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7-simplex	Runcinated 7-simplex	Biruncinated 7-simplex
Runcitruncated 7-simplex	Biruncitruncated 7-simplex	Runcicantellated 7-simplex
Biruncicantellated 7-simplex	Runcicantitruncated 7-simplex	Biruncicantitruncated 7-simplex
Orthogonal projections in A7 Coxeter plane

In seven-dimensional geometry, a runcinated 7-simplex is a convex uniform 7-polytope with 3rd order truncations (runcination) of the regular 7-simplex.

There are 8 unique runcinations of the 7-simplex with permutations of truncations, and cantellations.

Runcinated 7-simplex

Runcinated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t0,3{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	2100
Vertices	280
Vertex figure
Coxeter group	A7, , order 40320
Properties	convex

Alternate names

• Small prismated octaexon (acronym: spo) (Jonathan Bowers)

Coordinates

The vertices of the runcinated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,1,1,2). This construction is based on facets of the runcinated 8-orthoplex.

Images

orthographic projections

Ak Coxeter plane	A7	A6	A5
Graph
Dihedral symmetry
Ak Coxeter plane	A4	A3	A2
Graph
Dihedral symmetry

Biruncinated 7-simplex

Biruncinated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t1,4{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	4200
Vertices	560
Vertex figure
Coxeter group	A7, , order 40320
Properties	convex

Alternate names

• Small biprismated octaexon (sibpo) (Jonathan Bowers)

Coordinates

The vertices of the biruncinated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,1,1,2,2). This construction is based on facets of the biruncinated 8-orthoplex.

Images

orthographic projections

Ak Coxeter plane	A7	A6	A5
Graph
Dihedral symmetry
Ak Coxeter plane	A4	A3	A2
Graph
Dihedral symmetry

Runcitruncated 7-simplex

runcitruncated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t0,1,3{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	4620
Vertices	840
Vertex figure
Coxeter group	A7, , order 40320
Properties	convex

Alternate names

• Prismatotruncated octaexon (acronym: patto) (Jonathan Bowers)

Coordinates

The vertices of the runcitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,1,2,3). This construction is based on facets of the runcitruncated 8-orthoplex.

Images

orthographic projections

Ak Coxeter plane	A7	A6	A5
Graph
Dihedral symmetry
Ak Coxeter plane	A4	A3	A2
Graph
Dihedral symmetry

Biruncitruncated 7-simplex

Biruncitruncated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t1,2,4{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	8400
Vertices	1680
Vertex figure
Coxeter group	A7, , order 40320
Properties	convex

Alternate names

• Biprismatotruncated octaexon (acronym: bipto) (Jonathan Bowers)

Coordinates

The vertices of the biruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,1,2,3,3). This construction is based on facets of the biruncitruncated 8-orthoplex.

Images

orthographic projections

Ak Coxeter plane	A7	A6	A5
Graph
Dihedral symmetry
Ak Coxeter plane	A4	A3	A2
Graph
Dihedral symmetry

Runcicantellated 7-simplex

runcicantellated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t0,2,3{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	3360
Vertices	840
Vertex figure
Coxeter group	A7, , order 40320
Properties	convex

Alternate names

• Prismatorhombated octaexon (acronym: paro) (Jonathan Bowers)

Coordinates

The vertices of the runcicantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,2,2,3). This construction is based on facets of the runcicantellated 8-orthoplex.

Images

orthographic projections

Ak Coxeter plane	A7	A6	A5
Graph
Dihedral symmetry
Ak Coxeter plane	A4	A3	A2
Graph
Dihedral symmetry

Biruncicantellated 7-simplex

biruncicantellated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t1,3,4{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter group	A7, , order 40320
Properties	convex

Alternate names

• Biprismatorhombated octaexon (acronym: bipro) (Jonathan Bowers)

Coordinates

The vertices of the biruncicantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,2,2,3,3). This construction is based on facets of the biruncicantellated 8-orthoplex.

