Runcinated 8-simplex
Encyclopedia
8-simplex |
Runcinated 8-simplex |
Biruncinated 8-simplex |
Triruncinated 8-simplex |
Runcitruncated 8-simplex |
Biruncitruncated 8-simplex |
Triruncitruncated 8-simplex |
Runcicantellated 8-simplex |
Biruncicantellated 8-simplex |
Runcicantitruncated 8-simplex |
Biruncicantitruncated 8-simplex |
Triruncicantitruncated 8-simplex |
| Orthogonal projections in A8 Coxeter plane | |||
|---|---|---|---|
In eight-dimensional geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, a runcinated 8-simplex is a convex uniform 8-polytope with 3rd order truncations
Truncation (geometry)
In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new facet in place of each vertex.- Uniform truncation :...
(runcination
Runcination
In geometry, runcination is an operation that cuts a regular polytope simultaneously along the faces, edges and vertices, creating new facets in place of the original face, edge, and vertex centers....
) of the regular 8-simplex.
There are eleven unique runcinations of the 8-simplex, including permutations of truncation and cantellation. The triruncinated 8-simplex and triruncicantitruncated 8-simplex have a doubled symmetry, showing [18] order reflectional symmetry in the A8 Coxeter plane.
Runcinated 8-simplex
| Runcinated 8-simplex | |
|---|---|
| Type | uniform polyzetton |
| Schläfli symbol | t0,3{3,3,3,3,3,3,3} |
| Coxeter-Dynkin diagram Coxeter-Dynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors... s |
|
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 4536 |
| Vertices | 504 |
| Vertex figure Vertex figure In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:... |
|
| Coxeter group Coxeter group In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example... |
A8, [37], order 362880 |
| Properties | convex Convex polytope A convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the n-dimensional space Rn... |
Alternate names
- Runcinated enneazetton
- Small prismated enneazetton (Acronym: spene) (Jonathan Bowers)
Coordinates
The Cartesian coordinates of the vertices of the runcinated 8-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,0,1,1,1,2). This construction is based on facets of the runcinated 9-orthoplex.Biruncinated 8-simplex
| Biruncinated 8-simplex | |
|---|---|
| Type | uniform polyzetton |
| Schläfli symbol | t1,4{3,3,3,3,3,3,3} |
| Coxeter-Dynkin diagram Coxeter-Dynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors... |
|
| 7-faces | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 11340 |
| Vertices | 1260 |
| Vertex figure Vertex figure In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:... |
|
| Coxeter group Coxeter group In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example... |
A8, [37], order 362880 |
| Properties | convex Convex polytope A convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the n-dimensional space Rn... |
Alternate names
- Biruncinated enneazetton
- Small biprismated enneazetton (Acronym: spene) (Jonathan Bowers)
Coordinates
The Cartesian coordinates of the vertices of the biruncinated 8-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,1,1,2,2). This construction is based on facets of the biruncinated 9-orthoplex.Triruncinated 8-simplex
| Triruncinated 8-simplex | |
|---|---|
| Type | uniform polyzetton |
| Schläfli symbol | t2,5{3,3,3,3,3,3,3} |
| Coxeter-Dynkin diagram Coxeter-Dynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors... s |
|
| 7-faces | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 15120 |
| Vertices | 1680 |
| Vertex figure Vertex figure In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:... |
|
| Coxeter group Coxeter group In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example... |
A8, [37], order 362880 |
| Properties | convex Convex polytope A convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the n-dimensional space Rn... |
Alternate names
- Triruncinated enneazetton
- Small triprismated enneazetton (Acronym: sabpene) (Jonathan Bowers)