Truncated 8-orthoplexes

8-orthoplex	Truncated 8-orthoplex	Bitruncated 8-orthoplex	Tritruncated 8-orthoplex	Quadritruncated 8-cube
8-cube	Truncated 8-cube	Bitruncated 8-cube	Tritruncated 8-cube
Orthogonal projections in BC8 Coxeter plane

In eight-dimensional geometry, a truncated 8-orthoplex is a convex uniform 8-polytope, being a truncation of the regular 8-orthoplex.

There are 7 truncation for the 8-orthoplex. Vertices of the truncation 8-orthoplex are located as pairs on the edge of the 8-orthoplex. Vertices of the bitruncated 8-orthoplex are located on the triangular faces of the 8-orthoplex. Vertices of the tritruncated 7-orthoplex are located inside the tetrahedral cells of the 8-orthoplex. The final truncations are best expressed relative to the 8-cube.

Truncated 8-orthoplex

Truncated 8-orthoplex
Type	uniform 8-polytope
Schläfli symbol	t0,1{3,3,3,3,3,3,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	1456
Vertices	224
Vertex figure	Elongated 6-orthoplex pyramid
Coxeter groups	BC8, [3,3,3,3,3,3,4] D8, [3^5,1,1]
Properties	convex

Alternate names

• Truncated octacross (acronym tek) (Jonthan Bowers)^[1]

Construction

There are two Coxeter groups associated with the truncated 8-orthoplex, one with the C₈ or [4,3,3,3,3,3,3] Coxeter group, and a lower symmetry with the D₈ or [3^5,1,1] Coxeter group.

Coordinates

Cartesian coordinates for the vertices of a truncated 8-orthoplex, centered at the origin, are all 224 vertices are sign (4) and coordinate (56) permutations of

(±2,±1,0,0,0,0,0,0)

Images

orthographic projections

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

Bitruncated 8-orthoplex

Bitruncated 8-orthoplex
Type	uniform 8-polytope
Schläfli symbol	t1,2{3,3,3,3,3,3,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups	BC8, [3,3,3,3,3,3,4] D8, [3^5,1,1]
Properties	convex

Alternate names

• Bitruncated octacross (acronym batek) (Jonthan Bowers)^[2]

Coordinates

Cartesian coordinates for the vertices of a bitruncated 8-orthoplex, centered at the origin, are all sign and coordinate permutations of

(±2,±2,±1,0,0,0,0,0)

Images

orthographic projections

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

Tritruncated 8-orthoplex

Tritruncated 8-orthoplex
Type	uniform 8-polytope
Schläfli symbol	t2,3{3,3,3,3,3,3,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups	BC8, [3,3,3,3,3,3,4] D8, [3^5,1,1]
Properties	convex

Alternate names

• Tritruncated octacross (acronym tatek) (Jonthan Bowers)^[3]

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,±1,0,0,0,0)

Images

orthographic projections

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

Notes

1. Klitizing, (x3x3o3o3o3o3o4o - tek)
2. Klitizing, (o3x3x3o3o3o3o4o - batek)
3. Klitizing, (o3o3x3x3o3o3o4o - tatek)

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. (1966)
• Richard Klitzing, 8D, uniform polytopes (polyzetta) x3x3o3o3o3o3o4o - tek, o3x3x3o3o3o3o4o - batek, o3o3x3x3o3o3o4o - tatek

External links

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