Rectified 5-cubes

Orthogonal projections in A₅ Coxeter plane
File:5-cube t0.svg 5-cube File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	File:5-cube t1.svg Rectified 5-cube File:CDel node.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	File:5-cube t2.svg Birectified 5-cube Birectified 5-orthoplex File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
File:5-cube t4.svg 5-orthoplex File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png	File:5-cube t3.svg Rectified 5-orthoplex File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png

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-cube 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	File:CDel node.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png = File:CDel node 1.pngFile:CDel split1-43.pngFile:CDel nodes.pngFile:CDel 3b.pngFile:CDel nodeb.pngFile:CDel 3b.pngFile:CDel nodeb.png File:CDel nodes 11.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
4-faces	42	10 File:CDel node.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:4-cube t1.svg 32 File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:4-simplex t0.svg
Cells	200	40 File:CDel node.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png File:3-cube t1.svg 160 File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:3-simplex t0.svg
Faces	400	80 File:CDel node.pngFile:CDel 4.pngFile:CDel node 1.png File:2-cube.svg 320 File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png File:2-simplex t0.svg
Edges	320
Vertices	80
Vertex figure	File:Rectified 5-cube verf.png Tetrahedral prism
Coxeter group	B₅, [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, \pm 1, \pm 1, \pm 1, \pm 1)

Images

orthographic projections
Coxeter plane	B₅	B₄ / D₅	B₃ / D₄ / A₂
Graph	File:5-cube t1.svg	File:5-cube t1 B4.svg	File:5-cube t1 B3.svg
Dihedral symmetry	[10]	[8]	[6]
Coxeter plane	B₂	A₃
Graph	File:5-cube t1 B2.svg	File:5-cube t1 A3.svg
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	File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png = File:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes.pngFile:CDel 4a3b.pngFile:CDel nodes.png File:CDel nodes.pngFile:CDel split2.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
4-faces	42	10 File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png File:4-cube t2.svg 32 File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:4-simplex t1.svg
Cells	280	40 File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png File:3-cube t2.svg 160 File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png File:3-simplex t1.svg 80 File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:3-simplex t0.svg
Faces	640	320 File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png File:2-simplex t0.svg 320 File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png File:2-simplex t0.svg
Edges	480
Vertices	80
Vertex figure	File:Birectified penteract verf.png {3}×{4}
Coxeter group	B₅, [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 birectifying 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:

(0, 0, \pm 1, \pm 1, \pm 1)

Images

orthographic projections
Coxeter plane	B₅	B₄ / D₅	B₃ / D₄ / A₂
Graph	File:5-cube t2.svg	File:5-cube t2 B4.svg	File:5-cube t2 B3.svg
Dihedral symmetry	[10]	[8]	[6]
Coxeter plane	B₂	A₃
Graph	File:5-cube t2 B2.svg	File:5-cube t2 A3.svg
Dihedral symmetry	[4]	[4]

Related polytopes

2-isotopic hypercubes
Dim.	2	3	4	5	6	7	8	n
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	File:CDel label4.pngFile:CDel branch 11.png	File:CDel node 1.pngFile:CDel split1-43.pngFile:CDel nodes.png	File:CDel branch 11.pngFile:CDel 4a3b.pngFile:CDel nodes.png	File:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes.pngFile:CDel 4a3b.pngFile:CDel nodes.png	File:CDel branch 11.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 4a3b.pngFile:CDel nodes.png	File:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 4a3b.pngFile:CDel nodes.png	File:CDel branch 11.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 4a3b.pngFile:CDel nodes.png
Images	File:Truncated square.png	File:3-cube t1.svgFile:Cuboctahedron.png	File:4-cube t12.svgFile:Schlegel half-solid bitruncated 8-cell.png	File:5-cube t2.svgFile:5-cube t2 A3.svg	File:6-cube t23.svgFile:6-cube t23 A5.svg	File:7-cube t3.svgFile:7-cube t3 A5.svg	File:8-cube t34.svgFile:8-cube t34 A7.svg
Facets		{3} File:Regular polygon 3 annotated.svg {4} File:Regular polygon 4 annotated.svg	t{3,3} File:Uniform polyhedron-33-t01.png t{3,4} File:Uniform polyhedron-43-t12.png	r{3,3,3} File:Schlegel half-solid rectified 5-cell.png r{3,3,4} File:Schlegel wireframe 24-cell.png	2t{3,3,3,3} File:5-simplex t12.svg 2t{3,3,3,4} File:5-cube t23.svg	2r{3,3,3,3,3} File:6-simplex t2.svg 2r{3,3,3,3,4} File:6-cube t4.svg	3t{3,3,3,3,3,3} File:7-simplex t23.svg 3t{3,3,3,3,3,4} File:7-cube t45.svg
Vertex figure	( )v( )	File:Cuboctahedron vertfig.svg { }×{ }	File:Bitruncated 8-cell verf.png { }v{ }	File:Birectified penteract verf.png {3}×{4}	File:Tritruncated 6-cube verf.png {3}v{4}	{3,3}×{3,4}	{3,3}v{3,4}

Related polytopes

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

B5 polytopes
File:5-cube t4.svg β₅	File:5-cube t3.svg t₁β₅	File:5-cube t2.svg t₂γ₅	File:5-cube t1.svg t₁γ₅	File:5-cube t0.svg γ₅	File:5-cube t34.svg t_0,1β₅	File:5-cube t24.svg t_0,2β₅	File:5-cube t23.svg t_1,2β₅
File:5-cube t14.svg t_0,3β₅	File:5-cube t13.svg t_1,3γ₅	File:5-cube t12.svg t_1,2γ₅	File:5-cube t04.svg t_0,4γ₅	File:5-cube t03.svg t_0,3γ₅	File:5-cube t02.svg t_0,2γ₅	File:5-cube t01.svg t_0,1γ₅	File:5-cube t234.svg t_0,1,2β₅
File:5-cube t134.svg t_0,1,3β₅	File:5-cube t124.svg t_0,2,3β₅	File:5-cube t123.svg t_1,2,3γ₅	File:5-cube t034.svg t_0,1,4β₅	File:5-cube t024.svg t_0,2,4γ₅	File:5-cube t023.svg t_0,2,3γ₅	File:5-cube t014.svg t_0,1,4γ₅	File:5-cube t013.svg t_0,1,3γ₅
File:5-cube t012.svg t_0,1,2γ₅	File:5-cube t1234.svg t_0,1,2,3β₅	File:5-cube t0234.svg t_0,1,2,4β₅	File:5-cube t0134.svg t_0,1,3,4γ₅	File:5-cube t0124.svg t_0,1,2,4γ₅	File:5-cube t0123.svg t_0,1,2,3γ₅	File:5-cube t01234.svg 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, Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value). [1] Archived 2016-07-11 at the Wayback Machine
  - (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.
Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value). o3x3o3o4o - rin, o3o3x3o4o - nit

External links

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
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 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 • Polytope operations

Rectified 5-cubes

Contents

Rectified 5-cube

Alternate names

Construction

Coordinates

Images

Birectified 5-cube

Alternate names

Construction and coordinates

Images

Related polytopes

Related polytopes

Notes

References

External links

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Rectified 5-cubes

Rectified 5-cube

Alternate names

Construction

Coordinates

Images

Birectified 5-cube

Alternate names

Construction and coordinates

Images

Related polytopes

Related polytopes

Notes

References

External links

Navigation menu

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