Hexicated 7-orthoplexes

Last updated April 07, 2019

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

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.

Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer.

In seven-dimensional geometry, a 7-polytope is a polytope contained by 6-polytope facets. Each 5-polytope ridge being shared by exactly two 6-polytope facets.

In geometry, a 7-orthoplex, or 7-cross polytope, is a regular 7-polytope with 14 vertices, 84 edges, 280 triangle faces, 560 tetrahedron cells, 672 5-cells 4-faces, 448 5-faces, and 128 6-faces.

Hexitruncated 7-orthoplex

Hexitruncated 7-orthoplex
Type	Uniform 7-polytope
Schläfli symbol	t_0,1,6{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	t_0,2,6{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	t_0,1,2,6{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	t_0,1,3,6{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	t_0,2,3,6{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	t_0,1,4,6{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	t_0,1,2,3,6{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	t_0,1,2,4,6{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	t_0,1,3,4,6{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	t_0,1,2,5,6{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	t_0,1,2,3,4,6{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	t_0,1,2,3,5,6{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]

Notes

Related Research Articles

In five-dimensional geometry, a truncated 5-simplex is a convex uniform 5-polytope, being a truncation of the regular 5-simplex.

In six-dimensional geometry, a rectified 6-cube is a convex uniform 6-polytope, being a rectification of the regular 6-cube.

In five-dimensional geometry, a truncated 5-cube is a convex uniform 5-polytope, being a truncation of the regular 5-cube.

In six-dimensional geometry, a truncated 5-orthoplex is a convex uniform 5-polytope, being a truncation of the regular 5-orthoplex.

In six-dimensional geometry, a truncated 6-cube is a convex uniform 6-polytope, being a truncation of the regular 6-cube.

In six-dimensional geometry, a truncated 6-orthoplex is a convex uniform 6-polytope, being a truncation of the regular 6-orthoplex.

In seven-dimensional geometry, a truncated 7-orthoplex is a convex uniform 7-polytope, being a truncation of the regular 7-orthoplex.

In seven-dimensional geometry, a truncated 7-simplex is a convex uniform 7-polytope, being a truncation of the regular 7-simplex.

In eight-dimensional geometry, a stericated 8-simplex is a convex uniform 8-polytope with 4th order truncations (sterication) of the regular 8-simplex. There are 16 unique sterications for the 8-simplex including permutations of truncation, cantellation, and runcination.

In eight-dimensional geometry, a hexicated 8-simplex is a uniform 8-polytope, being a hexication of the regular 8-simplex.

In eight-dimensional geometry, a truncated 8-orthoplex is a convex uniform 8-polytope, being a truncation of the regular 8-orthoplex.

In five-dimensional geometry, a cantellated 5-orthoplex is a convex uniform 5-polytope, being a cantellation of the regular 5-orthoplex.

In six-dimensional geometry, a cantellated 5-cube is a convex uniform 5-polytope, being a cantellation of the regular 5-cube.

In six-dimensional geometry, a cantellated 6-cube is a convex uniform 6-polytope, being a cantellation of the regular 6-cube.

In eight-dimensional geometry, a truncated 8-cube is a convex uniform 8-polytope, being a truncation of the regular 8-cube.

In six-dimensional geometry, a runcinated 6-orthplex is a convex uniform 6-polytope with 3rd order truncations (runcination) of the regular 6-orthoplex.

In six-dimensional geometry, a stericated 6-cube is a convex uniform 6-polytope, constructed as a sterication of the regular 6-cube.

In six-dimensional geometry, a pentellated 6-orthoplex is a convex uniform 6-polytope with 5th order truncations of the regular 6-orthoplex.

In seven-dimensional geometry, a cantellated 7-orthoplex is a convex uniform 7-polytope, being a cantellation of the regular 7-orthoplex.

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

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, PhD (1966)
Klitzing, Richard. "7D uniform polytopes (polyexa)".

Harold Scott MacDonald Coxeter Canadian mathematician

Harold Scott MacDonald "Donald" Coxeter, FRS, FRSC, was a British-born Canadian geometer. Coxeter is regarded as one of the greatest geometers of the 20th century. He was born in London, received his BA (1929) and PhD (1931) from Cambridge, but lived in Canada from age 29. He was always called Donald, from his third name MacDonald. He was most noted for his work on regular polytopes and higher-dimensional geometries. He was a champion of the classical approach to geometry, in a period when the tendency was to approach geometry more and more via algebra.

International Standard Book Number Unique numeric book identifier

The International Standard Book Number (ISBN) is a numeric commercial book identifier which is intended to be unique. Publishers purchase ISBNs from an affiliate of the International ISBN Agency.

External links

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	A_n	B_n	I₂(p) / D_n	E₆ / E₇ / E₈ / F₄ / G₂	H_n
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform 4-polytope	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
Uniform n-polytope	n-simplex	n-orthoplex • n-cube	n-demicube	1_k2 • 2_k1 • k₂₁	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds

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Hexicated 7-orthoplexes

Contents

Hexitruncated 7-orthoplex

Alternate names

Images

Hexicantellated 7-orthoplex

Alternate names

Images

Hexicantitruncated 7-orthoplex

Alternate names

Images

Hexiruncitruncated 7-orthoplex

Alternate names

Images

Hexiruncicantellated 7-orthoplex

Alternate names

Images

Hexisteritruncated 7-orthoplex

Alternate names

Images

Hexiruncicantitruncated 7-orthoplex

Alternate names

Images

Hexistericantitruncated 7-orthoplex

Alternate names

Images

Hexisteriruncitruncated 7-orthoplex

Alternate names

Images

Hexipenticantitruncated 7-orthoplex

Alternate names

Images

Hexisteriruncicantitruncated 7-orthoplex

Alternate names

Images

Hexipentiruncicantitruncated 7-orthoplex

Alternate names

Images

Notes

Related Research Articles

References

External links