling.

Root vectors

Its 56 vertices represent the root vectors of the simple Lie group A7.

Alternate names

  • Expanded 7-simplex
  • Small petated hexadecaexon (acronym: suph) (Jonathan Bowers)[1]

Coordinates

The vertices of the hexicated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,1,1,1,1,2). This construction is based on facets of the hexicated 8-orthoplex, .

A second construction in 8-space, from the center of a rectified 8-orthoplex is given by coordinate permutations of:

(1,-1,0,0,0,0,0,0)

Images

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [[7]] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [[5]] [4] [[3]]

Hexitruncated 7-simplex

hexitruncated 7-simplex
Typeuniform 7-polytope
Schläfli symbolt0,1,6{36}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges1848
Vertices336
Vertex figure
Coxeter groupA7, [36], order 40320
Propertiesconvex

Alternate names

  • Petitruncated octaexon (acronym: puto) (Jonathan Bowers)[2]

Coordinates

The vertices of the hexitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,1,1,1,2,3). This construction is based on facets of the hexitruncated 8-orthoplex, .

Images

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]

Hexicantellated 7-simplex

Hexicantellated 7-simplex
Typeuniform 7-polytope
Schläfli symbolt0,2,6{36}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges5880
Vertices840
Vertex figure
Coxeter groupA7, [36], order 40320
Propertiesconvex

Alternate names

  • Petirhombated octaexon (acronym: puro) (Jonathan Bowers)[3]

Coordinates

The vertices of the hexicantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,1,1,2,2,3). This construction is based on facets of the hexicantellated 8-orthoplex, .

Images

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]

Hexiruncinated 7-simplex

Hexiruncinated 7-simplex
Typeuniform 7-polytope
Schläfli symbolt0,3,6{36}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges8400
Vertices1120
Vertex figure
Coxeter groupA7×2, [[36]], order 80640
Propertiesconvex

Alternate names

  • Petiprismated hexadecaexon (acronym: puph) (Jonathan Bowers)[4]

Coordinates

The vertices of the hexiruncinated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,1,2,2,2,3). This construction is based on facets of the hexiruncinated 8-orthoplex, .

Images

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [[7]] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [[5]] [4] [[3]]

Hexicantitruncated 7-simplex

Hexicantitruncated 7-simplex
Typeuniform 7-polytope
Schläfli symbolt0,1,2,6{36}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges8400
Vertices1680
Vertex figure
Coxeter groupA7, [36], order 40320
Propertiesconvex

Alternate names

  • Petigreatorhombated octaexon (acronym: pugro) (Jonathan Bowers)[5]

Coordinates

The vertices of the hexicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,1,1,2,3,4). This construction is based on facets of the hexicantitruncated 8-orthoplex, .

Images

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]

Hexiruncitruncated 7-simplex

Hexiruncitruncated 7-simplex
Typeuniform 7-polytope
Schläfli symbolt0,1,3,6{36}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges20160
Vertices3360
Vertex figure
Coxeter groupA7, [36], order 40320
Propertiesconvex

Alternate names

  • Petiprismatotruncated octaexon (acronym: pupato) (Jonathan Bowers)[6]

Coordinates

The vertices of the hexiruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,1,2,2,3,4). This construction is based on facets of the hexiruncitruncated 8-orthoplex, .

Images

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]

Hexiruncicantellated 7-simplex

Hexiruncicantellated 7-simplex
Typeuniform 7-polytope
Schläfli symbolt0,2,3,6{36}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges16800
Vertices3360
Vertex figure
Coxeter groupA7, [36], order 40320
Propertiesconvex

In seven-dimensional geometry, a hexiruncicantellated 7-simplex is a uniform 7-polytope.

Alternate names

  • Petiprismatorhombated octaexon (acronym: pupro) (Jonathan Bowers)[7]

Coordinates

The vertices of the hexiruncicantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,1,2,3,3,4). This construction is based on facets of the hexiruncicantellated 8-orthoplex, .

Images

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]

Hexisteritruncated 7-simplex

hexisteritruncated 7-simplex
Typeuniform 7-polytope
Schläfli symbolt0,1,4,6{36}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges20160
Vertices3360
Vertex figure
Coxeter groupA7, [36], order 40320
Propertiesconvex

Alternate names

  • Peticellitruncated octaexon (acronym: pucto) (Jonathan Bowers)[8]

Coordinates

The vertices of the hexisteritruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,2,2,2,3,4). This construction is based on facets of the hexisteritruncated 8-orthoplex, .

