# Regular (InstanceAttribute, 4)

### From Hi.gher. Space

A **regular polytope** is a polytope whose hypercells are transitive within each dimension.

In two dimensions, there are infinitely many regular polytopes, each one having a different number of sides.

In three dimensions and above, there are three distinct sets of regular polytopes: the simplices, which is self-dual, and the hypercubes and cross polytopes which are dual to each other. In three or four dimensions only, there are two more regular polytopes: the dodecahedron and the icosahedron, which are dual to each other. In four dimensions and no other dimension, there is also a sixth regular polytope, with several unique properties: the icositetrachoron, which is self-dual.

For the three main aforementioned sets, the simplices are all rotopes, the cross polytopes are all bracketopes and the hypercubes are all both rotopes and bracketopes.

Note that it does not make sense to speak of regularity in dimensions less than two. Also, since shapes can have curved hypercells, there are infinitely many regular *shapes* in any dimension, which is why we specify that regularity usually applies only to polytopes. There are also infinitely many hyperbolic tilings in each dimension although those are not a primary focus of this site.