No in either case.

The first step in proving this in the rational polygon case is to observe that if you flip over and over around a single vertex of a rational polygon, you eventually get back to where you started. This implies a finite reflection group.

Now take your rational polygon and hit by that full reflection group. You get some large thing, with many pairs of parallel sides. Identify parallel sides, and bam, you now have a compact surface of some genus and the billiard flow on the polygon is equivalent geodesic flow on this surface -- you can do math. The most famous case is that of the square -- ancient results of Hopf -- Hedlund, Caroline Series,... -- makes all of this comprehensible.

For irrational polygons, the reflection group is infinite -- the induced surface is not compact. Hard to do math. It seems Teichmuller methods (aka rational methods) can't be extended/won't be able to deal with this -- some known failed attempts from really top people.

For polyhedra, even if you have rational dihedral angles you do not necessarily have a finite reflection group. Again your 3d "surface" is not compact.

So in either case, both are major open problems.

No in either case.

The first step in proving this in the rational polygon case is to observe that if you flip over and over around a single vertex of a rational polygon, you eventually get back to where you started. This implies a finite reflection group.

Now take your rational polygon and hit by that full reflection group. You get some large thing, with many pairs of parallel sides. Identify parallel sides, and bam, you now have a compact surface of some genus and the billiard flow on the polygon is equivalent geodesic flow on this surface -- you can do math. The most famous case is that of the square -- ancient results of Hopf -- Hedlund, Caroline Series,... -- makes all of this comprehensible. The equivalence is made clear in Katok Zelmyakov, also Katok survey Billiards as a playground.

For irrational polygons, the reflection group is infinite -- the induced surface is not compact. Hard to do math. It seems Teichmuller methods (aka rational methods) can't be extended/won't be able to deal with this -- some known failed attempts from really top people.

For polyhedra, even if you have rational dihedral angles you do not necessarily have a finite reflection group. Again your 3d "surface" is not compact.

So in either case, both are major open problems. How do you do Geodesic flow on a non compact manifold (flat metric with finite number of singularities) and get results on existence of periodic orbits/ergodcitiy/diffusion/etc...