Suppose we have compact space, like sphere or torus in particular dimension $d$. Is it possible to construct aperiodic tiling in such setting? It seems obvious, answer is yes, because we may just divide its surface on finite number of basic shapes ( "curved polygons") but every one of those may be different than another. It is relatively easy I suppose.
Some of curved polygons above may be isomorphic tiles according to Euclidean group, translation and rotation, action in $d+1$ dimensional space, in which case we may say they have the same shape.
What is the smallest number of basic tiles of the same shape for which we may get aperiodic tilling? In particular: is it possible to construct aperiodic Penrose tiling on sphere? Torus? On cylinder?
If answers are known, could you take some references?
Edit: of course you may always use two different, "curved polygons" so probably question need some refactoring. Tiling I am asking should be defined as some kind of limit, where tiles are small in comparison to surface of the compact space of sphere or torus. So to make it nontrivial and mathematically interesting, more details are needed. From the other side, intuitive meaning here's clear. We want to exclude trivial cases of very few, very deformed polygons, and focus rather on very small tiles, building something like mesh rather than just dividing surface in finite number if shapes.