Truncated 5-orthoplexes

5-orthoplex	Truncated 5-orthoplex	Bitruncated 5-orthoplex
5-cube	Truncated 5-cube	Bitruncated 5-cube
Orthogonal projections in BC5 Coxeter plane

In six-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

Truncated 5-orthoplex
Type	uniform 5-polytope
Schläfli symbol	t{3,3,3,4} t{3,3^1,1}
Coxeter-Dynkin diagrams
4-faces	42
Cells	240
Faces	400
Edges	280
Vertices	80
Vertex figure	Elongated octahedral pyramid
Coxeter groups	BC5, [3,3,3,4] D5, [3^2,1,1]
Properties	convex

Alternate names

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

Coordinates

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)

Images

The trunacted 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

Bitruncated 5-orthoplex
Type	uniform 5-polytope
Schläfli symbol	2t{3,3,3,4} 2t{3,3^1,1}
Coxeter-Dynkin diagrams
4-faces	42
Cells	280
Faces	720
Edges	720
Vertices	240
Vertex figure	square-pyramidal pyramid
Coxeter groups	BC5, [3,3,3,4] D5, [3^2,1,1]
Properties	convex

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

Alternate names

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

Coordinates

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)

Images

The bitrunacted 5-orthoplex is constructed by a bitruncation 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]

Related polytopes

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

Notes

1. Klitzing, (x3x3o3o4o - tot)
2. Klitzing, (x3x3x3o4o - gart)

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. "5D uniform polytopes (polytera)". x3x3o3o4o - tot, x3x3x3o4o - gart

External links

• Weisstein, Eric W. "Hypercube". MathWorld.

• Olshevsky, George. "Measure polytope". Glossary for Hyperspace. Archived from the original on 4 February 2007.
• Polytopes of Various Dimensions
• Multi-dimensional Glossary

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	An	Bn	I2(p) / Dn	E6 / E7 / E8 / E9 / E10 / F4 / G2	Hn
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform 4-polytope	5-cell	16-cell • Tesseract	Demitesseract	24-cell	120-cell • 600-cell
Uniform 5-polytope	5-simplex	5-orthoplex • 5-cube	5-demicube
Uniform 6-polytope	6-simplex	6-orthoplex • 6-cube	6-demicube	122 • 221
Uniform 7-polytope	7-simplex	7-orthoplex • 7-cube	7-demicube	132 • 231 • 321
Uniform 8-polytope	8-simplex	8-orthoplex • 8-cube	8-demicube	142 • 241 • 421
Uniform 9-polytope	9-simplex	9-orthoplex • 9-cube	9-demicube
Uniform 10-polytope	10-simplex	10-orthoplex • 10-cube	10-demicube
Uniform n-polytope	n-simplex	n-orthoplex • n-cube	n-demicube	1k2 • 2k1 • k21	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds