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Runcinated 6-simplexes
| Orthogonal projections in A6 Coxeter plane |
|---|
In six-dimensional geometry, a runcinated 6-simplex is a convex uniform 6-polytope constructed as a runcination (3rd order truncations) of the regular 6-simplex.
There are 8 unique runcinations of the 6-simplex with permutations of truncations, and cantellations.
Runcinated 6-simplex
| Runcinated 6-simplex |
|---|
| Type |
| Schläfli symbol |
| Coxeter-Dynkin diagrams |
| 5-faces |
| 4-faces |
| Cells |
| Faces |
| Edges |
| Vertices |
| Vertex figure |
| Coxeter group |
| Properties |
Alternate names
- Small prismated heptapeton (Acronym: spil) (Jonathan Bowers)
Coordinates
The vertices of the runcinated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,1,1,2). This construction is based on facets of the runcinated 7-orthoplex.
Images
Biruncinated 6-simplex
| biruncinated 6-simplex |
|---|
| Type |
| Schläfli symbol |
| Coxeter-Dynkin diagrams |
| 5-faces |
| 4-faces |
| Cells |
| Faces |
| Edges |
| Vertices |
| Vertex figure |
| Coxeter group |
| Properties |
Alternate names
- Small biprismated tetradecapeton (Acronym: sibpof) (Jonathan Bowers)
Coordinates
The vertices of the biruncinated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,1,1,2,2). This construction is based on facets of the biruncinated 7-orthoplex.
Images
Runcitruncated 6-simplex
| Runcitruncated 6-simplex |
|---|
| Type |
| Schläfli symbol |
| Coxeter-Dynkin diagrams |
| 5-faces |
| 4-faces |
| Cells |
| Faces |
| Edges |
| Vertices |
| Vertex figure |
| Coxeter group |
| Properties |
Alternate names
- Prismatotruncated heptapeton (Acronym: patal) (Jonathan Bowers)
Coordinates
The vertices of the runcitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,1,2,3). This construction is based on facets of the runcitruncated 7-orthoplex.
Images
Biruncitruncated 6-simplex
| biruncitruncated 6-simplex |
|---|
| Type |
| Schläfli symbol |
| Coxeter-Dynkin diagrams |
| 5-faces |
| 4-faces |
| Cells |
| Faces |
| Edges |
| Vertices |
| Vertex figure |
| Coxeter group |
| Properties |
Alternate names
- Biprismatorhombated heptapeton (Acronym: bapril) (Jonathan Bowers)
Coordinates
The vertices of the biruncitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,1,2,3,3). This construction is based on facets of the biruncitruncated 7-orthoplex.
Images
Runcicantellated 6-simplex
| Runcicantellated 6-simplex |
|---|
| Type |
| Schläfli symbol |
| Coxeter-Dynkin diagrams |
| 5-faces |
| 4-faces |
| Cells |
| Faces |
| Edges |
| Vertices |
| Vertex figure |
| Coxeter group |
| Properties |
Alternate names
- Prismatorhombated heptapeton (Acronym: pril) (Jonathan Bowers)
Coordinates
The vertices of the runcicantellated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,2,2,3). This construction is based on facets of the runcicantellated 7-orthoplex.
Images
Runcicantitruncated 6-simplex
| Runcicantitruncated 6-simplex |
|---|
| Type |
| Schläfli symbol |
| Coxeter-Dynkin diagrams |
| 5-faces |
| 4-faces |
| Cells |
| Faces |
| Edges |
| Vertices |
| Vertex figure |
| Coxeter group |
| Properties |
Alternate names
- Runcicantitruncated heptapeton
- Great prismated heptapeton (Acronym: gapil) (Jonathan Bowers)
Coordinates
The vertices of the runcicantitruncated 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 runcicantitruncated 7-orthoplex.
Images
Biruncicantitruncated 6-simplex
| biruncicantitruncated 6-simplex |
|---|
| Type |
| Schläfli symbol |
| Coxeter-Dynkin diagrams |
| 5-faces |
| 4-faces |
| Cells |
| Faces |
| Edges |
| Vertices |
| Vertex figure |
| Coxeter group |
| Properties |
Alternate names
- Biruncicantitruncated heptapeton
- Great biprismated tetradecapeton (Acronym: gibpof) (Jonathan Bowers)
Coordinates
The vertices of the biruncicantittruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,2,3,4,4). This construction is based on facets of the biruncicantitruncated 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.
- x3o3o3x3o3o - spil, o3x3o3o3x3o - sibpof, x3x3o3x3o3o - patal, o3x3x3o3x3o - bapril, x3o3x3x3o3o - pril, x3x3x3x3o3o - gapil, o3x3x3x3x3o - gibpof
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