Runcic 7-cubes


7-demicube


Runcic 7-cube


Runcicantic 7-cube

Orthogonal projections in D7 Coxeter plane

In seven-dimensional geometry, a runcic 7-cube is a convex uniform 7-polytope, related to the uniform 7-demicube. There are 2 unique forms.

Runcic 7-cube

Runcic 7-cube
Typeuniform 7-polytope
Schläfli symbol t0,2{3,34,1}
h3{4,35}
Coxeter-Dynkin diagram
5-faces
4-faces
Cells
Faces
Edges16800
Vertices2240
Vertex figure
Coxeter groupsD7, [34,1,1]
Propertiesconvex

A runcic 7-cube, h3{4,35}, has half the vertices of a runcinated 7-cube, t0,3{4,35}.

Alternate names

Cartesian coordinates

The Cartesian coordinates for the vertices of a cantellated demihepteract centered at the origin are coordinate permutations:

(±1,±1,±1,±3,±3,±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]

Runcicantic 7-cube

Runcicantic 7-cube
Typeuniform 7-polytope
Schläfli symbol t0,1,2{3,34,1}
h2,3{4,35}
Coxeter-Dynkin diagram
5-faces
4-faces
Cells
Faces
Edges23520
Vertices6720
Vertex figure
Coxeter groupsD6, [33,1,1]
Propertiesconvex

A runcicantic 7-cube, h2,3{4,35}, has half the vertices of a runcicantellated 7-cube, t0,1,3{4,35}.

Alternate names

Cartesian coordinates

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

(±1,±1,±1,±1,±3,±5,±5)

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]

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 BC6 symmetry, and 32 are unique:

Notes

  1. Klitzing, (x3o3o *b3x3o3o3o - sirhesa)
  2. Klitzing, (x3x3o *b3x3o3o3o - girhesa)

References

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / E9 / E10 / 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
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