Classification and Description
Regular polytopes are classified primarily according to their dimensionality.
They can be further classified according to symmetry. For example the cube and the regular octahedron share the same symmetry, as do the regular dodecahedron and icosahedron. Indeed, symmetry groups are sometimes named after regular polytopes, for example the tetrahedral and icosahedral symmetries.
Three special classes of regular polytope exist in every dimensionality:
- Regular simplex
- Measure polytope (Hypercube)
- Cross polytope (Orthoplex)
In two dimensions there are infinitely many regular polygons. In three and four dimensions there are several more regular polyhedra and polychora besides these three. In five dimensions and above, these are the only ones. See also the list of regular polytopes.
The idea of a polytope is sometimes generalised to include related kinds of geometrical object. Some of these have regular examples, as discussed in the section on historical discovery below.
Read more about this topic: Regular Polytope
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