Stericated 7-orthoplexes

In seven-dimensional geometry, a stericated 7-orthoplex is a convex uniform 7-polytope with 4th order truncations (sterication) of the regular 7-orthoplex.

Orthogonal projections in B6 Coxeter plane
7-orthoplex	Stericated 7-orthoplex	Steritruncated 7-orthoplex
Bisteritruncated 7-orthoplex	Stericantellated 7-orthoplex	Stericantitruncated 7-orthoplex
Bistericantitruncated 7-orthoplex	Steriruncinated 7-orthoplex	Steriruncitruncated 7-orthoplex
Steriruncicantellated 7-orthoplex	Bisteriruncitruncated 7-orthoplex	Steriruncicantitruncated 7-orthoplex

There are 24 unique sterication for the 7-orthoplex with permutations of truncations, cantellations, and runcinations. 14 are more simply constructed from the 7-cube.

This polytope is one of 127 uniform 7-polytopes with B₇ symmetry.

Stericated 7-orthoplex

Stericated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t0,4{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups	B7, [4,35]
Properties	convex

Alternate names

• Small cellated hecatonicosoctaexon (acronym: ) (Jonathan Bowers)^[1]

Images

orthographic projections

Coxeter plane	B7 / A6	B6 / D7	B5 / D6 / A4
Graph
Dihedral symmetry	[14]	[12]	[10]
Coxeter plane	B4 / D5	B3 / D4 / A2	B2 / D3
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A5	A3
Graph
Dihedral symmetry	[6]	[4]

Steritruncated 7-orthoplex

steritruncated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t0,1,4{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups	B7, [4,35]
Properties	convex

Alternate names

• Cellitruncated hecatonicosoctaexon (acronym: ) (Jonathan Bowers)^[2]

Images

orthographic projections

Coxeter plane	B7 / A6	B6 / D7	B5 / D6 / A4
Graph
Dihedral symmetry	[14]	[12]	[10]
Coxeter plane	B4 / D5	B3 / D4 / A2	B2 / D3
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A5	A3
Graph
Dihedral symmetry	[6]	[4]

Bisteritruncated 7-orthoplex

bisteritruncated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t1,2,5{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups	B7, [4,35]
Properties	convex

Alternate names

• Bicellitruncated hecatonicosoctaexon (acronym: ) (Jonathan Bowers)^[3]

Images

orthographic projections

Coxeter plane	B7 / A6	B6 / D7	B5 / D6 / A4
Graph
Dihedral symmetry	[14]	[12]	[10]
Coxeter plane	B4 / D5	B3 / D4 / A2	B2 / D3
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A5	A3
Graph
Dihedral symmetry	[6]	[4]

Stericantellated 7-orthoplex

Stericantellated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t0,2,4{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups	B7, [4,35]
Properties	convex

Alternate names

• Cellirhombated hecatonicosoctaexon (acronym: ) (Jonathan Bowers)^[4]

Images

orthographic projections

Coxeter plane	B7 / A6	B6 / D7	B5 / D6 / A4
Graph
Dihedral symmetry	[14]	[12]	[10]
Coxeter plane	B4 / D5	B3 / D4 / A2	B2 / D3
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A5	A3
Graph
Dihedral symmetry	[6]	[4]

Stericantitruncated 7-orthoplex

stericantitruncated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t0,1,2,4{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups	B7, [4,35]
Properties	convex

Alternate names

• Celligreatorhombated hecatonicosoctaexon (acronym: ) (Jonathan Bowers)^[5]

Images

orthographic projections

Coxeter plane	B7 / A6	B6 / D7	B5 / D6 / A4
Graph
Dihedral symmetry	[14]	[12]	[10]
Coxeter plane	B4 / D5	B3 / D4 / A2	B2 / D3
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A5	A3
Graph
Dihedral symmetry	[6]	[4]

Bistericantitruncated 7-orthoplex

bistericantitruncated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t1,2,3,5{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups	B7, [4,35]
Properties	convex

Alternate names

• Bicelligreatorhombated hecatonicosoctaexon (acronym: ) (Jonathan Bowers)^[6]

Images

orthographic projections

Coxeter plane	B7 / A6	B6 / D7	B5 / D6 / A4
Graph
Dihedral symmetry	[14]	[12]	[10]
Coxeter plane	B4 / D5	B3 / D4 / A2	B2 / D3
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A5	A3
Graph
Dihedral symmetry	[6]	[4]

Steriruncinated 7-orthoplex

Steriruncinated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t0,3,4{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups	B7, [4,35]
Properties	convex

Alternate names

• Celliprismated hecatonicosoctaexon (acronym: ) (Jonathan Bowers)^[7]

Images

orthographic projections

Coxeter plane	B7 / A6	B6 / D7	B5 / D6 / A4
Graph	too complex
Dihedral symmetry	[14]	[12]	[10]
Coxeter plane	B4 / D5	B3 / D4 / A2	B2 / D3
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A5	A3
Graph
Dihedral symmetry	[6]	[4]

Steriruncitruncated 7-orthoplex

steriruncitruncated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t0,1,3,4{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups	B7, [4,35]
Properties	convex

Alternate names

• Celliprismatotruncated hecatonicosoctaexon (acronym: ) (Jonathan Bowers)^[8]

Images

orthographic projections

Coxeter plane	B7 / A6	B6 / D7	B5 / D6 / A4
Graph
Dihedral symmetry	[14]	[12]	[10]
Coxeter plane	B4 / D5	B3 / D4 / A2	B2 / D3
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A5	A3
Graph
Dihedral symmetry	[6]	[4]

Steriruncicantellated 7-orthoplex

steriruncicantellated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t0,2,3,4{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups	B7, [4,35]
Properties	convex

Alternate names

• Celliprismatorhombated hecatonicosoctaexon (acronym: ) (Jonathan Bowers)^[9]

Images

orthographic projections

Coxeter plane	B7 / A6	B6 / D7	B5 / D6 / A4
Graph
Dihedral symmetry	[14]	[12]	[10]
Coxeter plane	B4 / D5	B3 / D4 / A2	B2 / D3
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A5	A3
Graph
Dihedral symmetry	[6]	[4]

Steriruncicantitruncated 7-orthoplex

steriruncicantitruncated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t0,1,2,3,4{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups	B7, [4,35]
Properties	convex

Alternate names

• Great cellated hecatonicosoctaexon (acronym: ) (Jonathan Bowers)^[10]

Images

orthographic projections

Coxeter plane	B7 / A6	B6 / D7	B5 / D6 / A4
Graph
Dihedral symmetry	[14]	[12]	[10]
Coxeter plane	B4 / D5	B3 / D4 / A2	B2 / D3
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A5	A3
Graph
Dihedral symmetry	[6]	[4]

Notes

1. Klitizing, (x3o3o3o3x3o4o - )
2. Klitizing, (x3x3o3o3x3o4o - )
3. Klitizing, (o3x3x3o3o3x4o - )
4. Klitizing, (x3o3x3o3x3o4o - )
5. Klitizing, (x3x3x3o3x3o4o - )
6. Klitizing, (o3x3x3x3o3x4o - )
7. Klitizing, (x3o3o3x3x3o4o - )
8. Klitizing, (x3x3x3o3x3o4o - )
9. Klitizing, (x3o3x3x3x3o4o - )
10. Klitizing, (x3x3x3x3x3o4o - )

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)".

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