Hexicated 7-cubes

Orthogonal projections in B4 Coxeter plane
7-cube	Hexicated 7-cube	Hexitruncated 7-cube	Hexicantellated 7-cube
Hexiruncinated 7-cube	Hexicantitruncated 7-cube	Hexiruncitruncated 7-cube	Hexiruncicantellated 7-cube
Hexisteritruncated 7-cube	Hexistericantellated 7-cube	Hexipentitruncated 7-cube	Hexiruncicantitruncated 7-cube
Hexistericantitruncated 7-cube	Hexisteriruncitruncated 7-cube	Hexisteriruncicantellated 7-cube	Hexipenticantitruncated 7-cube
Hexipentiruncitruncated 7-cube	Hexisteriruncicantitruncated 7-cube	Hexipentiruncicantitruncated 7-cube	Hexipentistericantitruncated 7-cube
Hexipentisteriruncicantitruncated 7-cube (Omnitruncated 7-cube)

In seven-dimensional geometry, a hexicated 7-cube is a convex uniform 7-polytope, including 6th-order truncations (hexication) from the regular 7-cube.

There are 32 hexications for the 7-cube, including all permutations of truncations, cantellations, runcinations, sterications, and pentellations. 20 are represented here, while 12 are more easily constructed from the 7-orthoplex.

The simple hexicated 7-cube is also called an expanded 7-cube, with only the first and last nodes ringed, is constructed by an expansion operation applied to the regular 7-cube. The highest form, the hexipentisteriruncicantitruncated 7-cube is more simply called a omnitruncated 7-cube with all of the nodes ringed.

These polytope are among a family of 127 uniform 7-polytopes with B₇ symmetry.

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

In seven-dimensional geometry, a hexicated 7-cube is a convex uniform 7-polytope, a hexication (6th order truncation) of the regular 7-cube, or alternately can be seen as an expansion operation.

• Small petated hepteract (acronym: ) (Jonathan Bowers)

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]

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

• Petitruncated hepteract (acronym: ) (Jonathan Bowers)

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]

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

• Petirhombated hepteract (acronym: ) (Jonathan Bowers)

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]

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

• Petiprismated hepteract (acronym: ) (Jonathan Bowers)

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]

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

• Petigreatorhombated hepteract (acronym: ) (Jonathan Bowers)

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]

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

• Petiprismatotruncated hepteract (acronym: ) (Jonathan Bowers)

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]

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

In seven-dimensional geometry, a hexiruncicantellated 7-cube is a uniform 7-polytope.

• Petiprismatorhombated hepteract (acronym: ) (Jonathan Bowers)

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]

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

• Peticellitruncated hepteract (acronym: ) (Jonathan Bowers)

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]

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

• Peticellirhombihepteract (acronym: ) (Jonathan Bowers)

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]

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

• Petiteritruncated hepteract (acronym: ) (Jonathan Bowers)

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]

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

• Petigreatoprismated hepteract (acronym: ) (Jonathan Bowers)

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	too complex	too complex
Dihedral symmetry	[6]	[4]

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

• Peticelligreatorhombated hepteract (acronym: ) (Jonathan Bowers)

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]

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

• Peticelliprismatotruncated hepteract (acronym: ) (Jonathan Bowers)

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]

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

• Peticelliprismatorhombihepteract (acronym: ) (Jonathan Bowers)

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]

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

• Petiterigreatorhombated hepteract (acronym: ) (Jonathan Bowers)

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]

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

• Great petacellated hepteract (acronym: ) (Jonathan Bowers)

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]

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

• Great petacellated hepteract (acronym: ) (Jonathan Bowers)

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]

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

• Petiterigreatoprismated hepteract (acronym: ) (Jonathan Bowers)

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]

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

• Petitericelligreatorhombihepteract (acronym: putcagroh) (Jonathan Bowers)

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]

Omnitruncated 7-cube
Type	uniform 7-polytope
Schläfli symbol	t0,1,2,3,4,5,6{36}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups	B7, [4,35]
Properties	convex

The omnitruncated 7-cube is the largest uniform 7-polytope in the B₇ symmetry of the regular 7-cube. It can also be called the hexipentisteriruncicantitruncated 7-cube which is the long name for the omnitruncation for 7 dimensions, with all reflective mirrors active.

• Great petated hepteract (Acronym: ) (Jonathan Bowers)

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]

• 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, PhD (1966)
• Klitzing, Richard. "7D uniform polytopes (polyexa)". x3o3o3o3o3o4x - , x3x3o3o3o3o3x- , x3o3o3x3o3o4x - , x3x3x3o3o3o4x - , x3x3o3x3o3o4x - , x3o3x3x3o3o4x - , x3o3x3o3o3x4x - , x3o3x3o3x3o4x - , x3x3o3o3o3x4x - , x3x3x3x3o3o4x - , x3x3x3o3x3o4x - , x3x3o3x3x3o4x - , x3o3x3x3x3o4x - , x3x3x3oxo3x4x - , x3x3x3x3x3o4x - , x3x3x3o3x3x4x - , x3x3o3x3x3x4x - , x3x3x3x3x3x4x -

• Polytopes of Various Dimensions
• Multi-dimensional Glossary