Truncated 8-cubes (Redirected from Truncated 8-cube)


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, [3,3,3,3,3,3,4]
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

orthographic projections
B8 B7
[16] [14]
B6 B5
[12] [10]
B4 B3 B2
[8] [6] [4]
A7 A5 A3
[8] [6] [4]

Related polytopes

The truncated 8-cube, is seventh in a sequence of truncated hypercubes:

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, [3,3,3,3,3,3,4]
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

orthographic projections
B8 B7
[16] [14]
B6 B5
[12] [10]
B4 B3 B2
[8] [6] [4]
A7 A5 A3
[8] [6] [4]

Related polytopes

The bitruncated 8-cube is sixth in a sequence of bitruncated hypercubes:

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, [3,3,3,3,3,3,4]
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

orthographic projections
B8 B7
[16] [14]
B6 B5
[12] [10]
B4 B3 B2
[8] [6] [4]
A7 A5 A3
[8] [6] [4]

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, [3,3,3,3,3,3,4]
D8, [35,1,1]
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

orthographic projections
B8 B7
[16] [14]
B6 B5
[12] [10]
B4 B3 B2
[8] [6] [4]
A7 A5 A3
[8] [6] [4]

Related polytopes

2-isotopic hypercubes
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

  1. ^ Klitizing, (o3o3o3o3o3o3x4x – tocto)
  2. ^ Klitizing, (o3o3o3o3o3x3x4o – bato)
  3. ^ Klitizing, (o3o3o3o3x3x3o4o – tato)
  4. ^ Klitizing, (o3o3o3x3x3o3o4o – oke)

This page was last updated at 2023-10-10 02:40 UTC. Update now. View original page.

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