Truncated 7-cubes

7-cube t0


7-cube


Truncated 7-cube


Bitruncated 7-cube


Tritruncated 7-cube


7-orthoplex


Truncated 7-orthoplex


Bitruncated 7-orthoplex


Tritruncated 7-orthoplex

Orthogonal projections in B7 Coxeter plane

In seven-dimensional geometry, a truncated 7-cube is a convex uniform 7-polytope, being a truncation of the regular 7-cube.

There are 6 truncations for the 7-cube. Vertices of the truncated 7-cube are located as pairs on the edge of the 7-cube. Vertices of the bitruncated 7-cube are located on the square FACES of the 7-cube. Vertices of the tritruncated 7-cube are located inside the cubic cells of the 7-cube. The final three truncations are best expressed relative to the 7-orthoplex. {{-}}

Truncated 7-cube

Truncated 7-cube

Type

Schläfli symbol

Coxeter-Dynkin diagrams

6-faces

5-faces

4-faces

Cells

Faces

Edges

Vertices

Vertex figure

Coxeter groups

Properties

Alternate names

  • Truncated hepteract (Jonathan Bowers)

Coordinates

Cartesian coordinates for the vertices of a truncated 7-cube, centered at the origin, are all sign and coordinate permutations of

(1,1+√2,1+√2,1+√2,1+√2,1+√2,1+√2)

The truncated 7-cube, is sixth in a sequence of truncated hypercubes:

Bitruncated 7-cube

Bitruncated 7-cube

Type

Schläfli symbol

Coxeter-Dynkin diagrams

6-faces

5-faces

4-faces

Cells

Faces

Edges

Vertices

Vertex figure

Coxeter groups

Properties

Alternate names

  • Bitruncated hepteract (Jonathan Bowers)

Coordinates

Cartesian coordinates for the vertices of a bitruncated 7-cube, centered at the origin, are all sign and coordinate permutations of

(±2,±2,±2,±2,±2,±1,0)

The bitruncated 7-cube is fifth in a sequence of bitruncated hypercubes:

Tritruncated 7-cube

Tritruncated 7-cube

Type

Schläfli symbol

Coxeter-Dynkin diagrams

6-faces

5-faces

4-faces

Cells

Faces

Edges

Vertices

Vertex figure

Coxeter groups

Properties

Alternate names

  • Tritruncated hepteract (Jonathan Bowers)

Coordinates

Cartesian coordinates for the vertices of a tritruncated 7-cube, centered at the origin, are all sign and coordinate permutations of

(±2,±2,±2,±2,±1,0,0)
7-cube t01
7-cube t12
7-cube t23
7-cube t34
7-cube t45
7-cube t56
7-cube t6

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,
      • (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.

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