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 steric 5-cube or (steric 5-demicube or sterihalf 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 ...
. There are unique 4 steric forms of the 5-cube. Steric 5-cubes have half the vertices of
stericated 5-cubes
In five-dimensional geometry, a stericated 5-cube is a convex uniform 5-polytope with fourth-order truncations (sterication) of the regular 5-cube.
There are eight degrees of sterication for the 5-cube, including permutations of runcination, c ...
.
Steric 5-cube
Alternate names
* Steric penteract, runcinated demipenteract
* Small prismated hemipenteract (siphin) (Jonathan Bowers)
Cartesian coordinates
The
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 80 vertices of a steric 5-cube centered at the origin are the permutations of
: (±1,±1,±1,±1,±3)
with an odd number of plus signs.
Images
Related polytopes
Stericantic 5-cube
Alternate names
* Prismatotruncated hemipenteract (pithin) (Jonathan Bowers)
Cartesian coordinates
The
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 480 vertices of a stericantic 5-cube centered at the origin are coordinate permutations:
: (±1,±1,±3,±3,±5)
with an odd number of plus signs.
Images
Steriruncic 5-cube
Alternate names
* Prismatorhombated hemipenteract (pirhin) (Jonathan Bowers)
Cartesian coordinates
The
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 320 vertices of a steriruncic 5-cube centered at the origin are coordinate permutations:
: (±1,±1,±1,±3,±5)
with an odd number of plus signs.
Images
Steriruncicantic 5-cube
Alternate names
* Great prismated hemipenteract (giphin) (Jonathan Bowers)
Cartesian coordinates
The
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 960 vertices of a steriruncicantic 5-cube centered at the origin are coordinate permutations:
: (±1,±1,±3,±5,±7)
with an odd number of plus signs.
Images
Related polytopes
This polytope is based on the
5-demicube
In five-dimensional geometry, a demipenteract or 5-demicube is a semiregular 5-polytope, constructed from a ''5-hypercube'' (penteract) with alternated vertices removed.
It was discovered by Thorold Gosset. Since it was the only semiregular 5- ...
, a part of a dimensional family of
uniform polytope
In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. The uniform polytopes in two dimensions are the regular polygons (the definition is different in 2 dimensions to exclude vert ...
s called
demihypercube
In geometry, demihypercubes (also called ''n-demicubes'', ''n-hemicubes'', and ''half measure polytopes'') are a class of ''n''-polytopes constructed from alternation of an ''n''-hypercube, labeled as ''hγn'' for being ''half'' of the hype ...
s for being
alternation of the
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, ...
family.
There are 23
uniform polytera (uniform 5-polytope) that can be constructed from the D symmetry of the 5-demicube, of which are unique to this family, and 15 are shared within the
5-cube
In five-dimensional geometry, a 5-cube is a name for a five-dimensional hypercube with 32 vertices, 80 edges, 80 square faces, 40 cubic cells, and 10 tesseract 4-faces.
It is represented by Schläfli symbol or , constructed as 3 tesseracts, ...
family.
References
Further reading
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External links
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Polytopes of Various Dimensions
{{Polytopes
5-polytopes