Pentiruncic 7-cube

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7-demicube
(half 7-cube, h{4,35})
Pentic 7-cube
h5{4,35}
Penticantic 7-cube
h2,5{4,35}


Pentiruncic 7-cube
h3,5{4,35}
Pentiruncicantic 7-cube
h2,3,5{4,35}
Pentisteric 7-cube
h4,5{4,35}


Pentistericantic 7-cube
h2,4,5{4,35}
Pentisteriruncic 7-cube
h3,4,5{4,35}
Penticsteriruncicantic 7-cube
h2,3,4,5{4,35}

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
Typeuniform 7-polytope
Schläfli symbolt0,4{3,34,1}
h5{4,35}
Coxeter-Dynkin diagram
6-faces
5-faces
4-faces
Cells
Faces
Edges13440
Vertices1344
Vertex figure
Coxeter groupsD7, [34,1,1]
Propertiesconvex

Alternate names

  • Small cellated demihepteract (acronym: sochesa)[1]

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
Dihedral
symmetry		 [14/2] [12] [10]
Coxeter plane D5 D4 D3
Graph
Dihedral
symmetry		 [8] [6] [4]
Coxeter
plane		 A5 A3
Graph
Dihedral
symmetry		 [6] [4]
Dimensional family of pentic n-cubes
n678
[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
Coxeter
=
=
=
Schläfli h5{4,34} h5{4,35} h5{4,36}

Penticantic 7-cube

Alternate names

  • Cellitruncated demihepteract (acronym: cothesa)[2]

Images

Orthographic projections
Coxeter
plane		 B7 D7 D6
Graph
Dihedral
symmetry		 [14/2] [12] [10]
Coxeter plane D5 D4 D3
Graph
Dihedral
symmetry		 [8] [6] [4]
Coxeter
plane		 A5 A3
Graph
Dihedral
symmetry		 [6] [4]

Pentiruncic 7-cube

Alternate names

  • Cellirhombated demihepteract (acronym: crohesa)[3]

Images

Orthographic projections
Coxeter
plane		 B7 D7 D6
Graph
Dihedral
symmetry		 [14/2] [12] [10]
Coxeter plane D5 D4 D3
Graph
Dihedral
symmetry		 [8] [6] [4]
Coxeter
plane		 A5 A3
Graph
Dihedral
symmetry		 [6] [4]

Pentiruncicantic 7-cube

Alternate names

  • Celligreatorhombated demihepteract (acronym: cagrohesa)[4]

Images

Orthographic projections
Coxeter
plane		 B7 D7 D6
Graph
Dihedral
symmetry		 [14/2] [12] [10]
Coxeter plane D5 D4 D3
Graph
Dihedral
symmetry		 [8] [6] [4]
Coxeter
plane		 A5 A3
Graph
Dihedral
symmetry		 [6] [4]

Pentisteric 7-cube

Alternate names

  • Celliprismated demihepteract (acronym: caphesa)[5]

Images

Orthographic projections
Coxeter
plane		 B7 D7 D6
Graph
Dihedral
symmetry		 [14/2] [12] [10]
Coxeter plane D5 D4 D3
Graph
Dihedral
symmetry		 [8] [6] [4]
Coxeter
plane		 A5 A3
Graph
Dihedral
symmetry		 [6] [4]

Pentistericantic 7-cube

Alternate names

  • Celliprismatotruncated demihepteract (acronym: capthesa)[6]

Images

Orthographic projections
Coxeter
plane		 B7 D7 D6
Graph
Dihedral
symmetry		 [14/2] [12] [10]
Coxeter plane D5 D4 D3
Graph
Dihedral
symmetry		 [8] [6] [4]
Coxeter
plane		 A5 A3
Graph
Dihedral
symmetry		 [6] [4]

Pentisteriruncic 7-cube

Alternate names

  • Celliprismatorhombated demihepteract (acronym: coprahesa)[7]

Images

Orthographic projections
Coxeter
plane		 B7 D7 D6
Graph
Dihedral
symmetry		 [14/2] [12] [10]
Coxeter plane D5 D4 D3
Graph
Dihedral
symmetry		 [8] [6] [4]
Coxeter
plane		 A5 A3
Graph
Dihedral
symmetry		 [6] [4]

Pentisteriruncicantic 7-cube

Alternate names

  • Great cellated demihepteract (acronym: gochesa)[8]

Images

Orthographic projections
Coxeter
plane		 B7 D7 D6
Graph
Dihedral
symmetry		 [14/2] [12] [10]
Coxeter plane D5 D4 D3
Graph
Dihedral
symmetry		 [8] [6] [4]
Coxeter
plane		 A5 A3
Graph
Dihedral
symmetry		 [6] [4]

These polytopes are based on the 7-demicube, a member 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

t0(141)
t0,1(141)
t0,2(141)
t0,3(141)
t0,4(141)
t0,5(141)
t0,1,2(141)
t0,1,3(141)

t0,1,4(141)
t0,1,5(141)
t0,2,3(141)
t0,2,4(141)
t0,2,5(141)
t0,3,4(141)
t0,3,5(141)
t0,4,5(141)

t0,1,2,3(141)
t0,1,2,4(141)
t0,1,2,5(141)
t0,1,3,4(141)
t0,1,3,5(141)
t0,1,4,5(141)
t0,2,3,4(141)
t0,2,3,5(141)

t0,2,4,5(141)
t0,3,4,5(141)
t0,1,2,3,4(141)
t0,1,2,3,5(141)
t0,1,2,4,5(141)
t0,1,3,4,5(141)
t0,2,3,4,5(141)
t0,1,2,3,4,5(141)

Notes

  1. Klitzing, (x3o3o *b3o3o3x3o - sochesa)
  2. Klitzing, (x3x3o *b3o3o3x3o - cothesa)
  3. Klitzing, (x3o3o *b3x3o3x3o - crohesa)
  4. Klitzing, (x3x3o *b3x3o3x3o - cagrohesa)
  5. Klitzing, (x3o3o *b3o3x3x3o - caphesa)
  6. Klitzing, (x3x3o *b3o3x3x3o - capthesa)
  7. Klitzing, (x3o3o *b3x3x3x3o - coprahesa)
  8. Klitzing, (x3x3o *b3x3x3x3o - gochesa)

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 Ivić Weiss, Wiley-Interscience Publication, 1995, wiley.com, 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, Ph.D.
  • Klitzing, Richard. "7D uniform polytopes (polyexa) with acronyms".
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 • Polytope operations