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Stericated 6-simplexes

Orthogonal projections in A6 Coxeter plane

In six-dimensional geometry, a stericated 6-simplex is a convex uniform 6-polytope with 4th order truncations (sterication) of the regular 6-simplex.

There are 8 unique sterications for the 6-simplex with permutations of truncations, cantellations, and runcinations.

Stericated 6-simplex

Stericated 6-simplex
Type
Schläfli symbol
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter group
Properties

Alternate names

  • Small cellated heptapeton (Acronym: scal) (Jonathan Bowers)

Coordinates

The vertices of the stericated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,1,1,1,2). This construction is based on facets of the stericated 7-orthoplex.

Images

Steritruncated 6-simplex

Steritruncated 6-simplex
Type
Schläfli symbol
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter group
Properties

Alternate names

  • Cellitruncated heptapeton (Acronym: catal) (Jonathan Bowers)

Coordinates

The vertices of the steritruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,1,1,2,3). This construction is based on facets of the steritruncated 7-orthoplex.

Images

Stericantellated 6-simplex

Stericantellated 6-simplex
Type
Schläfli symbol
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter group
Properties

Alternate names

  • Cellirhombated heptapeton (Acronym: cral) (Jonathan Bowers)

Coordinates

The vertices of the stericantellated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,1,2,2,3). This construction is based on facets of the stericantellated 7-orthoplex.

Images

Stericantitruncated 6-simplex

stericantitruncated 6-simplex
Type
Schläfli symbol
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter group
Properties

Alternate names

  • Celligreatorhombated heptapeton (Acronym: cagral) (Jonathan Bowers)

Coordinates

The vertices of the stericanttruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,2,3,4). This construction is based on facets of the stericantitruncated 7-orthoplex.

Images

Steriruncinated 6-simplex

steriruncinated 6-simplex
Type
Schläfli symbol
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter group
Properties

Alternate names

  • Celliprismated heptapeton (Acronym: copal) (Jonathan Bowers)

Coordinates

The vertices of the steriruncinated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,2,2,3,3). This construction is based on facets of the steriruncinated 7-orthoplex.

Images

Steriruncitruncated 6-simplex

steriruncitruncated 6-simplex
Type
Schläfli symbol
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter group
Properties

Alternate names

  • Celliprismatotruncated heptapeton (Acronym: captal) (Jonathan Bowers)

Coordinates

The vertices of the steriruncittruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,2,3,4). This construction is based on facets of the steriruncitruncated 7-orthoplex.

Images

Steriruncicantellated 6-simplex

steriruncicantellated 6-simplex
Type
Schläfli symbol
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter group
Properties

Alternate names

  • Bistericantitruncated 6-simplex as t1,2,3,5{3,3,3,3,3}
  • Celliprismatorhombated heptapeton (Acronym: copril) (Jonathan Bowers)

Coordinates

The vertices of the steriruncitcantellated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,2,3,4). This construction is based on facets of the steriruncicantellated 7-orthoplex.

Images

Steriruncicantitruncated 6-simplex

Steriuncicantitruncated 6-simplex
Type
Schläfli symbol
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter group
Properties

Alternate names

  • Great cellated heptapeton (Acronym: gacal) (Jonathan Bowers)

Coordinates

The vertices of the steriruncicantittruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,2,3,4,5). This construction is based on facets of the steriruncicantitruncated 7-orthoplex.

Images

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, 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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