In six-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 cantellated 5-cube is a convex
uniform 5-polytope
In geometry, a uniform 5-polytope is a five-dimensional uniform polytope. By definition, a uniform 5-polytope is vertex-transitive and constructed from uniform 4-polytope Facet (geometry), facets.
The complete set of convex uniform 5-polytopes ...
, being a
cantellation
In geometry, a cantellation is a 2nd-order truncation in any dimension that bevels a regular polytope at its edges and at its vertices, creating a new facet in place of each edge and of each vertex. Cantellation also applies to regular tiling ...
of the regular
5-cube.
There are 6 unique cantellation for the 5-cube, including truncations. Half of them are more easily constructed from the dual
5-orthoplex
In five-dimensional geometry, a 5-orthoplex, or 5-cross polytope, is a five-dimensional polytope with 10 vertices, 40 edges, 80 triangle faces, 80 tetrahedron cells, 32 5-cell 4-faces.
It has two constructed forms, the first being regular with ...
Cantellated 5-cube
Alternate names
* Small rhombated penteract (Acronym: sirn) (Jonathan Bowers)
Coordinates
The
Cartesian coordinate
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 ...
s of the vertices of a ''cantellated 5-cube'' having edge length 2 are all permutations of:
:
Images
Bicantellated 5-cube
In
five-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 bicantellated 5-cube is a
uniform 5-polytope
In geometry, a uniform 5-polytope is a five-dimensional uniform polytope. By definition, a uniform 5-polytope is vertex-transitive and constructed from uniform 4-polytope Facet (geometry), facets.
The complete set of convex uniform 5-polytopes ...
.
Alternate names
* Bicantellated penteract, bicantellated 5-orthoplex, or bicantellated pentacross
* Small birhombated penteractitriacontiditeron (Acronym: sibrant) (Jonathan Bowers)
Coordinates
The
Cartesian coordinate
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 ...
s of the vertices of a ''bicantellated 5-cube'' having edge length 2 are all permutations of:
:(0,1,1,2,2)
Images
Cantitruncated 5-cube
Alternate names
* Tricantitruncated 5-orthoplex / tricantitruncated pentacross
* Great rhombated penteract (girn) (Jonathan Bowers)
Coordinates
The
Cartesian coordinate
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 ...
s of the vertices of an cantitruncated 5-cube having an edge length of 2 are given by all permutations of coordinates and sign of:
:
Images
Related polytopes
It is third in a series of cantitruncated hypercubes:
Bicantitruncated 5-cube
Alternate names
* Bicantitruncated penteract
* Bicantitruncated pentacross
* Great birhombated penteractitriacontiditeron (Acronym: gibrant) (Jonathan Bowers)
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 bicantitruncated 5-cube, centered at the origin, are all sign and coordinate
permutation
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or proc ...
s of
: (±3,±3,±2,±1,0)
Images
Related polytopes
These polytopes are from a set of 31
uniform 5-polytopes generated from the regular
5-cube or
5-orthoplex
In five-dimensional geometry, a 5-orthoplex, or 5-cross polytope, is a five-dimensional polytope with 10 vertices, 40 edges, 80 triangle faces, 80 tetrahedron cells, 32 5-cell 4-faces.
It has two constructed forms, the first being regular with ...
.
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, editied 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.
* o3o3x3o4x - sirn, o3x3o3x4o - sibrant, o3o3x3x4x - girn, o3x3x3x4o - gibrant
External links
*
Polytopes of Various Dimensions Jonathan Bowers
*
(spid), Jonathan Bowers
{{Polytopes
5-polytopes