Pentellated 7-simplexes

In seven-dimensional geometry, a pentellated 7-simplex is a convex uniform 7-polytope with 5th order truncations (pentellation) of the regular 7-simplex.

7-simplex	Pentellated 7-simplex	Pentitruncated 7-simplex	Penticantellated 7-simplex
Penticantitruncated 7-simplex	Pentiruncinated 7-simplex	Pentiruncitruncated 7-simplex	Pentiruncicantellated 7-simplex
Pentiruncicantitruncated 7-simplex	Pentistericated 7-simplex	Pentisteritruncated 7-simplex	Pentistericantellated 7-simplex
Pentistericantitruncated 7-simplex	Pentisteriruncinated 7-simplex	Pentisteriruncitruncated 7-simplex	Pentisteriruncicantellated 7-simplex
Pentisteriruncicantitruncated 7-simplex

There are 16 unique pentellations of the 7-simplex with permutations of truncations, cantellations, runcinations, and sterications.

Pentellated 7-simplex

Pentellated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t_0,5{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	1260
Vertices	168
Vertex figure
Coxeter groups	A₇, [3,3,3,3,3,3]
Properties	convex

Alternate names

Small terated octaexon (acronym: seto) (Jonathan Bowers)[1]

Coordinates

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

Images

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

Pentitruncated 7-simplex

pentitruncated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t_0,1,5{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	5460
Vertices	840
Vertex figure
Coxeter groups	A₇, [3,3,3,3,3,3]
Properties	convex

Alternate names

Teritruncated octaexon (acronym: teto) (Jonathan Bowers)[2]

Coordinates

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

Images

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

Penticantellated 7-simplex

Penticantellated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t_0,2,5{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	11760
Vertices	1680
Vertex figure
Coxeter groups	A₇, [3,3,3,3,3,3]
Properties	convex

Alternate names

Terirhombated octaexon (acronym: tero) (Jonathan Bowers)[3]

Coordinates

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

Images

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

Penticantitruncated 7-simplex

penticantitruncated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t_0,1,2,5{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups	A₇, [3,3,3,3,3,3]
Properties	convex

Alternate names

Terigreatorhombated octaexon (acronym: tegro) (Jonathan Bowers)[4]

Coordinates

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

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

Pentiruncinated 7-simplex

pentiruncinated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t_0,3,5{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	10920
Vertices	1680
Vertex figure
Coxeter groups	A₇, [3,3,3,3,3,3]
Properties	convex

Alternate names

Teriprismated octaexon (acronym: tepo) (Jonathan Bowers)[5]

Coordinates

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

Images

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

Pentiruncitruncated 7-simplex

pentiruncitruncated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t_0,1,3,5{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	27720
Vertices	5040
Vertex figure
Coxeter groups	A₇, [3,3,3,3,3,3]
Properties	convex

Alternate names

Teriprismatotruncated octaexon (acronym: tapto) (Jonathan Bowers)[6]

Coordinates

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

Images

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

Pentiruncicantellated 7-simplex

pentiruncicantellated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t_0,2,3,5{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	25200
Vertices	5040
Vertex figure
Coxeter groups	A₇, [3,3,3,3,3,3]
Properties	convex

Alternate names

Teriprismatorhombated octaexon (acronym: tapro) (Jonathan Bowers)[7]

Coordinates

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

Images

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

Pentiruncicantitruncated 7-simplex

pentiruncicantitruncated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t_0,1,2,3,5{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	45360
Vertices	10080
Vertex figure
Coxeter groups	A₇, [3,3,3,3,3,3]
Properties	convex

Alternate names

Terigreatoprismated octaexon (acronym: tegapo) (Jonathan Bowers)[8]

Coordinates

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

Images

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

Pentistericated 7-simplex

pentistericated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t_0,4,5{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	4200
Vertices	840
Vertex figure
Coxeter groups	A₇, [3,3,3,3,3,3]
Properties	convex

