How rich is the class of vertex- and edge-transitive polytopes? There are only a few regular polytopes (five in 3D, six in 4D, three in any dimension above). In contrast, the class of uniform polytopes becomes very rich with higher dimensions.
The class of vertex- and edge-transitive polytopes is in the middle between these two classes (in 3D, they are called quasi-regular). Recently I had a result only applying to polytopes from this class, and now I wonder how relevant this result is. Does it only apply to a small finite number per dimension, or only a few infinite families? Or are there many "new" such polytopes in every higher dimension?
Note: I am only interested in convex polytopes in Euclidean spaces.
 A: I can't provide you with a full classification of those, but at least with a subset of convex polytopes which all are being contained. And which itself might be "rich enough"?
Just consider any Coxeter group (reflection group) diagram with just a single node (at any position) being ringed. All their corresponding polytopes (according to Wythoff's kaleidoscopical construction rule) would clearly pass: Any such polytope clearly would be uniform and thence is vertex transitive (via construction by a single seed point, i.e. all vertices are each other's images). Moreover each ringed node symbol represents a class of symmetry equivalent edges (transversal to the according mirror class). Thence, when just a single such is being ringed, the resulting polytope is being bound to be edge transitive as well. - Wrt. linear diagrams those are just all the (multi-)rectified versions of the (convex) regular polytopes. But then there are bifurcated diagrams as well.
You might increase that set further, by observing that for those (undecorated) Coxeter group diagrams, which have (as a diagram) some further outer symmetry, an according symmetrical decoration by ringed nodes could result within still edge transitive polytopes again. This is because then the being used edge classes (i.e. those node ringings) due to that outer symmetry become unified after all. - This not only allows for such figures as the decachoron, but for instance all multiprisms of polytopes of the above mentioned subclass. Therefore already within 4D you would encompass the infinite set of all regular n-gonal duoprisms.
--- rk
