 8-cube    
|
 Truncated 8-cube
|
 Bitruncated 8-cube
| ||
 Quadritruncated 8-cube
|
 Tritruncated 8-cube
|
 Tritruncated 8-orthoplex
| ||
 Bitruncated 8-orthoplex
|
 Truncated 8-orthoplex
|
 8-orthoplex
| ||
| Orthogonal projections in B8 Coxeter plane | ||||
|---|---|---|---|---|
In eight-dimensional geometry, a truncated 8-cube is a convex uniform 8-polytope, being a truncation of the regular 8-cube.
There are unique 7 degrees of truncation for the 8-cube. Vertices of the truncation 8-cube are located as pairs on the edge of the 8-cube. Vertices of the bitruncated 8-cube are located on the square faces of the 8-cube. Vertices of the tritruncated 7-cube are located inside the cubic cells of the 8-cube. The final truncations are best expressed relative to the 8-orthoplex.
Truncated 8-cube
| Truncated 8-cube | |
|---|---|
| Type | uniform 8-polytope |
| Schläfli symbol | t{4,3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams |
|
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | |
| Vertices | |
| Vertex figure | ( )v{3,3,3,3,3} |
| Coxeter groups | B8, |
| Properties | convex |
Alternate names
- Truncated octeract (acronym tocto) (Jonathan Bowers)
Coordinates
Cartesian coordinates for the vertices of a truncated 8-cube, centered at the origin, are all 224 vertices are sign (4) and coordinate (56) permutations of
- (±2,±2,±2,±2,±2,±2,±1,0)
Images
| B8 | B7 | ||||
|---|---|---|---|---|---|
|
|
| ||||
| B6 | B5 | ||||
|
| ||||
| B4 | B3 | B2 | |||
|
|
| |||
| A7 | A5 | A3 | |||
|
|
| |||
Related polytopes
The truncated 8-cube, is seventh in a sequence of truncated hypercubes:
| Image | 
|
 
|
 
|
 
|
 
|
 
|
|
... |
|---|---|---|---|---|---|---|---|---|
| Name | Octagon | Truncated cube | Truncated tesseract | Truncated 5-cube | Truncated 6-cube | Truncated 7-cube | Truncated 8-cube | |
| Coxeter diagram |
|
|
|
|
|
|
| |
| Vertex figure | ( )v( ) |  ( )v{ } |
 ( )v{3} |
 ( )v{3,3} |
( )v{3,3,3} | ( )v{3,3,3,3} | ( )v{3,3,3,3,3} |
Bitruncated 8-cube
| Bitruncated 8-cube | |
|---|---|
| Type | uniform 8-polytope |
| Schläfli symbol | 2t{4,3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams |
|
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | |
| Vertices | |
| Vertex figure | { }v{3,3,3,3} |
| Coxeter groups | B8, |
| Properties | convex |
Alternate names
- Bitruncated octeract (acronym bato) (Jonathan Bowers)
Coordinates
Cartesian coordinates for the vertices of a truncated 8-cube, centered at the origin, are all the sign coordinate permutations of
- (±2,±2,±2,±2,±2,±1,0,0)
Images
| B8 | B7 | ||||
|---|---|---|---|---|---|
|
|
| ||||
| B6 | B5 | ||||
|
| ||||
| B4 | B3 | B2 | |||
|
|
| |||
| A7 | A5 | A3 | |||
|
|
| |||
Related polytopes
The bitruncated 8-cube is sixth in a sequence of bitruncated hypercubes:
| Image |  
|
 
