Dear Übersketch,

I'd think you're after a longer and more detailed reply.

ubersketch wrote:Yeah, but in most contexts, it shouldn’t matter whether you are saying pseudo-vertex transitive or vertex transitive.

In the contrary, there is a huge difference between those just local and the full global concept! - Or would you agree in calling pi a rational number? Simply because of

3 = 3/1,

3.1 = 31/10,

3.14 = 314/100 = 157/50

etc.

all are rational numbers?

Or consider the quasiperiodic tilings. Those implement a higher degree module (symmetry of spectrum of difraction) than the number of basis vectors of the tiled space. E.g. the well-known Penrose tiling implements just 2 types of rhombs. But that very building rule of quasiperiodicity does not allow any random combination of those rhombs, rather there are some few allowed vertex surroundings only. But note, the number of those local patches depends from the radius of those patches! For any given radius (edge size assumed to be unity throughout) there will be a finite set of local "allowed" configurations. But in order to get the resulting tiling to be identified uniquely (not just whole equivalence class of such Penrose tilings) the size of the patches would have to be infinite. - This is what I tried to mention wrt. Miller's solid resp. the Archimedean rhombicuboctahedron.

Different authors address that difference between "pseudo (= seemingly) vertex transitivity", which considers just local patches only, and (true) vertex transitivity, which considers the global object, as "equigonal" vs. "isogonal". Just to mention.

ubersketch wrote:Also you haven’t quite answered my questions, but thanksgiving for reminding me how I got the name in the first place.

Well, that highly depends on the definition of "weakly uniform". I for one used that very term in the dawn of the 20th century for the relaxation of the 3rd rule of uniformity, i.e. that one, which asks for dimensional hierarchicallity (= all its elements have to be uniform in turn). In fact, I'd found a figure, that very tutcup, which does follow other rules of uniformity, except of that 3rd one. In the sequel by a joined research of different authors brought up a great number of convex and even non-convex polychora of that type (e.g. cf. Jonathan Bowers' (aka PolyhedronDude's) according

listing). So that working title just took about 4 years to settle down into an own specific term: the term "scaliform" got born. - In that sense, for sure, tutcup will be "weakly uniform" right by concept.

If you'd like to relax uniformity in a

different sense, and call that again "weakly uniform", then you will have to investigate for new, whether it does follow your rule set or not.

But be warned. There are infinitely many choices for rule sets. So you cannot aim to address those all by an own label (term). It rather is whether the result set of Elements (polytopes) could be achieved to enumerate all contained elements (polytopes) explicitely, or at least whether you can include/exclude specific well-known groups of elements (polytopes), or the set of rules in any other way opens an "interesting" direction of research (e.g. like the "CRF"s) - only then that rule set deserves a specific own name. ("Interesting", btw. is not being attributed by the inventor, rather by his audience.)

--- rk