# Stericated 6-simplexes

(Redirected from Stericated 6-simplex)
 Orthogonal projections in A6 Coxeter plane 6-simplex Stericated 6-simplex Steritruncated 6-simplex Stericantellated 6-simplex Stericantitruncated 6-simplex Steriruncinated 6-simplex Steriruncitruncated 6-simplex Steriruncicantellated 6-simplex Steriruncicantitruncated 6-simplex

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 uniform 6-polytope
Schläfli symbol t0,4{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces 105
4-faces 700
Cells 1470
Faces 1400
Edges 630
Vertices 105
Vertex figure
Coxeter group A6, [35], order 5040
Properties convex

### Alternate names

• Small cellated heptapeton (Acronym: scal) (Jonathan Bowers)[1]

### 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

orthographic projections
Ak Coxeter plane A6 A5 A4
Graph
Dihedral symmetry [7] [6] [5]
Ak Coxeter plane A3 A2
Graph
Dihedral symmetry [4] [3]

## Steritruncated 6-simplex

Steritruncated 6-simplex
Type uniform 6-polytope
Schläfli symbol t0,1,4{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces 105
4-faces 945
Cells 2940
Faces 3780
Edges 2100
Vertices 420
Vertex figure
Coxeter group A6, [35], order 5040
Properties convex

### Alternate names

• Cellirhombated heptapeton (Acronym: catal) (Jonathan Bowers)[2]

### 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

orthographic projections
Ak Coxeter plane A6 A5 A4
Graph
Dihedral symmetry [7] [6] [5]
Ak Coxeter plane A3 A2
Graph
Dihedral symmetry [4] [3]

## Stericantellated 6-simplex

Stericantellated 6-simplex
Type uniform 6-polytope
Schläfli symbol t0,2,4{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces 105
4-faces 1050
Cells 3465
Faces 5040
Edges 3150
Vertices 630
Vertex figure
Coxeter group A6, [35], order 5040
Properties convex

### Alternate names

• Cellirhombated heptapeton (Acronym: cral) (Jonathan Bowers)[3]

### 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

orthographic projections
Ak Coxeter plane A6 A5 A4
Graph
Dihedral symmetry [7] [6] [5]
Ak Coxeter plane A3 A2
Graph
Dihedral symmetry [4] [3]

## Stericantitruncated 6-simplex

stericantitruncated 6-simplex
Type uniform 6-polytope
Schläfli symbol t0,1,2,4{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces 105
4-faces 1155
Cells 4410
Faces 7140
Edges 5040
Vertices 1260
Vertex figure
Coxeter group A6, [35], order 5040
Properties convex

### Alternate names

• Celligreatorhombated heptapeton (Acronym: cagral) (Jonathan Bowers)[4]

### 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

orthographic projections
Ak Coxeter plane A6 A5 A4
Graph
Dihedral symmetry [7] [6] [5]
Ak Coxeter plane A3 A2
Graph
Dihedral symmetry [4] [3]

## Steriruncinated 6-simplex

steriruncinated 6-simplex
Type uniform 6-polytope
Schläfli symbol t0,3,4{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces 105
4-faces 700
Cells 1995
Faces 2660
Edges 1680
Vertices 420
Vertex figure
Coxeter group A6, [35], order 5040
Properties convex

### Alternate names

• Celliprismated heptapeton (Acronym: copal) (Jonathan Bowers)[5]

### 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

orthographic projections
Ak Coxeter plane A6 A5 A4
Graph
Dihedral symmetry [7] [6] [5]
Ak Coxeter plane A3 A2
Graph
Dihedral symmetry [4] [3]

## Steriruncitruncated 6-simplex

steriruncitruncated 6-simplex
Type uniform 6-polytope
Schläfli symbol t0,1,3,4{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces 105
4-faces 945
Cells 3360
Faces 5670
Edges 4410
Vertices 1260
Vertex figure
Coxeter group A6, [35], order 5040
Properties convex

### Alternate names

• Celliprismatotruncated heptapeton (Acronym: captal) (Jonathan Bowers)[6]

### 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

orthographic projections
Ak Coxeter plane A6 A5 A4
Graph
Dihedral symmetry [7] [6] [5]
Ak Coxeter plane A3 A2
Graph
Dihedral symmetry [4] [3]

## Steriruncicantellated 6-simplex

steriruncicantellated 6-simplex
Type uniform 6-polytope
Schläfli symbol t0,2,3,4{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces 105
4-faces 1050
Cells 3675
Faces 5880
Edges 4410
Vertices 1260
Vertex figure
Coxeter group A6, [35], order 5040
Properties convex

### Alternate names

• Bistericantitruncated 6-simplex as t1,2,3,5{3,3,3,3,3}
• Celliprismatorhombated heptapeton (Acronym: copril) (Jonathan Bowers)[7]

### 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

orthographic projections
Ak Coxeter plane A6 A5 A4
Graph
Dihedral symmetry [7] [6] [5]
Ak Coxeter plane A3 A2
Graph
Dihedral symmetry [4] [3]

## Steriruncicantitruncated 6-simplex

Steriuncicantitruncated 6-simplex
Type uniform 6-polytope
Schläfli symbol t0,1,2,3,4{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces 105
4-faces 1155
Cells 4620
Faces 8610
Edges 7560
Vertices 2520
Vertex figure
Coxeter group A6, [35], order 5040
Properties convex

### Alternate names

• Great cellated heptapeton (Acronym: gacal) (Jonathan Bowers)[8]

### 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

orthographic projections
Ak Coxeter plane A6 A5 A4
Graph
Dihedral symmetry [7] [6] [5]
Ak Coxeter plane A3 A2
Graph
Dihedral symmetry [4] [3]

## Related uniform 6-polytopes

The truncated 6-simplex is one of 35 uniform 6-polytopes based on the [3,3,3,3,3] Coxeter group, all shown here in A6 Coxeter plane orthographic projections.

## Notes

1. ^ Klitzing, (x3o3o3o3x3o - scal)
2. ^ Klitzing, (x3x3o3o3x3o - catal)
3. ^ Klitzing, (x3o3x3o3x3o - cral)
4. ^ Klitzing, (x3x3x3o3x3o - cagral)
5. ^ Klitzing, (x3o3o3x3x3o - copal)
6. ^ Klitzing, (x3x3o3x3x3o - captal)
7. ^ Klitzing, ( x3o3x3x3x3o - copril)
8. ^ Klitzing, (x3x3x3x3x3o - gacal)

## 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. "6D uniform polytopes (polypeta)".