Images

orthographic projections

Ak Coxeter plane	A7	A6	A5
Graph
Dihedral symmetry
Ak Coxeter plane	A4	A3	A2
Graph
Dihedral symmetry

Runcicantitruncated 7-simplex

runcicantitruncated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t0,1,2,3{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	5880
Vertices	1680
Vertex figure
Coxeter group	A7, , order 40320
Properties	convex

Alternate names

• Great prismated octaexon (acronym: gapo) (Jonathan Bowers)

Coordinates

The vertices of the runcicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,2,3,4). This construction is based on facets of the runcicantitruncated 8-orthoplex.

Images

orthographic projections

Ak Coxeter plane	A7	A6	A5
Graph
Dihedral symmetry
Ak Coxeter plane	A4	A3	A2
Graph
Dihedral symmetry

Biruncicantitruncated 7-simplex

biruncicantitruncated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t1,2,3,4{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	11760
Vertices	3360
Vertex figure
Coxeter group	A7, , order 40320
Properties	convex

Alternate names

• Great biprismated octaexon (acronym: gibpo) (Jonathan Bowers)

Coordinates

The vertices of the biruncicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,2,3,4,4). This construction is based on facets of the biruncicantitruncated 8-orthoplex.

Images

orthographic projections

Ak Coxeter plane	A7	A6	A5
Graph
Dihedral symmetry
Ak Coxeter plane	A4	A3	A2
Graph
Dihedral symmetry

Related polytopes

These polytopes are among 71 uniform 7-polytopes with A₇ symmetry.

A7 polytopes
t0	t1	t2	t3	t0,1	t0,2	t1,2	t0,3
t1,3	t2,3	t0,4	t1,4	t2,4	t0,5	t1,5	t0,6
t0,1,2	t0,1,3	t0,2,3	t1,2,3	t0,1,4	t0,2,4	t1,2,4	t0,3,4
t1,3,4	t2,3,4	t0,1,5	t0,2,5	t1,2,5	t0,3,5	t1,3,5	t0,4,5
t0,1,6	t0,2,6	t0,3,6	t0,1,2,3	t0,1,2,4	t0,1,3,4	t0,2,3,4	t1,2,3,4
t0,1,2,5	t0,1,3,5	t0,2,3,5	t1,2,3,5	t0,1,4,5	t0,2,4,5	t1,2,4,5	t0,3,4,5
t0,1,2,6	t0,1,3,6	t0,2,3,6	t0,1,4,6	t0,2,4,6	t0,1,5,6	t0,1,2,3,4	t0,1,2,3,5
t0,1,2,4,5	t0,1,3,4,5	t0,2,3,4,5	t1,2,3,4,5	t0,1,2,3,6	t0,1,2,4,6	t0,1,3,4,6	t0,2,3,4,6
t0,1,2,5,6	t0,1,3,5,6	t0,1,2,3,4,5	t0,1,2,3,4,6	t0,1,2,3,5,6	t0,1,2,4,5,6	t0,1,2,3,4,5,6

Notes

1. Klitzing, (x3o3o3x3o3o3o - spo)
2. Klitzing, (o3x3o3o3x3o3o - sibpo)
3. Klitzing, (x3x3o3x3o3o3o - patto)
4. Klitzing, (o3x3x3o3x3o3o - bipto)
5. Klitzing, (x3o3x3x3o3o3o - paro)
6. Klitzing, (x3x3x3x3o3o3o - gapo)
7. Klitzing, (o3x3x3x3x3o3o- gibpo)

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
    • (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I,
    • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II,
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III,
• Norman Johnson Uniform Polytopes, Manuscript (1991)
  • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
• Klitzing, Richard. "7D uniform polytopes (polyexa)". x3o3o3x3o3o3o - spo, o3x3o3o3x3o3o - sibpo, x3x3o3x3o3o3o - patto, o3x3x3o3x3o3o - bipto, x3o3x3x3o3o3o - paro, x3x3x3x3o3o3o - gapo, o3x3x3x3x3o3o- gibpo

External links

• Polytopes of Various Dimensions
• Multi-dimensional Glossary

• 7-polytopes