Runcinated 5-demicube
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In five-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 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.


Steric 5-cube


Alternate names

* Steric penteract, runcinated demipenteract * Small prismated hemipenteract (siphin) (Jonathan Bowers)


Cartesian coordinates

The
Cartesian coordinates 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 ...
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 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 ...
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 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 ...
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 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 ...
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 space, five-dimensional geometry, a demipenteract or 5-demicube is a semiregular 5-polytope, constructed from a ''5-hypercube'' (penteract) with Alternation (geometry), alternated vertices removed. It was discovered by Thorold ...
, 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 Facet (mathematics), facets. Here, "vertex-transitive" means that it has symmetries taking every vertex to every other vertex; the sam ...
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 ( ); 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 ...
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

* * * * * * *


External links

*
Polytopes of Various Dimensions


{{Polytopes 5-polytopes