Images

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]

Hexistericantellated 7-simplex

hexistericantellated 7-simplex
Typeuniform 7-polytope
Schläfli symbolt0,2,4,6{36}
Coxeter-Dynkin diagrams
6-facest0,2,4{3,3,3,3,3}

{}xt0,2,4{3,3,3,3}
{3}xt0,2{3,3,3}
t0,2{3,3}xt0,2{3,3}

5-faces
4-faces
Cells
Faces
Edges30240
Vertices5040
Vertex figure
Coxeter groupA7×2, [[36]], order 80640
Propertiesconvex

Alternate names

  • Peticellirhombihexadecaexon (acronym: pucroh) (Jonathan Bowers)[9]

Coordinates

The vertices of the hexistericantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,2,2,3,3,4). This construction is based on facets of the hexistericantellated 8-orthoplex, .

Images

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [[7]] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [[5]] [4] [[3]]

Hexipentitruncated 7-simplex

Hexipentitruncated 7-simplex
Typeuniform 7-polytope
Schläfli symbolt0,1,5,6{36}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges8400
Vertices1680
Vertex figure
Coxeter groupA7×2, [[36]], order 80640
Propertiesconvex

Alternate names

  • Petiteritruncated hexadecaexon (acronym: putath) (Jonathan Bowers)[10]

Coordinates

The vertices of the hexipentitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,2,2,2,2,3,4). This construction is based on facets of the hexipentitruncated 8-orthoplex, .

Images

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [[7]] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [[5]] [4] [[3]]

Hexiruncicantitruncated 7-simplex

Hexiruncicantitruncated 7-simplex
Typeuniform 7-polytope
Schläfli symbolt0,1,2,3,6{36}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges30240
Vertices6720
Vertex figure
Coxeter groupA7, [36], order 40320
Propertiesconvex

Alternate names

  • Petigreatoprismated octaexon (acronym: pugopo) (Jonathan Bowers)[11]

Coordinates

The vertices of the hexiruncicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,2,2,3,4,5). This construction is based on facets of the hexiruncicantitruncated 8-orthoplex, .

Images

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [[7]] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [[5]] [4] [[3]]

Hexistericantitruncated 7-simplex

Hexistericantitruncated 7-simplex
Typeuniform 7-polytope
Schläfli symbolt0,1,2,4,6{36}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges50400
Vertices10080
Vertex figure
Coxeter groupA7, [36], order 40320
Propertiesconvex

Alternate names

  • Peticelligreatorhombated octaexon (acronym: pucagro) (Jonathan Bowers)[12]

Coordinates

The vertices of the hexistericantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,2,2,3,4,5). This construction is based on facets of the hexistericantitruncated 8-orthoplex, .

Images

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [[7]] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [[5]] [4] [[3]]

Hexisteriruncitruncated 7-simplex

Hexisteriruncitruncated 7-simplex
Typeuniform 7-polytope
Schläfli symbolt0,1,3,4,6{36}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges45360
Vertices10080
Vertex figure
Coxeter groupA7, [36], order 40320
Propertiesconvex

Alternate names

  • Peticelliprismatotruncated octaexon (acronym: pucpato) (Jonathan Bowers)[13]

Coordinates

The vertices of the hexisteriruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,2,3,3,4,5). This construction is based on facets of the hexisteriruncitruncated 8-orthoplex, .

Images

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]

Hexisteriruncicantellated 7-simplex

Hexisteriruncicantellated 7-simplex
Typeuniform 7-polytope
Schläfli symbolt0,2,3,4,6{36}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges45360
Vertices10080
Vertex figure
Coxeter groupA7×2, [[36]], order 80640
Propertiesconvex

Alternate names

  • Peticelliprismatorhombihexadecaexon (acronym: pucproh) (Jonathan Bowers)[14]

Coordinates

The vertices of the hexisteriruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,2,3,4,4,5). This construction is based on facets of the hexisteriruncitruncated 8-orthoplex, .

