Pentic 7-cubes

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

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

Cartesian coordinates

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

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

Related polytopes

Dimensional family of pentic n-cubes
n 6 7 8
[1+,4,3n-2]
= [3,3n-3,1] 		[1+,4,34]
= [3,33,1] 		[1+,4,35]
= [3,34,1] 		[1+,4,36]
= [3,35,1]
Cantic
figure 		File:6-demicube t04 D6.svg File:7-demicube t04 D7.svg File:8-demicube t04 D8.svg
Coxeter File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png
= File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png 		File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png
= File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png 		File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
= File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
Schläfli h5{4,34} h5{4,35} h5{4,36}

Penticantic 7-cube

Images

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

Pentiruncic 7-cube

Images

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

Pentiruncicantic 7-cube

Images

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

Pentisteric 7-cube

Images

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

Pentistericantic 7-cube

Images

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

Pentisteriruncic 7-cube

Images

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

Pentisteriruncicantic 7-cube

Images

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

Related polytopes

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:

D7 polytopes
File:7-demicube t0 D7.svg
t0(141) 		File:7-demicube t01 D7.svg
t0,1(141) 		File:7-demicube t02 D7.svg
t0,2(141) 		File:7-demicube t03 D7.svg
t0,3(141) 		File:7-demicube t04 D7.svg
t0,4(141) 		File:7-demicube t05 D7.svg
t0,5(141) 		File:7-demicube t012 D7.svg
t0,1,2(141) 		File:7-demicube t013 D7.svg
t0,1,3(141)
File:7-demicube t014 D7.svg
t0,1,4(141) 		File:7-demicube t015 D7.svg
t0,1,5(141) 		File:7-demicube t023 D7.svg
t0,2,3(141) 		File:7-demicube t024 D7.svg
t0,2,4(141) 		File:7-demicube t025 D7.svg
t0,2,5(141) 		File:7-demicube t034 D7.svg
t0,3,4(141) 		File:7-demicube t035 D7.svg
t0,3,5(141) 		File:7-demicube t045 D7.svg
t0,4,5(141)
File:7-demicube t0123 D7.svg
t0,1,2,3(141) 		File:7-demicube t0124 D7.svg
t0,1,2,4(141) 		File:7-demicube t0125 D7.svg
t0,1,2,5(141) 		File:7-demicube t0134 D7.svg
t0,1,3,4(141) 		File:7-demicube t0135 D7.svg
t0,1,3,5(141) 		File:7-demicube t0145 D7.svg
t0,1,4,5(141) 		File:7-demicube t0234 D7.svg
t0,2,3,4(141) 		File:7-demicube t0235 D7.svg
t0,2,3,5(141)
File:7-demicube t0245 D7.svg
t0,2,4,5(141) 		File:7-demicube t0345 D7.svg
t0,3,4,5(141) 		File:7-demicube t01234 D7.svg
t0,1,2,3,4(141) 		File:7-demicube t01235 D7.svg
t0,1,2,3,5(141) 		File:7-demicube t01245 D7.svg
t0,1,2,4,5(141) 		File:7-demicube t01345 D7.svg
t0,1,3,4,5(141) 		File:7-demicube t02345 D7.svg
t0,2,3,4,5(141) 		File:7-demicube t012345 D7.svg
t0,1,2,3,4,5(141)

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

External links

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 Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform polychoron Pentachoron 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 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
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