Rectified 5-cubes
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
=
4-faces
42
10 32
Cells
200
40 160
Faces
400
80 320
Edges
320
Vertices
80
Vertex figure
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
2
{\displaystyle {\sqrt {2}}}
is given by all permutations of:
(
0
,
± ± -->
1
,
± ± -->
1
,
± ± -->
1
,
± ± -->
1
)
{\displaystyle (0,\ \pm 1,\ \pm 1,\ \pm 1,\ \pm 1)}
Images
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 32
Cells
280
40 160 80
Faces
640
320 320
Edges
480
Vertices
80
Vertex figure
{3}×{4}
Coxeter group
B5 , [4,33 ], order 3840 D5 , [32,1,1 ], 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 Cr5 2 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
2
{\displaystyle {\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
,
± ± -->
1
,
± ± -->
1
,
± ± -->
1
)
{\displaystyle \left(0,\ 0,\ \pm 1,\ \pm 1,\ \pm 1\right)}
Images
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
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
( )v( )
{ }×{ }
{ }v{ }
{3}×{4}
{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
β5
t1 β5
t2 γ5
t1 γ5
γ5
t0,1 β5
t0,2 β5
t1,2 β5
t0,3 β5
t1,3 γ5
t1,2 γ5
t0,4 γ5
t0,3 γ5
t0,2 γ5
t0,1 γ5
t0,1,2 β5
t0,1,3 β5
t0,2,3 β5
t1,2,3 γ5
t0,1,4 β5
t0,2,4 γ5
t0,2,3 γ5
t0,1,4 γ5
t0,1,3 γ5
t0,1,2 γ5
t0,1,2,3 β5
t0,1,2,4 β5
t0,1,3,4 γ5
t0,1,2,4 γ5
t0,1,2,3 γ5
t0,1,2,3,4 γ5
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 [1]
(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)" . o3x3o3o4o - rin, o3o3x3o4o - nit
External links