In ten-dimensional
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a rectified 10-cube is a convex
uniform 10-polytope
In ten-dimensional geometry, a 10-polytope is a 10-dimensional polytope whose boundary consists of 9-polytope facets, exactly two such facets meeting at each 8-polytope ridge.
A uniform 10-polytope is one which is vertex-transitive, and construct ...
, being a
rectification
Rectification has the following technical meanings:
Mathematics
* Rectification (geometry), truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points
* Rectifiable curve, in mathematics
* Recti ...
of the regular
10-cube.
There are 10 rectifications of the ''10-cube'', with the zeroth being the 10-cube itself. Vertices of the ''rectified 10-cube'' are located at the edge-centers of the ''10-cube''. Vertices of the ''birectified 10-cube'' are located in the square face centers of the ''10-cube''. Vertices of the ''trirectified 10-cube'' are located in the
cubic cell centers of the 10-cube. The others are more simply constructed relative to the 10-cube dual polytope, the
10-orthoplex.
These polytopes are part of a family 1023
uniform 10-polytope
In ten-dimensional geometry, a 10-polytope is a 10-dimensional polytope whose boundary consists of 9-polytope facets, exactly two such facets meeting at each 8-polytope ridge.
A uniform 10-polytope is one which is vertex-transitive, and construct ...
s with BC
10 symmetry.
Rectified 10-cube
Alternate names
* Rectified dekeract (Acronym rade) (Jonathan Bowers)
Cartesian coordinates
Cartesian coordinates
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
for the vertices of a rectified 10-cube, centered at the origin, edge length
are all permutations of:
: (±1,±1,±1,±1,±1,±1,±1,±1,±1,0)
Images
Birectified 10-cube
Alternate names
* Birectified dekeract (Acronym brade) (Jonathan Bowers)
Cartesian coordinates
Cartesian coordinates
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
for the vertices of a birectified 10-cube, centered at the origin, edge length
are all permutations of:
: (±1,±1,±1,±1,±1,±1,±1,±1,0,0)
Images
Trirectified 10-cube
Alternate names
* Tririrectified dekeract (Acronym trade) (Jonathan Bowers)
Cartesian coordinates
Cartesian coordinates
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
for the vertices of a triirectified 10-cube, centered at the origin, edge length
are all permutations of:
: (±1,±1,±1,±1,±1,±1,±1,0,0,0)
Images
Quadrirectified 10-cube
Alternate names
* Quadrirectified dekeract
* Quadrirectified decacross (Acronym terade) (Jonathan Bowers)
[Klitzing, (o3o3o3o3o3x3o3o3o4o - terade)]
Cartesian coordinates
Cartesian coordinates
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
for the vertices of a quadrirectified 10-cube, centered at the origin, edge length
are all permutations of:
: (±1,±1,±1,±1,±1,±1,0,0,0,0)
Images
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,
*** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'',
ath. Zeit. 46 (1940) 380–407, MR 2,10*** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'',
ath. Zeit. 188 (1985) 559-591*** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'',
ath. Zeit. 200 (1988) 3-45*
Norman Johnson ''Uniform Polytopes'', Manuscript (1991)
** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. (1966)
* x3o3o3o3o3o3o3o3o4o - ka, o3x3o3o3o3o3o3o3o4o - rake, o3o3x3o3o3o3o3o3o4o - brake, o3o3o3x3o3o3o3o3o4o - trake, o3o3o3o3x3o3o3o3o4o - terake, o3o3o3o3o3x3o3o3o4o - terade, o3o3o3o3o3o3x3o3o4o - trade, o3o3o3o3o3o3o3x3o4o - brade, o3o3o3o3o3o3o3o3x4o - rade, o3o3o3o3o3o3o3o3o4x - deker
External links
Polytopes of Various Dimensions
{{Polytopes
10-polytopes