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	t0,5{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

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

Pentitruncated 7-simplex

pentitruncated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t0,1,5{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

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

Penticantellated 7-simplex

Penticantellated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t0,2,5{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

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

Penticantitruncated 7-simplex

penticantitruncated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t0,1,2,5{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

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

Pentiruncinated 7-simplex

pentiruncinated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t0,3,5{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

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

Pentiruncitruncated 7-simplex

pentiruncitruncated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t0,1,3,5{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

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

Pentiruncicantellated 7-simplex

pentiruncicantellated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t0,2,3,5{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

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

Pentiruncicantitruncated 7-simplex

pentiruncicantitruncated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t0,1,2,3,5{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

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

Pentistericated 7-simplex

pentistericated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t0,4,5{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

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

Pentisteritruncated 7-simplex

pentisteritruncated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t0,1,4,5{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

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

Pentistericantellated 7-simplex

pentistericantellated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t0,2,4,5{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

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

Pentistericantitruncated 7-simplex

pentistericantitruncated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t0,1,2,4,5{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

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

Pentisteriruncinated 7-simplex

Pentisteriruncinated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t0,3,4,5{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

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

Pentisteriruncitruncated 7-simplex

pentisteriruncitruncated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t0,1,3,4,5{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

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

Pentisteriruncicantellated 7-simplex

pentisteriruncicantellated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t0,2,3,4,5{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

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

Pentisteriruncicantitruncated 7-simplex

pentisteriruncicantitruncated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t0,1,2,3,4,5{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

Ak Coxeter plane	A7	A6	A5
Graph
Dihedral symmetry	[8]	[[7]]	[6]
Ak 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.

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, (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)

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

• Polytopes of Various Dimensions
• Multi-dimensional Glossary

v t e 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