Rectified 5-cubes

Orthogonal projections in A₅ Coxeter plane
5-cube	Rectified 5-cube	Birectified 5-cube Birectified 5-orthoplex
5-orthoplex	Rectified 5-orthoplex	Birectified 5-cube Birectified 5-orthoplex

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

There are 5 degrees of rectifications of a 5-polytope, the zeroth here being the 5-cube, and the 4th and last being the 5-orthoplex. Vertices of the rectified 5-cube are located at the edge-centers of the 5-cube. Vertices of the birectified 5-ocube are located in the square face centers of the 5-cube.

Rectified 5-cube

Rectified 5-cube rectified penteract (rin)
Type	uniform 5-polytope
Schläfli symbol	r{4,3,3,3}
Coxeter diagram	=
4-faces	42
Cells	200
Faces	400
Edges	320
Vertices	80
Vertex figure	tetrahedral prism
Coxeter group	BC₅, [4,3³], order 3840
Dual
Base point	(0,1,1,1,1,1)√2
Circumradius	sqrt(2) = 1.414214
Properties	convex, isogonal

Alternate names

Rectified penteract (acronym: rin) (Jonathan Bowers)

Construction

The rectified 5-cube may be constructed from the 5-cube by truncating its vertices at the midpoints of its edges.

Coordinates

The Cartesian coordinates of the vertices of the rectified 5-cube with edge length $\sqrt{2}$ is given by all permutations of:

(0,\ \pm1,\ \pm1,\ \pm1,\ \pm1)

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]

Birectified 5-cube

Birectified 5-cube birectified penteract (nit)
Type	uniform 5-polytope
Schläfli symbol	2r{4,3,3,3}
Coxeter diagram	=
4-faces	42	10 {3,4,3} 32 t1{3,3,3}
Cells	280
Faces	640
Edges	480
Vertices	80
Vertex figure	{3}×{4}
Coxeter group	BC₅, [4,3³], order 3840 D₅, [3^2,1,1], order 1920
Dual
Base point	(0,0,1,1,1,1)√2
Circumradius	sqrt(3/2) = 1.224745
Properties	convex, isogonal

E. L. Elte identified it in 1912 as a semiregular polytope, identifying it as Cr₅² as a second rectification of a 5-dimensional cross polytope.

Alternate names

Birectified 5-cube/penteract
Birectified pentacross/5-orthoplex/triacontiditeron
Penteractitriacontiditeron (acronym: nit) (Jonathan Bowers)
Rectified 5-demicube/demipenteract

Construction and coordinates

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

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

\left(0,\ 0,\ \pm1,\ \pm1,\ \pm1\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

2-isotopic hypercubes
Dim.	2	3	4	5	6	7	8
Name	t{4}	r{4,3}	2t{4,3,3}	2r{4,3,3,3}	3t{4,3,3,3,3}	3r{4,3,3,3,3,3}	4t{4,3,3,3,3,3,3}
Coxeter diagram
Images								...
Facets		{3} {4}	t{3,3} t{3,4}	r{3,3,3} r{3,3,4}	2t{3,3,3,3} 2t{3,3,3,4}	2r{3,3,3,3,3} 2r{3,3,3,3,4}	3t{3,3,3,3,3,3} 3t{3,3,3,3,3,4}
Vertex figure		Rectangle	Disphenoid	{3}×{4} duoprism		{3,3}×{3,4} duoprism

Related polytopes

Thes polytopes are a part of 31 uniform polytera generated from the regular 5-cube or 5-orthoplex.

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

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 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.
Richard Klitzing, 5D, uniform polytopes (polytera) o3x3o3o4o - rin, o3o3x3o4o - nit

External links

Weisstein, Eric W., "Hypercube", MathWorld.
Olshevsky, George, Measure polytope at Glossary for Hyperspace.
Polytopes of Various Dimensions
Multi-dimensional Glossary

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	A_n	B_n	I₂(p) / D_n	E₆ / E₇ / E₈ / F₄ / G₂	H_n
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	1_k2 • 2_k1 • k₂₁	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds

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