In six-dimensional
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, a truncated 6-cube (or truncated hexeract) is a convex
uniform 6-polytope, being a
truncation
In mathematics and computer science, truncation is limiting the number of digits right of the decimal point.
Truncation and floor function
Truncation of positive real numbers can be done using the floor function. Given a number x \in \mathbb ...
of the regular
6-cube.
There are 5 truncations for the 6-cube. Vertices of the truncated 6-cube are located as pairs on the edge of the 6-cube. Vertices of the bitruncated 6-cube are located on the square faces of the 6-cube. Vertices of the tritruncated 6-cube are located inside the
cubic
Cubic may refer to:
Science and mathematics
* Cube (algebra), "cubic" measurement
* Cube, a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex
** Cubic crystal system, a crystal system w ...
cells of the 6-cube.
Truncated 6-cube
Alternate names
* Truncated hexeract (Acronym: tox) (Jonathan Bowers)
Construction and coordinates
The truncated 6-cube may be constructed by
truncating the vertices of the
6-cube at
of the edge length. A regular
5-simplex replaces each original vertex.
The
Cartesian coordinate
In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...
s of the vertices of a ''truncated 6-cube'' having edge length 2 are the permutations of:
:
Images
Related polytopes
The ''
truncated 6-cube'', is fifth in a sequence of truncated
hypercube
In geometry, a hypercube is an ''n''-dimensional analogue of a square ( ) and a cube ( ); the special case for is known as a ''tesseract''. It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel l ...
s:
Bitruncated 6-cube
Alternate names
* Bitruncated hexeract (Acronym: botox) (Jonathan Bowers)
Construction and coordinates
The
Cartesian coordinate
In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...
s of the vertices of a ''bitruncated 6-cube'' having edge length 2 are the permutations of:
:
Images
Related polytopes
The ''
bitruncated 6-cube'' is fourth in a sequence of bitruncated
hypercube
In geometry, a hypercube is an ''n''-dimensional analogue of a square ( ) and a cube ( ); the special case for is known as a ''tesseract''. It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel l ...
s:
Tritruncated 6-cube
Alternate names
* Tritruncated hexeract (Acronym: xog) (Jonathan Bowers)
Construction and coordinates
The
Cartesian coordinate
In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...
s of the vertices of a ''tritruncated 6-cube'' having edge length 2 are the permutations of:
:
Images
Related polytopes
The table below contains a set of 63
Uniform 6-polytopes generated from the B
6 Coxeter plane, including the regular
6-cube or
6-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 Ivic Weiss, Wiley-Interscience Publication, 1995
wiley.com
*** (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.
* o3o3o3o3x4x - tox, o3o3o3x3x4o - botox, o3o3x3x3o4o - xog
External links
*
Polytopes of Various Dimensions
{{Polytopes
6-polytopes