6-cube |
Truncated 6-cube |
Bitruncated 6-cube |
Tritruncated 6-cube |
6-orthoplex |
Truncated 6-orthoplex |
Bitruncated 6-orthoplex | |
Orthogonal projections in B6 Coxeter plane |
---|
In six-dimensional geometry, a truncated 6-cube (or truncated hexeract) is a convex uniform 6-polytope, being a truncation of the regular 6-cube.
There are 5 truncations for the 6-cube. Vertices of the truncated 6-cube are located as pairs on the edge of the 6-cube. Vertices of the bitruncated 6-cube are located on the square faces of the 6-cube. Vertices of the tritruncated 6-cube are located inside the cubic cells of the 6-cube.
Truncated 6-cube | |
---|---|
Type | uniform 6-polytope |
Class | B6 polytope |
Schläfli symbol | t{4,3,3,3,3} |
Coxeter-Dynkin diagrams | |
5-faces | 76 |
4-faces | 464 |
Cells | 1120 |
Faces | 1520 |
Edges | 1152 |
Vertices | 384 |
Vertex figure |
( )v{3,3,3} |
Coxeter groups | B6, [3,3,3,3,4] |
Properties | convex |
The truncated 6-cube may be constructed by truncating the vertices of the 6-cube at of the edge length. A regular 5-simplex replaces each original vertex.
The Cartesian coordinates of the vertices of a truncated 6-cube having edge length 2 are the permutations of:
Coxeter plane | B6 | B5 | B4 |
---|---|---|---|
Graph | |||
Dihedral symmetry | [12] | [10] | [8] |
Coxeter plane | B3 | B2 | |
Graph | |||
Dihedral symmetry | [6] | [4] | |
Coxeter plane | A5 | A3 | |
Graph | |||
Dihedral symmetry | [6] | [4] |
The truncated 6-cube, is fifth 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 6-cube | |
---|---|
Type | uniform 6-polytope |
Class | B6 polytope |
Schläfli symbol | 2t{4,3,3,3,3} |
Coxeter-Dynkin diagrams | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | |
Vertices | |
Vertex figure |
{ }v{3,3} |
Coxeter groups | B6, [3,3,3,3,4] |
Properties | convex |
The Cartesian coordinates of the vertices of a bitruncated 6-cube having edge length 2 are the permutations of:
Coxeter plane | B6 | B5 | B4 |
---|---|---|---|
Graph | |||
Dihedral symmetry | [12] | [10] | [8] |
Coxeter plane | B3 | B2 | |
Graph | |||
Dihedral symmetry | [6] | [4] | |
Coxeter plane | A5 | A3 | |
Graph | |||
Dihedral symmetry | [6] | [4] |
The bitruncated 6-cube is fourth 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 6-cube | |
---|---|
Type | uniform 6-polytope |
Class | B6 polytope |
Schläfli symbol | 3t{4,3,3,3,3} |
Coxeter-Dynkin diagrams | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | |
Vertices | |
Vertex figure |
{3}v{4} [3] |
Coxeter groups | B6, [3,3,3,3,4] |
Properties | convex |
The Cartesian coordinates of the vertices of a tritruncated 6-cube having edge length 2 are the permutations of:
Coxeter plane | B6 | B5 | B4 |
---|---|---|---|
Graph | |||
Dihedral symmetry | [12] | [10] | [8] |
Coxeter plane | B3 | B2 | |
Graph | |||
Dihedral symmetry | [6] | [4] | |
Coxeter plane | A5 | A3 | |
Graph | |||
Dihedral symmetry | [6] | [4] |
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} |
These polytopes are from a set of 63 Uniform 6-polytopes generated from the B6 Coxeter plane, including the regular 6-cube or 6-orthoplex.
6-cube |
Truncated 6-cube |
Bitruncated 6-cube |
Tritruncated 6-cube |
6-orthoplex |
Truncated 6-orthoplex |
Bitruncated 6-orthoplex | |
Orthogonal projections in B6 Coxeter plane |
---|
In six-dimensional geometry, a truncated 6-cube (or truncated hexeract) is a convex uniform 6-polytope, being a truncation of the regular 6-cube.
There are 5 truncations for the 6-cube. Vertices of the truncated 6-cube are located as pairs on the edge of the 6-cube. Vertices of the bitruncated 6-cube are located on the square faces of the 6-cube. Vertices of the tritruncated 6-cube are located inside the cubic cells of the 6-cube.
Truncated 6-cube | |
---|---|
Type | uniform 6-polytope |
Class | B6 polytope |
Schläfli symbol | t{4,3,3,3,3} |
Coxeter-Dynkin diagrams | |
5-faces | 76 |
4-faces | 464 |
Cells | 1120 |
Faces | 1520 |
Edges | 1152 |
Vertices | 384 |
Vertex figure |
( )v{3,3,3} |
Coxeter groups | B6, [3,3,3,3,4] |
Properties | convex |
The truncated 6-cube may be constructed by truncating the vertices of the 6-cube at of the edge length. A regular 5-simplex replaces each original vertex.
The Cartesian coordinates of the vertices of a truncated 6-cube having edge length 2 are the permutations of:
Coxeter plane | B6 | B5 | B4 |
---|---|---|---|
Graph | |||
Dihedral symmetry | [12] | [10] | [8] |
Coxeter plane | B3 | B2 | |
Graph | |||
Dihedral symmetry | [6] | [4] | |
Coxeter plane | A5 | A3 | |
Graph | |||
Dihedral symmetry | [6] | [4] |
The truncated 6-cube, is fifth 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 6-cube | |
---|---|
Type | uniform 6-polytope |
Class | B6 polytope |
Schläfli symbol | 2t{4,3,3,3,3} |
Coxeter-Dynkin diagrams | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | |
Vertices | |
Vertex figure |
{ }v{3,3} |
Coxeter groups | B6, [3,3,3,3,4] |
Properties | convex |
The Cartesian coordinates of the vertices of a bitruncated 6-cube having edge length 2 are the permutations of:
Coxeter plane | B6 | B5 | B4 |
---|---|---|---|
Graph | |||
Dihedral symmetry | [12] | [10] | [8] |
Coxeter plane | B3 | B2 | |
Graph | |||
Dihedral symmetry | [6] | [4] | |
Coxeter plane | A5 | A3 | |
Graph | |||
Dihedral symmetry | [6] | [4] |
The bitruncated 6-cube is fourth 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 6-cube | |
---|---|
Type | uniform 6-polytope |
Class | B6 polytope |
Schläfli symbol | 3t{4,3,3,3,3} |
Coxeter-Dynkin diagrams | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | |
Vertices | |
Vertex figure |
{3}v{4} [3] |
Coxeter groups | B6, [3,3,3,3,4] |
Properties | convex |
The Cartesian coordinates of the vertices of a tritruncated 6-cube having edge length 2 are the permutations of:
Coxeter plane | B6 | B5 | B4 |
---|---|---|---|
Graph | |||
Dihedral symmetry | [12] | [10] | [8] |
Coxeter plane | B3 | B2 | |
Graph | |||
Dihedral symmetry | [6] | [4] | |
Coxeter plane | A5 | A3 | |
Graph | |||
Dihedral symmetry | [6] | [4] |
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} |
These polytopes are from a set of 63 Uniform 6-polytopes generated from the B6 Coxeter plane, including the regular 6-cube or 6-orthoplex.