Rectified 5-cubes
Open-knowledge reference entry
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
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
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/triacontaditeron
Penteractitriacontaditeron (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
Related polytopes
These polytopes are a part of 31 uniform polytera 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.
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