Truncated 7-orthoplexes

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

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

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
Type	uniform 7-polytope
Schläfli symbol	t{3^5,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells	3920
Faces	2520
Edges	924
Vertices	168
Vertex figure	Elongated pentacross pyramid
Coxeter groups	B7, [3^5,4] D7, [3^4,1,1]
Properties	convex

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 [3^4,1,1] Coxeter group.

Bitruncated 7-orthoplex

Bitruncated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	2t{3^5,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	4200
Vertices	840
Vertex figure
Coxeter groups	B7, [3^5,4] D7, [3^4,1,1]
Properties	convex

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
Type	uniform 7-polytope
Schläfli symbol	3t{3^5,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	10080
Vertices	2240
Vertex figure
Coxeter groups	B7, [3^5,4] D7, [3^4,1,1]
Properties	convex

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

External links

• Olshevsky, George, Cross polytope at Glossary for Hyperspace.
• Polytopes of Various Dimensions
• Multi-dimensional Glossary