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New result: you can build a universal computer using a single billiard ball on a carefully crafted table!
In other words, you can create a computer that can run any program, using just a single point moving around without friction in a region of the plane, and bouncing off the walls elastically.
Since the halting problem is undecidable, this means there are some yes-or-no questions about the eventual future behavior of this point that cannot be settled in a finite time by any computer program.
This is true even though the point's motion is computable to arbitrary accuracy for any given finite time. In fact, since the methodology here does *not* exploit the chaos that can occur for billiards on certain shaped tables, it's not even one of those cases where the point's motion is computable in principle but your knowledge of the initial conditions needs to be absurdly precise.
• Eva Miranda and Isaac Ramos, Classical billiards can compute, 
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arXiv.org
Classical billiards can compute
We show that two-dimensional billiard systems are Turing complete by encoding their dynamics within the framework of Topological Kleene Field Theor...