Stericated 8-simplexes

8-simplex	80px Stericated 8-simplex	80px Bistericated 8-simplex
80px Steritruncated 8-simplex	80px Bisteritruncated 8-simplex	80px Stericantellated 8-simplex	80px Bistericantellated 8-simplex
80px Stericantitruncated 8-simplex	80px Bistericantitruncated 8-simplex	80px Steriruncinated 8-simplex	80px Bisteriruncinated 8-simplex
80px Steriruncitruncated 8-simplex	80px Bisteriruncitruncated 8-simplex	80px Steriruncicantellated 8-simplex	80px Bisteriruncicantellated 8-simplex
80px Steriruncicantitruncated 8-simplex	80px Bisteriruncicantitruncated 8-simplex
Orthogonal projections in A8 Coxeter plane

In eight-dimensional geometry, a stericated 8-simplex is a convex uniform 8-polytope with 4th order truncations (sterication) of the regular 8-simplex. There are 16 unique sterications for the 8-simplex including permutations of truncation, cantellation, and runcination.

Stericated 8-simplex

Stericated 8-simplex
Type	uniform 8-polytope
Schläfli symbol	t0,4{3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges	6300
Vertices	630
Vertex figure
Coxeter group	A8, [37], order 362880
Properties	convex

The Cartesian coordinates of the vertices of the stericated 8-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,1,1,1,2). This construction is based on facets of the stericated 9-orthoplex.

Images

orthographic projections

Ak Coxeter plane	A8	A7	A6	A5
Graph	120pxpx	120pxpx	120pxpx	120pxpx
Dihedral symmetry	[9]	[8]	[7]	[6]
Ak Coxeter plane	A4	A3	A2
Graph	120pxpx	120pxpx	120pxpx
Dihedral symmetry	[5]	[4]	[3]

Bistericated 8-simplex

bistericated 8-simplex
Type	uniform 8-polytope
Schläfli symbol	t1,5{3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges	12600
Vertices	1260
Vertex figure
Coxeter group	A8, [37], order 362880
Properties	convex

Coordinates

The Cartesian coordinates of the vertices of the bistericated 8-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,1,1,1,2,2). This construction is based on facets of the bistericated 9-orthoplex.

Images

orthographic projections

Ak Coxeter plane	A8	A7	A6	A5
Graph	120pxpx	120pxpx	120pxpx	120pxpx
Dihedral symmetry	[9]	[8]	[7]	[6]
Ak Coxeter plane	A4	A3	A2
Graph	120pxpx	120pxpx	120pxpx
Dihedral symmetry	[5]	[4]	[3]

Steritruncated 8-simplex

Steritruncated 8-simplex
Type	uniform 8-polytope
Schläfli symbol	t0,1,4{3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter group	A8, [37], order 362880
Properties	convex

Images

orthographic projections

Ak Coxeter plane	A8	A7	A6	A5
Graph	120pxpx	120pxpx	120pxpx	120pxpx
Dihedral symmetry	[9]	[8]	[7]	[6]
Ak Coxeter plane	A4	A3	A2
Graph	120pxpx	120pxpx	120pxpx
Dihedral symmetry	[5]	[4]	[3]

Bisteritruncated 8-simplex

Bisteritruncated 8-simplex
Type	uniform 8-polytope
Schläfli symbol	t1,2,5{3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter group	A8, [37], order 362880
Properties	convex

Images

orthographic projections

Ak Coxeter plane	A8	A7	A6	A5
Graph	120pxpx	120pxpx	120pxpx	120pxpx
Dihedral symmetry	[9]	[8]	[7]	[6]
Ak Coxeter plane	A4	A3	A2
Graph	120pxpx	120pxpx	120pxpx
Dihedral symmetry	[5]	[4]	[3]

Stericantellated 8-simplex

Images

orthographic projections

Ak Coxeter plane	A8	A7	A6	A5
Graph	120pxpx	120pxpx	120pxpx	120pxpx
Dihedral symmetry	[9]	[8]	[7]	[6]
Ak Coxeter plane	A4	A3	A2
Graph	120pxpx	120pxpx	120pxpx
Dihedral symmetry	[5]	[4]	[3]

Bistericantellated 8-simplex

Images

orthographic projections

Ak Coxeter plane	A8	A7	A6	A5
Graph	120pxpx	120pxpx	120pxpx	120pxpx
Dihedral symmetry	[9]	[8]	[7]	[6]
Ak Coxeter plane	A4	A3	A2
Graph	120pxpx	120pxpx	120pxpx
Dihedral symmetry	[5]	[4]	[3]

