There is something about the classic definition of a uniform polytope that has always bugged me. There are three clauses to the definition: 1. The polytope must be vertex-transitive; informally, there is only one "kind" of vertex. 2. The edges must all be of the same length. 3. All the facets (that is, (n-1)-dimensional cells) of the polytope must themselves be uniform. Clause 3 establishes a recursive requirement that _all_ the cells be uniform, but these cells aren't required to be _identical_ until you get down to dimension 1 and clause 2 takes over. It feels to me like clause 2 is a weird addendum. I define a para-uniform polytope as follows: 1. The polytope must be vertex-transitive. 2. All its facets must themselves be para-uniform. This makes the edges behave like all the other cells: there are allowed to be different kinds. The para-uniform polygons, for instance, are all the equi-angular polygons that alternate two edge lengths. For odd numbers of sides, the two edge lengths are forced to be equal by global structure. Here's my question: do we get any novel structures? Or are all para-uniform polytopes just uniform polytopes with variously-stretched edges?