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Hexicated 7-orthoplexes

(Redirected from Hexiruncicantellated 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

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.

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
TypeUniform 7-polytope
Schläfli symbolt0,1,6{35,4
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges29568
Vertices5376
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

  • Petitruncated heptacross

Images

orthographic projections
Coxeter planeB7 / A6B6 / D7B5 / D6 / A4
Graph
Dihedral symmetry[14][12][10]
Coxeter planeB4 / D5B3 / D4 / A2B2 / D3
Graph
Dihedral symmetry[8][6][4]
Coxeter planeA5A3
Graph
Dihedral symmetry[6][4]

Hexicantellated 7-orthoplex

Hexicantellated 7-orthoplex
Typeuniform 7-polytope
Schläfli symbolt0,2,6{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges94080
Vertices13440
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

  • Petirhombated heptacross

Images

orthographic projections
Coxeter planeB7 / A6B6 / D7B5 / D6 / A4
Graph
Dihedral symmetry[14][12][10]
Coxeter planeB4 / D5B3 / D4 / A2B2 / D3
Graph
Dihedral symmetry[8][6][4]
Coxeter planeA5A3
Graph
Dihedral symmetry[6][4]

Hexicantitruncated 7-orthoplex

Hexicantitruncated 7-orthoplex
Typeuniform 7-polytope
Schläfli symbolt0,1,2,6{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges134400
Vertices26880
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

  • Petigreatorhombated heptacross

Images

orthographic projections
Coxeter planeB7 / A6B6 / D7B5 / D6 / A4
Graph
Dihedral symmetry[14][12][10]
Coxeter planeB4 / D5B3 / D4 / A2B2 / D3
Graph
Dihedral symmetry[8][6][4]
Coxeter planeA5A3
Graph
Dihedral symmetry[6][4]

Hexiruncitruncated 7-orthoplex

Hexiruncitruncated 7-orthoplex
Typeuniform 7-polytope
Schläfli symbolt0,1,3,6{35,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges322560
Vertices53760
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

  • Petiprismatotruncated heptacross

Images

orthographic projections
Coxeter planeB7 / A6B6 / D7B5 / D6 / A4
Graph
Dihedral symmetry[14][12][10]
Coxeter planeB4 / D5B3 / D4 / A2B2 / D3
Graph
Dihedral symmetry[8][6][4]
Coxeter planeA5A3
Graph
Dihedral symmetry[6][4]

Hexiruncicantellated 7-orthoplex

Hexiruncicantellated 7-orthoplex
Typeuniform 7-polytope
Schläfli symbolt0,2,3,6{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges268800
Vertices53760
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

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

Alternate names

  • Petiprismatorhombated heptacross

Images

orthographic projections
Coxeter planeB7 / A6B6 / D7B5 / D6 / A4
Graph
Dihedral symmetry[14][12][10]
Coxeter planeB4 / D5B3 / D4 / A2B2 / D3
Graph
Dihedral symmetry[8][6][4]
Coxeter planeA5A3
Graph
Dihedral symmetry[6][4]

Hexisteritruncated 7-orthoplex

hexisteritruncated 7-orthoplex
Typeuniform 7-polytope
Schläfli symbolt0,1,4,6{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges322560
Vertices53760
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

  • Peticellitruncated heptacross

Images

orthographic projections
Coxeter planeB7 / A6B6 / D7B5 / D6 / A4
Graph
Dihedral symmetry[14][12][10]
Coxeter planeB4 / D5B3 / D4 / A2B2 / D3
Graph
Dihedral symmetry[8][6][4]
Coxeter planeA5A3
Graph
Dihedral symmetry[6][4]

Hexiruncicantitruncated 7-orthoplex

Hexiruncicantitruncated 7-orthoplex
Typeuniform 7-polytope
Schläfli symbolt0,1,2,3,6{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges483840
Vertices107520
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

  • Petigreatoprismated heptacross

Images

orthographic projections
Coxeter planeB7 / A6B6 / D7B5 / D6 / A4
Graph
Dihedral symmetry[14][12][10]
Coxeter planeB4 / D5B3 / D4 / A2B2 / D3
Graph
Dihedral symmetry[8][6][4]
Coxeter planeA5A3
Graph
Dihedral symmetry[6][4]

