Truncated 5-cubes

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

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

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	( )v{3,3}
Coxeter group	B₅, [3,3,3,4], order 3840
Properties	convex

Alternate names

Truncated penteract (Acronym: tan) (Jonathan Bowers)

Construction and coordinates

The truncated 5-cube may be constructed by truncating the vertices of the 5-cube at $1/({\sqrt {2}}+2)$ 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:

\left(\pm 1,\ \pm (1+{\sqrt {2}}),\ \pm (1+{\sqrt {2}}),\ \pm (1+{\sqrt {2}}),\ \pm (1+{\sqrt {2}})\right)

Images

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]

Related polytopes

The truncated 5-cube, is fourth in a sequence of truncated hypercubes:

Truncated hypercubes
Image								...
Name	Octagon	Truncated cube	Truncated tesseract	Truncated 5-cube	Truncated 6-cube	Truncated 7-cube	Truncated 8-cube
Coxeter diagram
Vertex figure	( )v( )	( )v{ }	( )v{3}	( )v{3,3}	( )v{3,3,3}	( )v{3,3,3,3}	( )v{3,3,3,3,3}

Bitruncated 5-cube

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	{ }v{3}
Coxeter groups	B₅, [3,3,3,4], order 3840
Properties	convex

Alternate names

Bitruncated penteract (Acronym: bittin) (Jonathan Bowers)

Construction and coordinates

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

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

\left(0,\ \pm 1,\ \pm 2,\ \pm 2,\ \pm 2\right)

Images

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

The bitruncated 5-cube is third in a sequence of bitruncated hypercubes:

Bitruncated hypercubes
Image							...
Name	Bitruncated cube	Bitruncated tesseract	Bitruncated 5-cube	Bitruncated 6-cube	Bitruncated 7-cube	Bitruncated 8-cube
Coxeter
Vertex figure	( )v{ }	{ }v{ }	{ }v{3}	{ }v{3,3}	{ }v{3,3,3}	{ }v{3,3,3,3}

Related 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.

Notes

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 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". o3o3o3x4x - tan, o3o3x3x4o - bittin

External links

v t e Fundamental convex regular and uniform polytopes
Family	$A n$	$B n$	$I 2 (p)$ / $D n$	$E 6$ / $E 7$ / $E 8$ / $F 4$ / $G 2$	$H n$
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	Pentachoron	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 n-polytope	n-simplex	n-orthoplex • n-cube	n-demicube	1_k2 • 2_k1 • k₂₁	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds • Polytope operations

[FOOTNOTEKlitzing[httpsbendwavyorgklitzingincmatstanhtm_(o3o3o3x4x_-_tan)]-1] Klitzing, (o3o3o3x4x - tan).

[FOOTNOTEKlitzing[httpsbendwavyorgklitzingincmatsbittinhtm_(o3o3x3x4o_-_bittin)]-2] Klitzing, (o3o3x3x4o - bittin).

B5 polytopes
β₅	t₁β₅	t₂γ₅	t₁γ₅	γ₅	t_0,1β₅	t_0,2β₅	t_1,2β₅
t_0,3β₅	t_1,3γ₅	t_1,2γ₅	t_0,4γ₅	t_0,3γ₅	t_0,2γ₅	t_0,1γ₅	t_0,1,2β₅
t_0,1,3β₅	t_0,2,3β₅	t_1,2,3γ₅	t_0,1,4β₅	t_0,2,4γ₅	t_0,2,3γ₅	t_0,1,4γ₅	t_0,1,3γ₅
t_0,1,2γ₅	t_0,1,2,3β₅	t_0,1,2,4β₅	t_0,1,3,4γ₅	t_0,1,2,4γ₅	t_0,1,2,3γ₅	t_0,1,2,3,4γ₅