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Truncated 8-orthoplexes
| Orthogonal projections in B8 Coxeter plane |
|---|
In eight-dimensional geometry, a truncated 8-orthoplex is a convex uniform 8-polytope, being a truncation of the regular 8-orthoplex.
There are 7 truncation for the 8-orthoplex. Vertices of the truncation 8-orthoplex are located as pairs on the edge of the 8-orthoplex. Vertices of the bitruncated 8-orthoplex are located on the triangular faces of the 8-orthoplex. Vertices of the tritruncated 7-orthoplex are located inside the tetrahedral cells of the 8-orthoplex. The final truncations are best expressed relative to the 8-cube.
Truncated 8-orthoplex
| Truncated 8-orthoplex |
|---|
| Type |
| Schläfli symbol |
| Coxeter-Dynkin diagrams |
| 7-faces |
| 6-faces |
| 5-faces |
| 4-faces |
| Cells |
| Faces |
| Edges |
| Vertices |
| Vertex figure |
| Coxeter groups |
| Properties |
Alternate names
- Truncated octacross (acronym: tek) (Jonthan Bowers)
Construction
There are two Coxeter groups associated with the truncated 8-orthoplex, one with the C8 or [4,3,3,3,3,3,3] Coxeter group, and a lower symmetry with the D8 or [35,1,1] Coxeter group.
Coordinates
Cartesian coordinates for the vertices of a truncated 8-orthoplex, centered at the origin, are all 224 vertices are sign (4) and coordinate (56) permutations of : (±2,±1,0,0,0,0,0,0)
Images
Bitruncated 8-orthoplex
| Bitruncated 8-orthoplex |
|---|
| Type |
| Schläfli symbol |
| Coxeter-Dynkin diagrams |
| 7-faces |
| 6-faces |
| 5-faces |
| 4-faces |
| Cells |
| Faces |
| Edges |
| Vertices |
| Vertex figure |
| Coxeter groups |
| Properties |
Alternate names
- Bitruncated octacross (acronym: batek) (Jonthan Bowers)
Coordinates
Cartesian coordinates for the vertices of a bitruncated 8-orthoplex, centered at the origin, are all sign and coordinate permutations of : (±2,±2,±1,0,0,0,0,0)
Images
Tritruncated 8-orthoplex
| Tritruncated 8-orthoplex |
|---|
| Type |
| Schläfli symbol |
| Coxeter-Dynkin diagrams |
| 7-faces |
| 6-faces |
| 5-faces |
| 4-faces |
| Cells |
| Faces |
| Edges |
| Vertices |
| Vertex figure |
| Coxeter groups |
| Properties |
Alternate names
- Tritruncated octacross (acronym: tatek) (Jonthan Bowers)
Coordinates
Cartesian coordinates for the vertices of a bitruncated 8-orthoplex, centered at the origin, are all sign and coordinate permutations of : (±2,±2,±2,±1,0,0,0,0)
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. (1966)
- x3x3o3o3o3o3o4o - tek, o3x3x3o3o3o3o4o - batek, o3o3x3x3o3o3o4o - tatek
References
- Klitizing, (x3x3o3o3o3o3o4o - tek)
- Klitizing, (o3x3x3o3o3o3o4o - batek)
- Klitizing, (o3o3x3x3o3o3o4o - tatek)
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