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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 facets. The complete set of convex uniform 5-polytopes has not been determined, but many can be made as Wythoff constructions from a small set of symmetry groups.
A 2D orthogonal projection of a 5-cube. A five-dimensional space is a space with five dimensions. In mathematics, a sequence of N numbers can represent a location in an N-dimensional space. If interpreted physically, that is one more than the usual three spatial dimensions and the fourth dimension of time used in relativistic physics. [1]
A prismatic 5-polytope is uniform if its factors are uniform. The hypercube is prismatic (product of a square and a cube), but is considered separately because it has symmetries other than those inherited from its factors. A 4-space tessellation is the division of four-dimensional Euclidean space into a regular grid of polychoral facets ...
It is a part of a dimensional family of uniform polytopes called demihypercubes for being alternation of the hypercube family.. There are 23 Uniform 5-polytopes (uniform 5-polytopes) that can be constructed from the D 5 symmetry of the demipenteract, 8 of which are unique to this family, and 15 are shared within the penteractic family.
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 {4,3,3,3} or {4,3 3 }, constructed as 3 tesseracts, {4,3,3}, around each cubic ridge .
Because (n+1)-dimensional polytopes are tilings of n-dimensional spherical space, tilings of n-dimensional Euclidean and hyperbolic space are also considered to be (n+1)-dimensional. Hence, the tilings of two-dimensional space are grouped with the three-dimensional solids.
Tessellations of euclidean and hyperbolic space may also be considered regular polytopes. Note that an 'n'-dimensional polytope actually tessellates a space of one dimension less. For example, the (three-dimensional) platonic solids tessellate the 'two'-dimensional 'surface' of the sphere.
In six-dimensional geometry, a runcinated 5-simplex is a convex uniform 5-polytope with 3rd order truncations (Runcination) of the regular 5-simplex. There are 4 unique runcinations of the 5-simplex with permutations of truncations, and cantellations .