Hexistericantitruncated 7-orthoplex

Hexistericantitruncated 7-orthoplex
Typeuniform 7-polytope
Schläfli symbolt0,1,2,4,6{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges806400
Vertices161280
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

  • Peticelligreatorhombated heptacross

Images

orthographic projections
Coxeter planeB7 / A6B6 / D7B5 / D6 / A4
Graph
Dihedral symmetry[14][12][10]
Coxeter planeB4 / D5B3 / D4 / A2B2 / D3
Graph
Dihedral symmetry[8][6][4]
Coxeter planeA5A3
Graph
Dihedral symmetry[6][4]

Hexisteriruncitruncated 7-orthoplex

Hexisteriruncitruncated 7-orthoplex
Typeuniform 7-polytope
Schläfli symbolt0,1,3,4,6{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges725760
Vertices161280
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

  • Peticelliprismatotruncated heptacross

Images

orthographic projections
Coxeter planeB7 / A6B6 / D7B5 / D6 / A4
Graphtoo complex
Dihedral symmetry[14][12][10]
Coxeter planeB4 / D5B3 / D4 / A2B2 / D3
Graph
Dihedral symmetry[8][6][4]
Coxeter planeA5A3
Graph
Dihedral symmetry[6][4]

Hexipenticantitruncated 7-orthoplex

hexipenticantitruncated 7-orthoplex
Typeuniform 7-polytope
Schläfli symbolt0,1,2,5,6{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges483840
Vertices107520
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

  • Petiterigreatorhombated heptacross

Images

orthographic projections
Coxeter planeB7 / A6B6 / D7B5 / D6 / A4
Graph
Dihedral symmetry[14][12][10]
Coxeter planeB4 / D5B3 / D4 / A2B2 / D3
Graph
Dihedral symmetry[8][6][4]
Coxeter planeA5A3
Graph
Dihedral symmetry[6][4]

Hexisteriruncicantitruncated 7-orthoplex

Hexisteriruncicantitruncated 7-orthoplex
Typeuniform 7-polytope
Schläfli symbolt0,1,2,3,4,6{4,35}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges1290240
Vertices322560
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

  • Great petacellated heptacross

Images

orthographic projections
Coxeter planeB7 / A6B6 / D7B5 / D6 / A4
Graphtoo complex
Dihedral symmetry[14][12][10]
Coxeter planeB4 / D5B3 / D4 / A2B2 / D3
Graph
Dihedral symmetry[8][6][4]
Coxeter planeA5A3
Graph
Dihedral symmetry[6][4]

Hexipentiruncicantitruncated 7-orthoplex

Hexipentiruncicantitruncated 7-orthoplex
Typeuniform 7-polytope
Schläfli symbolt0,1,2,3,5,6{35,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges1290240
Vertices322560
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

  • Petiterigreatoprismated heptacross

Images

orthographic projections
Coxeter planeB7 / A6B6 / D7B5 / D6 / A4
Graphtoo complex
Dihedral symmetry[14][12][10]
Coxeter planeB4 / D5B3 / D4 / A2B2 / D3
Graph
Dihedral symmetry[8][6][4]
Coxeter planeA5A3
Graph
Dihedral symmetry[6][4]

Notes

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 [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)".
Fundamental convex regular and uniform polytopes in dimensions 2–10
FamilyAnBnI2(p) / DnE6 / E7 / E8 / F4 / G2Hn
Regular polygonTriangleSquarep-gonHexagonPentagon
Uniform polyhedronTetrahedronOctahedron • CubeDemicubeDodecahedron • Icosahedron
Uniform polychoronPentachoron16-cell • TesseractDemitesseract24-cell120-cell • 600-cell
Uniform 5-polytope5-simplex5-orthoplex • 5-cube5-demicube
Uniform 6-polytope6-simplex6-orthoplex • 6-cube6-demicube122 • 221
Uniform 7-polytope7-simplex7-orthoplex • 7-cube7-demicube132 • 231 • 321
Uniform 8-polytope8-simplex8-orthoplex • 8-cube8-demicube142 • 241 • 421
Uniform 9-polytope9-simplex9-orthoplex • 9-cube9-demicube
Uniform 10-polytope10-simplex10-orthoplex • 10-cube10-demicube
Uniform n-polytopen-simplexn-orthoplex • n-cuben-demicube1k2 • 2k1 • k21n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
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