Margulis and Mozes constructed aperiodic tiling system on the hyperbolic plane consisting of a single tile(hyperbolic polygon) whose area (or each inner angle) is irrational multiple of $\pi$. Having area irrational multiple of $\pi$ is not a necessary condition. So my question is, is there an aperiodic single tile whose area (or each angles)is rational multiple of $\pi$ ?
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$\begingroup$ I believe the binary tiling is aperiodic and its tiles can be arbitrarily wide (and thus the area can be arbitrary)? Or did I miss any requirements? $\endgroup$– Zeno RogueCommented Jul 5, 2019 at 20:27
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$\begingroup$ @Zeno, by single tile I meant a hyperbolic polygon whose sides are geodesic segements. The pentagon used in binary tiling is not really a hyperbolic one. $\endgroup$– ArunCommented Jul 6, 2019 at 6:07
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$\begingroup$ The simplest binary tiling has horocyclic segments indeed, but you can replace them with geodesic segments (i.e., connect their endpoints with a geodesic instead) and it still works. $\endgroup$– Zeno RogueCommented Jul 6, 2019 at 8:09
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$\begingroup$ Or if you don't need convexity, you could replace them with "puzzle piece polylines" which match in only the way you want, for a simple proof. $\endgroup$– Zeno RogueCommented Jul 6, 2019 at 8:15
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$\begingroup$ Thanks @zeno! what if we ask for only convex polygons? $\endgroup$– ArunCommented Jul 6, 2019 at 13:20
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