wendy wrote:A number of stary hyperbolic tilings of finite density are known. But there are not a lot of star-subgroups up there, and we mostly rely on Coxeter's theorm that starry symmetries are nade from regular ones.

There's a theorem that symmetries of nonconvex Wythoffian figures can always be described by Dynkin diagrams for convex figures? I'm not surprised; this seems true in practice but it's nice that there's a proof.

I wonder how well this extends to compound polytopes. It's probably still true if the compound is formed by replacing each instance of an element or element-figure (whose symmetry is given by the Dynkin diagram with one node removed) and replacing it by a compound that preserves the symmetry of each component. Such compounds are the only ones I consider to be "true extensions" of Wythoffian polytopes. For example, the rhombihedron (compound of 5 cubes) doesn't count because the triangular symmetry around each vertex of a cube degrades into chiral triangular symmetry in the compound.

There is a similar subclass of uniform polytopes that I call "locally uniform", where each element is not only uniform but also has the property that symmetries of the original shape can map a vertex from one element onto every other vertex of the same element, while still mapping the element onto itself. This filters out almost all polytopes whose vertex configurations can't be made by kaleidoscopical construction; it excludes the antiprisms and snubs in 3 dimensions and almost everything from Bowers' "miscellaneous" categories in 4 and 5 dimensions.