In eight-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 truncated 8-cube is a convex
uniform 8-polytope
In eight-dimensional geometry, an eight-dimensional polytope or 8-polytope is a polytope contained by 7-polytope facets. Each 6-polytope ridge being shared by exactly two 7-polytope facets.
A uniform 8-polytope is one which is vertex-transitive ...
, 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
8-cube
In geometry, an 8-cube is an eight-dimensional hypercube. It has 256 vertices, 1024 edges, 1792 square faces, 1792 cubic cells, 1120 tesseract 4-faces, 448 5-cube 5-faces, 112 6-cube 6-faces, and 16 7-cube 7-faces.
It is represented by ...
.
There are unique 7 degrees of truncation for the 8-cube. Vertices of the truncation 8-cube are located as pairs on the edge of the 8-cube. Vertices of the bitruncated 8-cube are located on the square faces of the 8-cube. Vertices of the tritruncated 7-cube are located inside the
cubic cells of the 8-cube. The final truncations are best expressed relative to the 8-orthoplex.
Truncated 8-cube
Alternate names
* Truncated octeract (acronym tocto) (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 i ...
for the vertices of a truncated 8-cube, centered at the origin, are all 224 vertices are sign (4) and coordinate (56)
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 p ...
s of
: (±2,±2,±2,±2,±2,±2,±1,0)
Images
Related polytopes
The ''
truncated 8-cube
In geometry, an 8-cube is an eight-dimensional hypercube. It has 256 vertices, 1024 edges, 1792 square faces, 1792 cubic cells, 1120 tesseract 4-faces, 448 5-cube 5-faces, 112 6-cube 6-faces, and 16 7-cube 7-faces.
It is represented by ...
'', is seventh in a sequence of truncated
hypercube
In geometry, a hypercube is an ''n''-dimensional analogue of a square () and a cube (). It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions ...
s:
Bitruncated 8-cube
Alternate names
* Bitruncated octeract (acronym bato) (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 i ...
for the vertices of a truncated 8-cube, centered at the origin, are all the sign 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 p ...
s of
: (±2,±2,±2,±2,±2,±1,0,0)
Images
Related polytopes
The ''
bitruncated
In geometry, a bitruncation is an operation on regular polytopes. It represents a truncation beyond rectification. The original edges are lost completely and the original faces remain as smaller copies of themselves.
Bitruncated regular polyt ...
8-cube
In geometry, an 8-cube is an eight-dimensional hypercube. It has 256 vertices, 1024 edges, 1792 square faces, 1792 cubic cells, 1120 tesseract 4-faces, 448 5-cube 5-faces, 112 6-cube 6-faces, and 16 7-cube 7-faces.
It is represented by ...
'' is sixth in a sequence of bitruncated
hypercube
In geometry, a hypercube is an ''n''-dimensional analogue of a square () and a cube (). It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions ...
s:
Tritruncated 8-cube
Alternate names
* Tritruncated octeract (acronym tato) (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 i ...
for the vertices of a truncated 8-cube, centered at the origin, are all the sign 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 p ...
s of
: (±2,±2,±2,±2,±1,0,0,0)
Images
Quadritruncated 8-cube
Alternate names
* Quadritruncated octeract (acronym oke) (Jonathan Bowers)
[Klitizing, (o3o3o3x3x3o3o4o – oke)]
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 i ...
for the vertices of a bitruncated 8-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 p ...
s of
: (±2,±2,±2,±2,±1,0,0,0)
Images
Related polytopes
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.
* o3o3o3o3o3o3x4x – tocto, o3o3o3o3o3x3x4o – bato, o3o3o3o3x3x3o4o – tato, o3o3o3x3x3o3o4o – oke
External links
Polytopes of Various Dimensions
{{Polytopes
8-polytopes