by Klitzing » Wed Jun 14, 2023 8:47 am
Coxeter et al. last century (1954) addressed the search for "uniform polyhedra", cf. their according paper, then in fact by means of using various facetings of the verfs of the already known convex cases. They then proclaimed that their listing is complete. It only was later (1975) by Skilling that this claim has been proven to be true. (He just added a single figure then, which however wasn't dyadic.)
The same way of exploiting the set of uniform polychora by means of faceting known vertex figures was then commenced by Bowers (after Dinogeorge was trying to do so by elaborating all Wythoffians). However this search project is still ongoing, as every now and then further uniform polychora pop up. - Within higher dimensions even less is known.
They search for scaliform polychora (they only occur as such from 4D on - within 3D they would coincide with the uniforms) was inaugurated by my own find of tutcup, then the first such at all. In those days I still called that one "weakely uniform". It was again Bowers, who proposed a positive name for those (since then they are called "scaliforms"), instead of that more diminuing working title. In those days soon all 4 convex scaliform polychora had been found, which up to now still seem to be all such. For a systematic way to exhaust scaliforms, esp. the non-convex ones, nothing is known to me. However, all alterprisms, gyroprisms, as well as all their higherdimensional analogues clearly will provide examples.
--- rk