Alternate names

Tericellated octaexon (acronym: teco) (Jonathan Bowers)[9]

Coordinates

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

Images

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

Pentisteritruncated 7-simplex

pentisteritruncated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t_0,1,4,5{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	15120
Vertices	3360
Vertex figure
Coxeter groups	A₇, [3,3,3,3,3,3]
Properties	convex

Alternate names

Tericellitruncated octaexon (acronym: tecto) (Jonathan Bowers)[10]

Coordinates

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

Images

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

Pentistericantellated 7-simplex

pentistericantellated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t_0,2,4,5{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	25200
Vertices	5040
Vertex figure
Coxeter groups	A₇, [3,3,3,3,3,3]
Properties	convex

Alternate names

Tericellirhombated octaexon (acronym: tecro) (Jonathan Bowers)[11]

Coordinates

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

Images

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

Pentistericantitruncated 7-simplex

pentistericantitruncated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t_0,1,2,4,5{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	40320
Vertices	10080
Vertex figure
Coxeter groups	A₇, [3,3,3,3,3,3]
Properties	convex

Alternate names

Tericelligreatorhombated octaexon (acronym: tecagro) (Jonathan Bowers)[12]

Coordinates

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

Images

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

Pentisteriruncinated 7-simplex

Pentisteriruncinated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t_0,3,4,5{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	15120
Vertices	3360
Vertex figure
Coxeter groups	A₇, [3,3,3,3,3,3]
Properties	convex

Alternate names

Bipenticantitruncated 7-simplex as t_1,2,3,6{3,3,3,3,3,3}
Tericelliprismated octaexon (acronym: tacpo) (Jonathan Bowers)[13]

Coordinates

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

Images

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

Pentisteriruncitruncated 7-simplex

pentisteriruncitruncated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t_0,1,3,4,5{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	40320
Vertices	10080
Vertex figure
Coxeter groups	A₇, [3,3,3,3,3,3]
Properties	convex

Alternate names

Tericelliprismatotruncated octaexon (acronym: tacpeto) (Jonathan Bowers)[14]

Coordinates

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

Images

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

Pentisteriruncicantellated 7-simplex

pentisteriruncicantellated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t_0,2,3,4,5{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	40320
Vertices	10080
Vertex figure
Coxeter groups	A₇, [3,3,3,3,3,3]
Properties	convex

Alternate names

Bipentiruncicantitruncated 7-simplex as t_1,2,3,4,6{3,3,3,3,3,3}
Tericelliprismatorhombated octaexon (acronym: tacpro) (Jonathan Bowers)[15]

Coordinates

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

Images

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

Pentisteriruncicantitruncated 7-simplex

pentisteriruncicantitruncated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t_0,1,2,3,4,5{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	70560
Vertices	20160
Vertex figure
Coxeter groups	A₇, [3,3,3,3,3,3]
Properties	convex

Alternate names

Great terated octaexon (acronym: geto) (Jonathan Bowers)[16]

Coordinates

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

Images

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

Related polytopes

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

Notes

Klitzing, (x3o3o3o3o3x3o - seto)
Klitzing, (x3x3o3o3o3x3o - teto)
Klitzing, (x3o3x3o3o3x3o - tero)
Klitzing, (x3x3x3oxo3x3o - tegro)
Klitzing, (x3o3o3x3o3x3o - tepo)
Klitzing, (x3x3o3x3o3x3o - tapto)
Klitzing, (x3o3x3x3o3x3o - tapro)
Klitzing, (x3x3x3x3o3x3o - tegapo)
Klitzing, (x3o3o3o3x3x3o - teco)
Klitzing, (x3x3o3o3x3x3o - tecto)
Klitzing, (x3o3x3o3x3x3o - tecro)
Klitzing, (x3x3x3o3x3x3o - tecagro)
Klitzing, (x3o3o3x3x3x3o - tacpo)
Klitzing, (x3x3o3x3x3x3o - tacpeto)
Klitzing, (x3o3x3x3x3x3o - tacpro)
Klitzing, (x3x3x3x3x3x3o - geto)

