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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.
An important uniform 5-polytope is the 5-demicube, h{4,3,3,3} has half the vertices of the 5-cube (16), bounded by alternating 5-cell and 16-cell hypercells. The expanded or stericated 5-simplex is the vertex figure of the A 5 lattice, . It and has a doubled symmetry from its symmetric Coxeter diagram.
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 ...
Regular n-polytopes have n orders of rectification.The zeroth rectification is the original form. The (n−1)-th rectification is the dual.A rectification reduces edges to vertices, a birectification reduces faces to vertices, a trirectification reduces cells to vertices, a quadirectification reduces 4-faces to vertices, a quintirectification reduced 5-faces to vertices, and so on.
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.
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.
Five-dimensional space, 5-polytope and uniform 5-polytope. 5-simplex, Rectified 5-simplex, Truncated 5-simplex, Cantellated 5-simplex, ...
The vertex arrangement of the 5-demicubic honeycomb is the D 5 lattice which is the densest known sphere packing in 5 dimensions. [1] The 40 vertices of the rectified 5-orthoplex vertex figure of the 5-demicubic honeycomb reflect the kissing number 40 of this lattice. [2] The D + 5 packing (also called D 2 5) can be constructed by the union of ...