Stericantitruncated 8-simplex

Images

orthographic projections

Ak Coxeter plane	A8	A7	A6	A5
Graph	120pxpx	120pxpx	120pxpx	120pxpx
Dihedral symmetry	[9]	[8]	[7]	[6]
Ak Coxeter plane	A4	A3	A2
Graph	120pxpx	120pxpx	120pxpx
Dihedral symmetry	[5]	[4]	[3]

Bistericantitruncated 8-simplex

Images

orthographic projections

Ak Coxeter plane	A8	A7	A6	A5
Graph	120pxpx	120pxpx	120pxpx	120pxpx
Dihedral symmetry	[9]	[8]	[7]	[6]
Ak Coxeter plane	A4	A3	A2
Graph	120pxpx	120pxpx	120pxpx
Dihedral symmetry	[5]	[4]	[3]

Steriruncinated 8-simplex

Images

orthographic projections

Ak Coxeter plane	A8	A7	A6	A5
Graph	120pxpx	120pxpx	120pxpx	120pxpx
Dihedral symmetry	[9]	[8]	[7]	[6]
Ak Coxeter plane	A4	A3	A2
Graph	120pxpx	120pxpx	120pxpx
Dihedral symmetry	[5]	[4]	[3]

Bisteriruncinated 8-simplex

Images

orthographic projections

Ak Coxeter plane	A8	A7	A6	A5
Graph	120pxpx	120pxpx	120pxpx	120pxpx
Dihedral symmetry	[9]	[8]	[7]	[6]
Ak Coxeter plane	A4	A3	A2
Graph	120pxpx	120pxpx	120pxpx
Dihedral symmetry	[5]	[4]	[3]

Steriruncitruncated 8-simplex

Images

orthographic projections

Ak Coxeter plane	A8	A7	A6	A5
Graph	120pxpx	120pxpx	120pxpx	120pxpx
Dihedral symmetry	[9]	[8]	[7]	[6]
Ak Coxeter plane	A4	A3	A2
Graph	120pxpx	120pxpx	120pxpx
Dihedral symmetry	[5]	[4]	[3]

Bisteriruncitruncated 8-simplex

Images

orthographic projections

Ak Coxeter plane	A8	A7	A6	A5
Graph	120pxpx	120pxpx	120pxpx	120pxpx
Dihedral symmetry	[9]	[8]	[7]	[6]
Ak Coxeter plane	A4	A3	A2
Graph	120pxpx	120pxpx	120pxpx
Dihedral symmetry	[5]	[4]	[3]

Steriruncicantellated 8-simplex

Images

orthographic projections

Ak Coxeter plane	A8	A7	A6	A5
Graph	120pxpx	120pxpx	120pxpx	120pxpx
Dihedral symmetry	[9]	[8]	[7]	[6]
Ak Coxeter plane	A4	A3	A2
Graph	120pxpx	120pxpx	120pxpx
Dihedral symmetry	[5]	[4]	[3]

Bisteriruncicantellated 8-simplex

Images

orthographic projections

Ak Coxeter plane	A8	A7	A6	A5
Graph	120pxpx	120pxpx	120pxpx	120pxpx
Dihedral symmetry	[9]	[8]	[7]	[6]
Ak Coxeter plane	A4	A3	A2
Graph	120pxpx	120pxpx	120pxpx
Dihedral symmetry	[5]	[4]	[3]

Steriruncicantitruncated 8-simplex

Images

orthographic projections

Ak Coxeter plane	A8	A7	A6	A5
Graph	120pxpx	120pxpx	120pxpx	120pxpx
Dihedral symmetry	[9]	[8]	[7]	[6]
Ak Coxeter plane	A4	A3	A2
Graph	120pxpx	120pxpx	120pxpx
Dihedral symmetry	[5]	[4]	[3]

Bisteriruncicantitruncated 8-simplex

Images

orthographic projections

Ak Coxeter plane	A8	A7	A6	A5
Graph	120pxpx	120pxpx	120pxpx	120pxpx
Dihedral symmetry	[9]	[8]	[7]	[6]
Ak Coxeter plane	A4	A3	A2
Graph	120pxpx	120pxpx	120pxpx
Dihedral symmetry	[5]	[4]	[3]

Related polytopes

This polytope is one of 135 uniform 8-polytopes with A₈ symmetry.

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.
• Richard Klitzing, 8D, uniform polytopes (polyzetta) x3o3o3o3x3o3o3o, o3x3o3o3o3x3o3o

External links

• Olshevsky, George, Cross polytope at Glossary for Hyperspace.
• Polytopes of Various Dimensions
• Multi-dimensional Glossary