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
  - (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.
Klitzing, Richard. "7D uniform polytopes (polyexa)". x3o3o3o3o3x3o - seto, x3x3o3o3o3x3o - teto, x3o3x3o3o3x3o - tero, x3x3x3oxo3x3o - tegro, x3o3o3x3o3x3o - tepo, x3x3o3x3o3x3o - tapto, x3o3x3x3o3x3o - tapro, x3x3x3x3o3x3o - tegapo, x3o3o3o3x3x3o - teco, x3x3o3o3x3x3o - tecto, x3o3x3o3x3x3o - tecro, x3x3x3o3x3x3o - tecagro, x3o3o3x3x3x3o - tacpo, x3x3o3x3x3x3o - tacpeto, x3o3x3x3x3x3o - tacpro, x3x3x3x3x3x3o - geto

External links

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	A_n	B_n	I₂(p) / D_n	E₆ / E₇ / E₈ / F₄ / G₂	H_n
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	1_k2 • 2_k1 • k₂₁	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds

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[1] Klitzing, (x3o3o3o3o3x3o - seto)

[2] Klitzing, (x3x3o3o3o3x3o - teto)

[3] Klitzing, (x3o3x3o3o3x3o - tero)

[4] Klitzing, (x3x3x3oxo3x3o - tegro)

[5] Klitzing, (x3o3o3x3o3x3o - tepo)

[6] Klitzing, (x3x3o3x3o3x3o - tapto)

[7] Klitzing, (x3o3x3x3o3x3o - tapro)

[8] Klitzing, (x3x3x3x3o3x3o - tegapo)

[9] Klitzing, (x3o3o3o3x3x3o - teco)

[10] Klitzing, (x3x3o3o3x3x3o - tecto)

[11] Klitzing, (x3o3x3o3x3x3o - tecro)

[12] Klitzing, (x3x3x3o3x3x3o - tecagro)

[13] Klitzing, (x3o3o3x3x3x3o - tacpo)

[14] Klitzing, (x3x3o3x3x3x3o - tacpeto)

[15] Klitzing, (x3o3x3x3x3x3o - tacpro)

[16] Klitzing, (x3x3x3x3x3x3o - geto)

A7 polytopes
t₀	t₁	t₂	t₃	t_0,1	t_0,2	t_1,2	t_0,3
t_1,3	t_2,3	t_0,4	t_1,4	t_2,4	t_0,5	t_1,5	t_0,6
t_0,1,2	t_0,1,3	t_0,2,3	t_1,2,3	t_0,1,4	t_0,2,4	t_1,2,4	t_0,3,4
t_1,3,4	t_2,3,4	t_0,1,5	t_0,2,5	t_1,2,5	t_0,3,5	t_1,3,5	t_0,4,5
t_0,1,6	t_0,2,6	t_0,3,6	t_0,1,2,3	t_0,1,2,4	t_0,1,3,4	t_0,2,3,4	t_1,2,3,4
t_0,1,2,5	t_0,1,3,5	t_0,2,3,5	t_1,2,3,5	t_0,1,4,5	t_0,2,4,5	t_1,2,4,5	t_0,3,4,5
t_0,1,2,6	t_0,1,3,6	t_0,2,3,6	t_0,1,4,6	t_0,2,4,6	t_0,1,5,6	t_0,1,2,3,4	t_0,1,2,3,5
t_0,1,2,4,5	t_0,1,3,4,5	t_0,2,3,4,5	t_1,2,3,4,5	t_0,1,2,3,6	t_0,1,2,4,6	t_0,1,3,4,6	t_0,2,3,4,6
t_0,1,2,5,6	t_0,1,3,5,6	t_0,1,2,3,4,5	t_0,1,2,3,4,6	t_0,1,2,3,5,6	t_0,1,2,4,5,6	t_{0,1,2,3,4,5,6}