Truncated 5-orthoplexes

5-orthoplex	Truncated 5-orthoplex	Bitruncated 5-orthoplex
5-cube	Truncated 5-cube	Bitruncated 5-cube
Orthogonal projections in B5 Coxeter plane

In five-dimensional geometry, a truncated 5-orthoplex is a convex uniform 5-polytope, being a truncation of the regular 5-orthoplex.

There are 4 unique truncations of the 5-orthoplex. Vertices of the truncation 5-orthoplex are located as pairs on the edge of the 5-orthoplex. Vertices of the bitruncated 5-orthoplex are located on the triangular faces of the 5-orthoplex. The third and fourth truncations are more easily constructed as second and first truncations of the 5-cube.

Truncated 5-orthoplex
Type	uniform 5-polytope
Schläfli symbol	t{3,3,3,4} t{3,31,1}
Coxeter-Dynkin diagrams
4-faces	42	10 32
Cells	240	160 80
Faces	400	320 80
Edges	280	240 40
Vertices	80
Vertex figure	( )v{3,4}
Coxeter groups	B5, [3,3,3,4], order 3840 D5, [32,1,1], order 1920
Properties	convex

• Truncated pentacross
• Truncated triacontaditeron (Acronym: tot) (Jonathan Bowers)^[1]

Cartesian coordinates for the vertices of a truncated 5-orthoplex, centered at the origin, are all 80 vertices are sign (4) and coordinate (20) permutations of

(±2,±1,0,0,0)

The truncated 5-orthoplex is constructed by a truncation operation applied to the 5-orthoplex. All edges are shortened, and two new vertices are added on each original edge.

orthographic projections

Coxeter plane	B5	B4 / D5	B3 / D4 / A2
Graph
Dihedral symmetry	[10]	[8]	[6]
Coxeter plane	B2	A3
Graph
Dihedral symmetry	[4]	[4]

Bitruncated 5-orthoplex
Type	uniform 5-polytope
Schläfli symbol	2t{3,3,3,4} 2t{3,31,1}
Coxeter-Dynkin diagrams
4-faces	42	10 32
Cells	280	40 160 80
Faces	720	320 320 80
Edges	720	480 240
Vertices	240
Vertex figure	{ }v{4}
Coxeter groups	B5, [3,3,3,4], order 3840 D5, [32,1,1], order 1920
Properties	convex

The bitruncated 5-orthoplex can tessellate space in the tritruncated 5-cubic honeycomb.

• Bitruncated pentacross
• Bitruncated triacontiditeron (acronym: bittit) (Jonathan Bowers)^[2]

Cartesian coordinates for the vertices of a truncated 5-orthoplex, centered at the origin, are all 80 vertices are sign and coordinate permutations of

(±2,±2,±1,0,0)

The bitrunacted 5-orthoplex is constructed by a bitruncation operation applied to the 5-orthoplex.

orthographic projections

Coxeter plane	B5	B4 / D5	B3 / D4 / A2
Graph
Dihedral symmetry	[10]	[8]	[6]
Coxeter plane	B2	A3
Graph
Dihedral symmetry	[4]	[4]

This polytope is one of 31 uniform 5-polytopes generated from the regular 5-cube or 5-orthoplex.

B5 polytopes
β5	t1β5	t2γ5	t1γ5	γ5	t0,1β5	t0,2β5	t1,2β5
t0,3β5	t1,3γ5	t1,2γ5	t0,4γ5	t0,3γ5	t0,2γ5	t0,1γ5	t0,1,2β5
t0,1,3β5	t0,2,3β5	t1,2,3γ5	t0,1,4β5	t0,2,4γ5	t0,2,3γ5	t0,1,4γ5	t0,1,3γ5
t0,1,2γ5	t0,1,2,3β5	t0,1,2,4β5	t0,1,3,4γ5	t0,1,2,4γ5	t0,1,2,3γ5	t0,1,2,3,4γ5

• 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 [1]
    • (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. "5D uniform polytopes (polytera)". x3x3o3o4o - tot, o3x3x3o4o - bittit

• Weisstein, Eric W. "Hypercube". MathWorld.
• Polytopes of Various Dimensions
• Multi-dimensional Glossary