In seven-dimensional
geometry, a truncated 7-orthoplex is a convex
uniform 7-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
7-orthoplex.
There are 6 truncations of the 7-orthoplex. Vertices of the truncation 7-orthoplex are located as pairs on the edge of the 7-orthoplex. Vertices of the bitruncated 7-orthoplex are located on the triangular faces of the 7-orthoplex. Vertices of the tritruncated 7-orthoplex are located inside the
tetrahedral cells of the 7-orthoplex. The final three truncations are best expressed relative to the 7-cube.
Truncated 7-orthoplex
Alternate names
* Truncated heptacross
* Truncated hecatonicosoctaexon (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 truncated 7-orthoplex, centered at the origin, are all 168 vertices are sign (4) and coordinate (42)
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
: (±2,±1,0,0,0,0,0)
Images
Construction
There are two
Coxeter groups associated with the ''truncated 7-orthoplex'', one with the C
7 or
,35Coxeter group, and a lower symmetry with the D
7 or
4,1,1Coxeter group.
Bitruncated 7-orthoplex
Alternate names
* Bitruncated heptacross
* Bitruncated hecatonicosoctaexon (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 bitruncated 7-orthoplex, 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
: (±2,±2,±1,0,0,0,0)
Images
Tritruncated 7-orthoplex
The tritruncated 7-orthoplex can tessellation space in the
quadritruncated 7-cubic honeycomb.
Alternate names
* Tritruncated heptacross
* Tritruncated hecatonicosoctaexon (Jonathan Bowers)
[Klitzing, (o3o3x3x3o3o4o - totaz)]
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 tritruncated 7-orthoplex, 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
: (±2,±2,±2,±1,0,0,0)
Images
Notes
References
*
H.S.M. Coxeter
Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.
Biography
Coxeter was born in Kensington t ...
:
** 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.
* x3x3o3o3o3o4o - tez, o3x3x3o3o3o4o - botaz, o3o3x3x3o3o4o - totaz
External links
Polytopes of Various Dimensions
{{polytopes
7-polytopes