# Truncated 7-orthoplexes

In seven-dimensional geometry, a truncated 7-orthoplex is a convex uniform 7-polytope, being a truncation of the regular 7-orthoplex.

 Orthogonal projections in B7 Coxeter plane 7-orthoplex Truncated 7-orthoplex Bitruncated 7-orthoplex Tritruncated 7-orthoplex 7-cube Truncated 7-cube Bitruncated 7-cube Tritruncated 7-cube

There are 6 truncations of the 7-orthoplex. Vertices of the truncation 7-orthoplex are located as pairs on the edge of the 7-orthoplex. Vertices of the bitruncated 7-orthoplex are located on the triangular faces of the 7-orthoplex. Vertices of the tritruncated 7-orthoplex are located inside the tetrahedral cells of the 7-orthoplex. The final three truncations are best expressed relative to the 7-cube.

## Truncated 7-orthoplex

Truncated 7-orthoplex
Typeuniform 7-polytope
Schläfli symbolt{35,4}
Coxeter-Dynkin diagrams

6-faces
5-faces
4-faces
Cells3920
Faces2520
Edges924
Vertices168
Vertex figure( )v{3,3,4}
Coxeter groupsB7, [35,4]
D7, [34,1,1]
Propertiesconvex

### Alternate names

• Truncated heptacross
• Truncated hecatonicosoctaexon (Jonathan Bowers)[1]

### Coordinates

Cartesian coordinates for the vertices of a truncated 7-orthoplex, centered at the origin, are all 168 vertices are sign (4) and coordinate (42) permutations of

(±2,±1,0,0,0,0,0)

### Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

### Construction

There are two Coxeter groups associated with the truncated 7-orthoplex, one with the C7 or [4,35] Coxeter group, and a lower symmetry with the D7 or [34,1,1] Coxeter group.

## Bitruncated 7-orthoplex

Bitruncated 7-orthoplex
Typeuniform 7-polytope
Schläfli symbol2t{35,4}
Coxeter-Dynkin diagrams

6-faces
5-faces
4-faces
Cells
Faces
Edges4200
Vertices840
Vertex figure{ }v{3,3,4}
Coxeter groupsB7, [35,4]
D7, [34,1,1]
Propertiesconvex

### Alternate names

• Bitruncated heptacross
• Bitruncated hecatonicosoctaexon (Jonathan Bowers)[2]

### Coordinates

Cartesian coordinates for the vertices of a bitruncated 7-orthoplex, centered at the origin, are all sign and coordinate permutations of

(±2,±2,±1,0,0,0,0)

### Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

## Tritruncated 7-orthoplex

The tritruncated 7-orthoplex can tessellation space in the quadritruncated 7-cubic honeycomb.

Tritruncated 7-orthoplex
Typeuniform 7-polytope
Schläfli symbol3t{35,4}
Coxeter-Dynkin diagrams

6-faces
5-faces
4-faces
Cells
Faces
Edges10080
Vertices2240
Vertex figure{3}v{3,4}
Coxeter groupsB7, [35,4]
D7, [34,1,1]
Propertiesconvex

### Alternate names

• Tritruncated heptacross
• Tritruncated hecatonicosoctaexon (Jonathan Bowers)[3]

### Coordinates

Cartesian coordinates for the vertices of a tritruncated 7-orthoplex, centered at the origin, are all sign and coordinate permutations of

(±2,±2,±2,±1,0,0,0)

### Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

## Notes

1. Klitzing, (x3x3o3o3o3o4o - tez)
2. Klitzing, (o3x3x3o3o3o4o - botaz)
3. Klitzing, (o3o3x3x3o3o4o - totaz)

## 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, [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.
• Klitzing, Richard. "7D uniform polytopes (polyexa)". x3x3o3o3o3o4o - tez, o3x3x3o3o3o4o - botaz, o3o3x3x3o3o4o - totaz