|
 
|
 
|
 
|
|
... |
|---|---|---|---|---|---|---|---|
| Name | Bitruncated cube | Bitruncated tesseract | Bitruncated 5-cube | Bitruncated 6-cube | Bitruncated 7-cube | Bitruncated 8-cube | |
| Coxeter |
|
|
|
|
|
| |
| Vertex figure |  ( )v{ } |
 { }v{ } |
 { }v{3} |
 { }v{3,3} |
{ }v{3,3,3} | { }v{3,3,3,3} |
Tritruncated 8-cube
| Tritruncated 8-cube | |
|---|---|
| Type | uniform 8-polytope |
| Schläfli symbol | 3t{4,3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams |
|
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | |
| Vertices | |
| Vertex figure | {4}v{3,3,3} |
| Coxeter groups | B8, |
| Properties | convex |
Alternate names
- Tritruncated octeract (acronym tato) (Jonathan Bowers)
Coordinates
Cartesian coordinates for the vertices of a truncated 8-cube, centered at the origin, are all the sign coordinate permutations of
- (±2,±2,±2,±2,±1,0,0,0)
Images
| B8 | B7 | ||||
|---|---|---|---|---|---|
|
|
| ||||
| B6 | B5 | ||||
|
| ||||
| B4 | B3 | B2 | |||
|
|
| |||
| A7 | A5 | A3 | |||
|
|
| |||
Quadritruncated 8-cube
| Quadritruncated 8-cube | |
|---|---|
| Type | uniform 8-polytope |
| Schläfli symbol | 4t{3,3,3,3,3,3,4} |
| Coxeter-Dynkin diagrams |
|
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | |
| Vertices | |
| Vertex figure | {3,4}v{3,3} |
| Coxeter groups | B8, D8, |
| Properties | convex |
Alternate names
- Quadritruncated octeract (acronym oke) (Jonathan 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,±2,±1,0,0,0)
Images
| B8 | B7 | ||||
|---|---|---|---|---|---|
|
|
| ||||
| B6 | B5 | ||||
|
| ||||
| B4 | B3 | B2 | |||
|
|
| |||
| A7 | A5 | A3 | |||
|
|
| |||
Related polytopes
| Dim. | 2 | 3 | 4 | 5 | 6 | 7 | 8 | n |
|---|---|---|---|---|---|---|---|---|
| Name | t{4} | r{4,3} | 2t{4,3,3} | 2r{4,3,3,3} | 3t{4,3,3,3,3} | 3r{4,3,3,3,3,3} | 4t{4,3,3,3,3,3,3} | ... |
| Coxeter diagram |
 
|
 
|
|
|
|
|
| |
| Images | 
|
 
|
|
 
|
 
|
 
|
| |
| Facets | {3}  {4} 
|
t{3,3}  t{3,4} 
|
r{3,3,3}  r{3,3,4} 
|
2t{3,3,3,3}  2t{3,3,3,4} 
|
2r{3,3,3,3,3}  2r{3,3,3,3,4} 
|
3t{3,3,3,3,3,3}  3t{3,3,3,3,3,4} 
| ||
| Vertex figure |
( )v( ) |  { }×{ } |
{ }v{ } |
 {3}×{4} |
 {3}v{4} |
{3,3}×{3,4} | {3,3}v{3,4} |
Notes
- Klitizing, (o3o3o3o3o3o3x4x – tocto)
- Klitizing, (o3o3o3o3o3x3x4o – bato)
- Klitizing, (o3o3o3o3x3x3o4o – tato)
- Klitizing, (o3o3o3x3x3o3o4o – oke)
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. "8D uniform polytopes (polyzetta)". o3o3o3o3o3o3x4x – tocto, o3o3o3o3o3x3x4o – bato, o3o3o3o3x3x3o4o – tato, o3o3o3x3x3o3o4o – oke
External links
| Fundamental convex regular and uniform polytopes in dimensions 2–10 | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Family | An | Bn | I2(p) / Dn | E6 / E7 / E8 / F4 / G2 | Hn | |||||||
| Regular polygon | Triangle | Square | p-gon | Hexagon | Pentagon | |||||||
| Uniform polyhedron | Tetrahedron | Octahedron • Cube | Demicube | Dodecahedron • Icosahedron | ||||||||
| Uniform polychoron | Pentachoron | 16-cell • Tesseract | Demitesseract | 24-cell | 120-cell • 600-cell | |||||||
| Uniform 5-polytope | 5-simplex | 5-orthoplex • 5-cube | 5-demicube | |||||||||
| Uniform 6-polytope | 6-simplex | 6-orthoplex • 6-cube | 6-demicube | 122 • 221 | ||||||||
| Uniform 7-polytope | 7-simplex | 7-orthoplex • 7-cube | 7-demicube | 132 • 231 • 321 | ||||||||
| Uniform 8-polytope | 8-simplex | 8-orthoplex • 8-cube | 8-demicube | 142 • 241 • 421 | ||||||||
| Uniform 9-polytope | 9-simplex | 9-orthoplex • 9-cube | 9-demicube | |||||||||
| Uniform 10-polytope | 10-simplex | 10-orthoplex • 10-cube | 10-demicube | |||||||||
| Uniform n-polytope | n-simplex | n-orthoplex • n-cube | n-demicube | 1k2 • 2k1 • k21 | n-pentagonal polytope | |||||||
| Topics: Polytope families • Regular polytope • List of regular polytopes and compounds | ||||||||||||