Images

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [[7]] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [[5]] [4] [[3]]

Hexipenticantitruncated 7-simplex

hexipenticantitruncated 7-simplex
Typeuniform 7-polytope
Schläfli symbolt0,1,2,5,6{36}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges30240
Vertices6720
Vertex figure
Coxeter groupA7, [36], order 40320
Propertiesconvex

Alternate names

  • Petiterigreatorhombated octaexon (acronym: putagro) (Jonathan Bowers)[15]

Coordinates

The vertices of the hexipenticantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,2,2,2,3,4,5). This construction is based on facets of the hexipenticantitruncated 8-orthoplex, .

Images

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]

Hexipentiruncitruncated 7-simplex

Hexipentiruncitruncated 7-simplex
Typeuniform 7-polytope
Schläfli symbolt0,1,3,5,6{36}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices10080
Vertex figure
Coxeter groupA7×2, [[36]], order 80640
Propertiesconvex

Alternate names

  • Petiteriprismatotruncated hexadecaexon (acronym: putpath) (Jonathan Bowers)[16]

Coordinates

The vertices of the hexipentiruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,2,2,3,4,4,5). This construction is based on facets of the hexipentiruncitruncated 8-orthoplex, .

Images

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [[7]] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [[5]] [4] [[3]]

Hexisteriruncicantitruncated 7-simplex

Hexisteriruncicantitruncated 7-simplex
Typeuniform 7-polytope
Schläfli symbolt0,1,2,3,4,6{36}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges80640
Vertices20160
Vertex figure
Coxeter groupA7, [36], order 40320
Propertiesconvex

Alternate names

  • Petigreatocellated octaexon (acronym: pugaco) (Jonathan Bowers)[17]

Coordinates

The vertices of the hexisteriruncicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,2,3,4,5,6). This construction is based on facets of the hexisteriruncicantitruncated 8-orthoplex, .

Images

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [[7]] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [[5]] [4] [[3]]

Hexipentiruncicantitruncated 7-simplex

Hexipentiruncicantitruncated 7-simplex
Typeuniform 7-polytope
Schläfli symbolt0,1,2,3,5,6{36}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges80640
Vertices20160
Vertex figure
Coxeter groupA7, [36], order 40320
Propertiesconvex

Alternate names

  • Petiterigreatoprismated octaexon (acronym: putgapo) (Jonathan Bowers)[18]

Coordinates

The vertices of the hexipentiruncicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,2,2,3,4,5,6). This construction is based on facets of the hexipentiruncicantitruncated 8-orthoplex, .

Images

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [[7]] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [[5]] [4] [[3]]

Hexipentistericantitruncated 7-simplex

Hexipentistericantitruncated 7-simplex
Typeuniform 7-polytope
Schläfli symbolt0,1,2,4,5,6{36}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges80640
Vertices20160
Vertex figure
Coxeter groupA7×2, [[36]], order 80640
Propertiesconvex

Alternate names

  • Petitericelligreatorhombihexadecaexon (acronym: putcagroh) (Jonathan Bowers)[19]

Coordinates

The vertices of the hexipentistericantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,2,3,3,4,5,6). This construction is based on facets of the hexipentistericantitruncated 8-orthoplex, .

Images

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [[7]] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [[5]] [4] [[3]]

Omnitruncated 7-simplex

Omnitruncated 7-simplex
Typeuniform 7-polytope
Schläfli symbolt0,1,2,3,4,5,6{36}
Coxeter-Dynkin diagrams
6-faces254
5-faces5796
4-faces40824
Cells126000
Faces191520
Edges141120
Vertices40320
Vertex figureIrr. 6-simplex
Coxeter groupA7×2, [[36]], order 80640
Propertiesconvex

The omnitruncated 7-simplex is composed of 40320 (8 factorial) vertices and is the largest uniform 7-polytope in the A7 symmetry of the regular 7-simplex. It can also be called the hexipentisteriruncicantitruncated 7-simplex which is the long name for the omnitruncation for 7 dimensions, with all reflective mirrors active.

The omnitruncated 7-simplex is the permutohedron of order 8. The omnitruncated 7-simplex is a zonotope, the Minkowski sum of eight line segments parallel to the eight lines through the origin and the eight vertices of the 7-simplex.

Like all uniform omnitruncated n-simplices, the omnitruncated 7-simplex can tessellate space by itself, in this case 7-dimensional space with three facets around each ridge. It has Coxeter-Dynkin diagram of .

Alternate names

  • Great petated hexadecaexon (Acronym: guph) (Jonathan Bowers)[20]

Coordinates

The vertices of the omnitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,2,3,4,5,6,7). This construction is based on facets of the hexipentisteriruncicantitruncated 8-orthoplex, t0,1,2,3,4,5,6{36,4}, .

