Pentic 7-cubes

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7-demicube t0 D7.svg
7-demicube
(half 7-cube, h{4,35})
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
7-demicube t04 D7.svg
Pentic 7-cube
h5{4,35}
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
7-demicube t014 D7.svg
Penticantic 7-cube
h2,5{4,35}
CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
7-demicube t024 D7.svg
Pentiruncic 7-cube
h3,5{4,35}
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
7-demicube t0124 D7.svg
Pentiruncicantic 7-cube
h2,3,5{4,35}
CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
7-demicube t034 D7.svg
Pentisteric 7-cube
h4,5{4,35}
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
7-demicube t0134 D7.svg
Pentistericantic 7-cube
h2,4,5{4,35}
CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
7-demicube t0234 D7.svg
Pentisteriruncic 7-cube
h3,4,5{4,35}
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
7-demicube t01234 D7.svg
Penticsteriruncicantic 7-cube
h2,3,4,5{4,35}
CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
Orthogonal projections in D7 Coxeter plane

In seven-dimensional geometry, a pentic 7-cube is a convex uniform 7-polytope, related to the uniform 7-demicube. There are 8 unique forms.

Pentic 7-cube[edit]

Pentic 7-cube
Type uniform 7-polytope
Schläfli symbol t0,4{3,34,1}
h5{4,35}
Coxeter-Dynkin diagram CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
CDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
5-faces
4-faces
Cells
Faces
Edges 13440
Vertices 1344
Vertex figure
Coxeter groups D7, [34,1,1]
Properties convex

Cartesian coordinates[edit]

The Cartesian coordinates for the vertices of a pentic 7-cube centered at the origin are coordinate permutations:

(±1,±1,±1,±1,±1,±3,±3)

with an odd number of plus signs.

Images[edit]

orthographic projections
Coxeter
plane
B7 D7 D6
Graph 7-demicube t04 B7.svg 7-demicube t04 D7.svg 7-demicube t04 D6.svg
Dihedral
symmetry
[14/2] [12] [10]
Coxeter plane D5 D4 D3
Graph 7-demicube t04 D5.svg 7-demicube t04 D4.svg 7-demicube t04 D3.svg
Dihedral
symmetry
[8] [6] [4]
Coxeter
plane
A5 A3
Graph 7-demicube t04 A5.svg 7-demicube t04 A3.svg
Dihedral
symmetry
[6] [4]

Related polytopes[edit]

Penticantic 7-cube[edit]

Images[edit]

orthographic projections
Coxeter
plane
B7 D7 D6
Graph 7-demicube t014 B7.svg 7-demicube t014 D7.svg 7-demicube t014 D6.svg
Dihedral
symmetry
[14/2] [12] [10]
Coxeter plane D5 D4 D3
Graph 7-demicube t014 D5.svg 7-demicube t014 D4.svg 7-demicube t014 D3.svg
Dihedral
symmetry
[8] [6] [4]
Coxeter
plane
A5 A3
Graph 7-demicube t014 A5.svg 7-demicube t014 A3.svg
Dihedral
symmetry
[6] [4]

Pentiruncic 7-cube[edit]

Images[edit]

orthographic projections
Coxeter
plane
B7 D7 D6
Graph 7-demicube t024 B7.svg 7-demicube t024 D7.svg 7-demicube t024 D6.svg
Dihedral
symmetry
[14/2] [12] [10]
Coxeter plane D5 D4 D3
Graph 7-demicube t024 D5.svg 7-demicube t024 D4.svg 7-demicube t024 D3.svg
Dihedral
symmetry
[8] [6] [4]
Coxeter
plane
A5 A3
Graph 7-demicube t024 A5.svg 7-demicube t024 A3.svg
Dihedral
symmetry
[6] [4]

Pentiruncicantic 7-cube[edit]

Images[edit]

orthographic projections
Coxeter
plane
B7 D7 D6
Graph 7-demicube t0124 B7.svg 7-demicube t0124 D7.svg 7-demicube t0124 D6.svg
Dihedral
symmetry
[14/2] [12] [10]
Coxeter plane D5 D4 D3
Graph 7-demicube t0124 D5.svg 7-demicube t0124 D4.svg 7-demicube t0124 D3.svg
Dihedral
symmetry
[8] [6] [4]
Coxeter
plane
A5 A3
Graph 7-demicube t0124 A5.svg 7-demicube t0124 A3.svg
Dihedral
symmetry
[6] [4]

Pentisteric 7-cube[edit]

Images[edit]

orthographic projections
Coxeter
plane
B7 D7 D6
Graph 7-demicube t034 B7.svg 7-demicube t034 D7.svg 7-demicube t034 D6.svg
Dihedral
symmetry
[14/2] [12] [10]
Coxeter plane D5 D4 D3
Graph 7-demicube t034 D5.svg 7-demicube t034 D4.svg 7-demicube t034 D3.svg
Dihedral
symmetry
[8] [6] [4]
Coxeter
plane
A5 A3
Graph 7-demicube t034 A5.svg 7-demicube t034 A3.svg
Dihedral
symmetry
[6] [4]

Pentistericantic 7-cube[edit]

Images[edit]

orthographic projections
Coxeter
plane
B7 D7 D6
Graph 7-demicube t0134 B7.svg 7-demicube t0134 D7.svg 7-demicube t0134 D6.svg
Dihedral
symmetry
[14/2] [12] [10]
Coxeter plane D5 D4 D3
Graph 7-demicube t0134 D5.svg 7-demicube t0134 D4.svg 7-demicube t0134 D3.svg
Dihedral
symmetry
[8] [6] [4]
Coxeter
plane
A5 A3
Graph 7-demicube t0134 A5.svg 7-demicube t0134 A3.svg
Dihedral
symmetry
[6] [4]

Pentisterirunic 7-cube[edit]

Images[edit]

orthographic projections
Coxeter
plane
B7 D7 D6
Graph 7-demicube t0234 B7.svg 7-demicube t0234 D7.svg 7-demicube t0234 D6.svg
Dihedral
symmetry
[14/2] [12] [10]
Coxeter plane D5 D4 D3
Graph 7-demicube t0234 D5.svg 7-demicube t0234 D4.svg 7-demicube t0234 D3.svg
Dihedral
symmetry
[8] [6] [4]
Coxeter
plane
A5 A3
Graph 7-demicube t0234 A5.svg 7-demicube t0234 A3.svg
Dihedral
symmetry
[6] [4]

Pentisteriruncicantic 7-cube[edit]

Images[edit]

orthographic projections
Coxeter
plane
B7 D7 D6
Graph 7-demicube t01234 B7.svg 7-demicube t01234 D7.svg 7-demicube t01234 D6.svg
Dihedral
symmetry
[14/2] [12] [10]
Coxeter plane D5 D4 D3
Graph 7-demicube t01234 D5.svg 7-demicube t01234 D4.svg 7-demicube t01234 D3.svg
Dihedral
symmetry
[8] [6] [4]
Coxeter
plane
A5 A3
Graph 7-demicube t01234 A5.svg 7-demicube t01234 A3.svg
Dihedral
symmetry
[6] [4]

Related polytopes[edit]

This polytope is based on the 7-demicube, a part of a dimensional family of uniform polytopes called demihypercubes for being alternation of the hypercube family.

There are 95 uniform polytopes with D7 symmetry, 63 are shared by the BC7 symmetry, and 32 are unique:

Notes[edit]

References[edit]

  • 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, Ph.D.
  • Klitzing, Richard. "7D uniform polytopes (polyexa)". 

External links[edit]

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 OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds