I think I can explain this. It looks like I'm turning into "that one guy who always answers when people have questions about geometry" .muzik wrote:I can't be the only one who finds it weird and confusing that there exists no star polyhedron {5/2,5/2}?

Wikipedia says it counts as a failed star polyhedron, since it doesn't eventually close.

I used to find the nonexistence of {5/2, 5/2} weird and confusing too, but it's really no stranger than the fact that {5/2, 4} doesn't exist. It just appears that it might because all of the subsymmetries implied by the Schlafli symbol (pentagonal and digonal/rectangular) exist as subsymmetries of dodecahedral symmetry.

Any nonconvex uniform polytope that has a kaleidoscopic (Wythoffian) construction--and therefore a Coxeter diagram--and does not overlap itself infinitely, must have a symmetry describable by a convex kaleidoscopic consruction (proof?). This means that nonconvex regular polytopes never "make" their own symmetry, they always follow the symmetries of convex polytopes. In this case, {5/2, 3} and {5/2, 5} both happen to close under the symmetry of {5, 3}, icosahedral symmetry. {5/2, 5/2}, however, does not, and there is no other symmetry that it can close under, as it would need to have a pentagonal subsymmetry. This is also why the only regular hyperbolic star tessellations are {n/2, n} and {n, n/2} for odd n greater than 6, as these are the only kaleidoscopic constructions that lend themselves to regular polytopes and a regular symmetry, namely {n, 3}.

Another way to think of it is that it doesn't exist because its conjugate* would be {5, 5}, which is hyperbolic and obviously doesn't close either.

Finding all the possible Coxeter diagrams that close under a certain symmetry is difficult (for me), but it can be done by realizing that their omnitruncates** are all facetings*** of each other if given the right edge lengths; each ring in the Coxeter diagram represents a different type of edge, and the edges of each type can have their length changed independently of those of any other type.

Dr. Richard Klitzing has a better way to find these, by "adding" the Coxeter diagrams (called Dynkin diagrams on the website), but I can't really understand it. It is described here: https://bendwavy.org/klitzing/explain/schwarz.htm

*A conjugate of a uniform polytope is one that, roughly, can be constructed by taking regular polygonal subsymmetries (usually on the faces or face figures, which are vertices in 3D) and "wrapping" the symmetry a different number of times. For example, the great dodecahedron {5, 5/2} can have its pentagonal faces doubly wrapped into pentagrammic faces and its pentagrammic vertex figures doubly wrapped into pentagonal ones. This produces the small stellated dodecahedron {5/2, 5}, thus the great dodecahedron and small stellated dodecahedron are conjugates. Taking a polytope's conjugate changes the circumradius and coordinates into their algebraic conjugates; numbers that are roots of the same polynomial.

It is also worth noting that the pentagonal faces and the pentagrammic vertices of the great dodecahedron must change together, as they can be "paired up"; each face lies directly under a vertex. On the other hand, in the square tiling {4, 4} (for example), squares and square-symmetry vertices cannot be paired up, meaning that they can be re-wound independently of each other. If the squares and/or vertices get re-wound thrice, it could lead to {4/3, 4}, {4, 4/3}, or {4/3, 4/3}--but these are all also the square tiling, so this creates no new tessellations.

**The omnitruncate of a Coxeter diagram is the shape made when every node in the diagram is ringed.

***Faceting: a polytope which shares the same vertices as another polytope.