Images

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [[7]] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [[5]] [4] [[3]]

These polytope are a part of 71 uniform 7-polytopes with A7 symmetry.

A7 polytopes

t0
t1
t2
t3
t0,1
t0,2
t1,2
t0,3

t1,3
t2,3
t0,4
t1,4
t2,4
t0,5
t1,5
t0,6

t0,1,2
t0,1,3
t0,2,3
t1,2,3
t0,1,4
t0,2,4
t1,2,4
t0,3,4

t1,3,4
t2,3,4
t0,1,5
t0,2,5
t1,2,5
t0,3,5
t1,3,5
t0,4,5

t0,1,6
t0,2,6
t0,3,6
t0,1,2,3
t0,1,2,4
t0,1,3,4
t0,2,3,4
t1,2,3,4

t0,1,2,5
t0,1,3,5
t0,2,3,5
t1,2,3,5
t0,1,4,5
t0,2,4,5
t1,2,4,5
t0,3,4,5

t0,1,2,6
t0,1,3,6
t0,2,3,6
t0,1,4,6
t0,2,4,6
t0,1,5,6
t0,1,2,3,4
t0,1,2,3,5

t0,1,2,4,5
t0,1,3,4,5
t0,2,3,4,5
t1,2,3,4,5
t0,1,2,3,6
t0,1,2,4,6
t0,1,3,4,6
t0,2,3,4,6

t0,1,2,5,6
t0,1,3,5,6
t0,1,2,3,4,5
t0,1,2,3,4,6
t0,1,2,3,5,6
t0,1,2,4,5,6
t0,1,2,3,4,5,6

Notes

  1. ↑ Klitzing, (x3o3o3o3o3o3x - suph)
  2. ↑ Klitzing, (x3x3o3o3o3o3x- puto)
  3. ↑ Klitzing, (x3o3x3o3o3o3x - puro)
  4. ↑ Klitzing, (x3o3o3x3o3o3x - puph)
  5. ↑ Klitzing, (x3o3o3o3x3o3x - pugro)
  6. ↑ Klitzing, (x3x3x3o3o3o3x - pupato)
  7. ↑ Klitzing, (x3o3x3x3o3o3x - pupro)
  8. ↑ Klitzing, (x3x3o3o3x3o3x - pucto)
  9. ↑ Klitzing, (x3o3x3o3x3o3x - pucroh)
  10. ↑ Klitzing, (x3x3o3o3o3x3x - putath)
  11. ↑ Klitzing, (x3x3x3x3o3o3x - pugopo)
  12. ↑ Klitzing, (x3x3x3o3x3o3x - pucagro)
  13. ↑ Klitzing, (x3x3o3x3x3o3x - pucpato)
  14. ↑ Klitzing, (x3o3x3x3x3o3x - pucproh)
  15. ↑ Klitzing, (x3x3x3o3o3x3x - putagro)
  16. ↑ Klitzing, (x3x3o3x3o3x3x - putpath)
  17. ↑ Klitzing, (x3x3x3x3x3o3x - pugaco)
  18. ↑ Klitzing, (x3x3x3x3o3x3x - putgapo)
  19. ↑ Klitzing, (x3x3x3o3x3x3x - putcagroh)
  20. ↑ Klitzing, (x3x3x3x3x3x3x - guph)

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, wiley.com
      • (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, PhD (1966)
  • Klitzing, Richard. "7D". x3o3o3o3o3o3x - suph, x3x3o3o3o3o3x- puto, x3o3x3o3o3o3x - puro, x3o3o3x3o3o3x - puph, x3o3o3o3x3o3x - pugro, x3x3x3o3o3o3x - pupato, x3o3x3x3o3o3x - pupro, x3x3o3o3x3o3x - pucto, x3o3x3o3x3o3x - pucroh, x3x3o3o3o3x3x - putath, x3x3x3x3o3o3x - pugopo, x3x3x3o3x3o3x - pucagro, x3x3o3x3x3o3x - pucpato, x3o3x3x3x3o3x - pucproh, x3x3x3o3o3x3x - putagro, x3x3x3x3o3x3x - putpath, x3x3x3x3x3o3x - pugaco, x3x3x3x3o3x3x - putgapo, x3x3x3o3x3x3x - putcagroh, x3x3x3x3x3x3x - guph
Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
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