Pentellated 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

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

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

Contents 1 Pentellated 7-simplex 1.1 Alternate names 1.2 Coordinates 1.3 Images 2 Pentitruncated 7-simplex 2.1 Alternate names 2.2 Coordinates 2.3 Images 3 Penticantellated 7-simplex 3.1 Alternate names 3.2 Coordinates 3.3 Images 4 Penticantitruncated 7-simplex 4.1 Alternate names 4.2 Coordinates 5 Pentiruncinated 7-simplex 5.1 Alternate names 5.2 Coordinates 5.3 Images 6 Pentiruncitruncated 7-simplex 6.1 Alternate names 6.2 Coordinates 6.3 Images 7 Pentiruncicantellated 7-simplex 7.1 Alternate names 7.2 Coordinates 7.3 Images 8 Pentiruncicantitruncated 7-simplex 8.1 Alternate names 8.2 Coordinates 8.3 Images 9 Pentistericated 7-simplex 9.1 Alternate names 9.2 Coordinates 9.3 Images 10 Pentisteritruncated 7-simplex 10.1 Alternate names 10.2 Coordinates 10.3 Images 11 Pentistericantellated 7-simplex 11.1 Alternate names 11.2 Coordinates 11.3 Images 12 Pentistericantitruncated 7-simplex 12.1 Alternate names 12.2 Coordinates 12.3 Images 13 Pentisteriruncinated 7-simplex 13.1 Alternate names 13.2 Coordinates 13.3 Images 14 Pentisteriruncitruncated 7-simplex 14.1 Alternate names 14.2 Coordinates 14.3 Images 15 Pentisteriruncicantellated 7-simplex 15.1 Alternate names 15.2 Coordinates 15.3 Images 16 Pentisteriruncicantitruncated 7-simplex 16.1 Alternate names 16.2 Coordinates 16.3 Images 17 Related polytopes 18 Notes 19 References 20 External links

Pentellated 7-simplex

Pentellated 7-simplex
Type	uniform polyexon
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 polyexon
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 polyexon
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 polyexon
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 polyexon
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 polyexon
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 polyexon
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 polyexon
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 polyexon
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 polyexon
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 polyexon
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 polyexon
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 polyexon
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 polyexon
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 polyexon
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 polyexon
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.

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

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, editied 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, 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

• Olshevsky, George, Cross polytope at Glossary for Hyperspace.
• Polytopes of Various Dimensions
• Multi-dimensional Glossary

Family	An	BCn	Dn	E6 / E7 / E8 / F4 / G2	Hn
Regular polygon	Triangle	Square		Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	5-cell	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
n-polytopes	n-simplex	n-orthoplex • n-cube	n-demicube	1k2 • 2k1 • k21	pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes