Hexicated 7-orthoplexes

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

Orthogonal projections in B4 Coxeter plane
7-orthoplex	Hexicated 7-orthoplex Hexicated 7-cube	Hexi-truncated 7-orthoplex	Hexi-cantellated 7-orthoplex	Hexicanti-truncated 7-orthoplex
Hexirunci-truncated 7-orthoplex	Hexirunci-cantellated 7-orthoplex	Hexisteri-truncated 7-orthoplex	Hexiruncicanti-truncated 7-orthoplex	Hexistericanti-truncated 7-orthoplex
Hexisterirunci-truncated 7-orthoplex	Hexipenticanti-truncated 7-orthoplex	Hexisteriruncicanti-truncated 7-orthoplex	Hexipentiruncicanti-truncated 7-orthoplex

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

Hexitruncated 7-orthoplex

Hexitruncated 7-orthoplex
Type	Uniform 7-polytope
Schläfli symbol	t0,1,6{35,4
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	29568
Vertices	5376
Vertex figure
Coxeter groups	B7, [4,35]
Properties	convex

Alternate names

• Petitruncated heptacross

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]

Hexicantellated 7-orthoplex

Hexicantellated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t0,2,6{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	94080
Vertices	13440
Vertex figure
Coxeter groups	B7, [4,35]
Properties	convex

Alternate names

• Petirhombated heptacross

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]

Hexicantitruncated 7-orthoplex

Hexicantitruncated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t0,1,2,6{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	134400
Vertices	26880
Vertex figure
Coxeter groups	B7, [4,35]
Properties	convex

Alternate names

• Petigreatorhombated heptacross

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]

Hexiruncitruncated 7-orthoplex

Hexiruncitruncated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t0,1,3,6{35,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	322560
Vertices	53760
Vertex figure
Coxeter groups	B7, [4,35]
Properties	convex

Alternate names

• Petiprismatotruncated heptacross

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]

Hexiruncicantellated 7-orthoplex

Hexiruncicantellated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t0,2,3,6{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	268800
Vertices	53760
Vertex figure
Coxeter groups	B7, [4,35]
Properties	convex

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

Alternate names

• Petiprismatorhombated heptacross

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]

Hexisteritruncated 7-orthoplex

hexisteritruncated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t0,1,4,6{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	322560
Vertices	53760
Vertex figure
Coxeter groups	B7, [4,35]
Properties	convex

Alternate names

• Peticellitruncated heptacross

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]

Hexiruncicantitruncated 7-orthoplex

Hexiruncicantitruncated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t0,1,2,3,6{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	483840
Vertices	107520
Vertex figure
Coxeter groups	B7, [4,35]
Properties	convex

Alternate names

• Petigreatoprismated heptacross

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]

Hexistericantitruncated 7-orthoplex

Hexistericantitruncated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t0,1,2,4,6{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	806400
Vertices	161280
Vertex figure
Coxeter groups	B7, [4,35]
Properties	convex

Alternate names

• Peticelligreatorhombated heptacross

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]

Hexisteriruncitruncated 7-orthoplex

Hexisteriruncitruncated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t0,1,3,4,6{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	725760
Vertices	161280
Vertex figure
Coxeter groups	B7, [4,35]
Properties	convex

Alternate names

• Peticelliprismatotruncated heptacross

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]

Hexipenticantitruncated 7-orthoplex

hexipenticantitruncated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t0,1,2,5,6{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	483840
Vertices	107520
Vertex figure
Coxeter groups	B7, [4,35]
Properties	convex

Alternate names

• Petiterigreatorhombated heptacross

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]

Hexisteriruncicantitruncated 7-orthoplex

Hexisteriruncicantitruncated 7-orthoplex
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	1290240
Vertices	322560
Vertex figure
Coxeter groups	B7, [4,35]
Properties	convex

Alternate names

• Great petacellated heptacross

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]

Hexipentiruncicantitruncated 7-orthoplex

Hexipentiruncicantitruncated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t0,1,2,3,5,6{35,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	1290240
Vertices	322560
Vertex figure
Coxeter groups	B7, [4,35]
Properties	convex

Alternate names

• Petiterigreatoprismated heptacross

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]