Pentellated 7-simplexes

From Wikipedia, the free encyclopedia

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.

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

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

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.

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
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

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

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.

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
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

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

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.

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
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

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

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
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

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

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.

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
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

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

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.

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
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

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

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.

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
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

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

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.

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
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

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

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.

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
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

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

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.

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
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

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

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.

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
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

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

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.

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
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

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

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.

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
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

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

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.

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
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

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

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.

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
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

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

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.

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 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

• 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 [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.
• 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

• 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

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Category:

• 7-polytopes