Truncated 5-cubes

120px 5-cube	120px Truncated 5-cube	120px Bitruncated 5-cube
120px 5-orthoplex	120px Truncated 5-orthoplex	120px Bitruncated 5-orthoplex
Orthogonal projections in B5 Coxeter plane

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

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

Truncated 5-cube
Type	uniform 5-polytope
Schläfli symbol	t{4,3,3,3}
Coxeter-Dynkin diagram
4-faces	42	10 32
Cells	200	40 160
Faces	400	80 320
Edges	400	80 320
Vertices	160
Vertex figure	60px ( )v{3,3}
Coxeter group	B5, [3,3,3,4], order 3840
Properties	convex

• Truncated penteract (Acronym: tan) (Jonathan Bowers)^[1]

The truncated 5-cube may be constructed by truncating the vertices of the 5-cube at ${\displaystyle 1/(\sqrt{2}+2)}$ of the edge length. A regular 5-cell is formed at each truncated vertex.

The Cartesian coordinates of the vertices of a truncated 5-cube having edge length 2 are all permutations of:

${\displaystyle (\pm 1,\pm (1+\sqrt{2}),\pm (1+\sqrt{2}),\pm (1+\sqrt{2}),\pm (1+\sqrt{2}))}$

The truncated 5-cube is constructed by a truncation applied to the 5-cube. 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]

The truncated 5-cube, is fourth in a sequence of truncated hypercubes: Template:Truncated hypercube polytopes

Bitruncated 5-cube
Type	uniform 5-polytope
Schläfli symbol	2t{4,3,3,3}
Coxeter-Dynkin diagrams
4-faces	42	10 32
Cells	280	40 160 80
Faces	720	80 320 320
Edges	800	320 480
Vertices	320
Vertex figure	80px { }v{3}
Coxeter groups	B5, [3,3,3,4], order 3840
Properties	convex

• Bitruncated penteract (Acronym: bittin) (Jonathan Bowers)^[2]

The bitruncated 5-cube may be constructed by bitruncating the vertices of the 5-cube at ${\displaystyle \sqrt{2}}$ of the edge length.

The Cartesian coordinates of the vertices of a bitruncated 5-cube having edge length 2 are all permutations of:

${\displaystyle (0,\pm 1,\pm 2,\pm 2,\pm 2)}$

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]

The bitruncated 5-cube is third in a sequence of bitruncated hypercubes: Template:Bitruncated hypercube polytopes

The truncated 5-cube and bitruncated 5-cube are from the family 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 Ivić Weiss, Wiley-Interscience Publication, 1995, wiley.com, 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. "5D uniform polytopes (polytera) with acronyms". https://bendwavy.org/klitzing/dimensions/polytera.htm.  o3o3o3x4x - tan, o3o3x3x4o - bittin

• Polytopes of Various Dimensions
• Multi-dimensional Glossary

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