John Carlos BaezDec 23

New result: you can build a universal computer using a single billiard ball on a carefully crafted table!

More precisely: you can create a computer that can run any program, using just a single point moving frictionlessly 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.

This result is not surprising to me - it would be much more surprising if you *couldn't* make a universal computer this way. Universal computation seems to be a very prevalent feature of sufficiently complex systems. But still it's very nice.

• Eva Miranda and Isaac Ramos, Classical billiards can compute, https://arxiv.org/abs/2512.19156.

Show threadPhil GibbsDec 23
@johncarlosbaez I find it surprising that such a paper can be written without even a passing mention of Feynman's very well known billiard ball model of reversible computing from the Feynman Lectures on Computation.
Show threadJohn Carlos Baez
@philg - did they really not mention that, while listing so much previous work on billiard ball physics?
Dec 23 at 8:37pmWeb
Show threadPhil GibbsDec 24

@johncarlosbaez I looked a little deeper and found that Feynman was just popularising the work of Edward Fredkin and Tommaso Toffoli who are cited in this paper even though Feynman isn't. Fredkin and Toffoli showed that classical billiard ball dynamics as a basis for computation is Turing complete. This already seems to cover the title of the new paper, which leaves me wondering what their new result is.

Perhaps there method is different and more robust? The abstract does not clarify. I am sure there must be something new and worthwhile there, but I wish they would state what it is. Looking a little further again, they seem to be looking at a system with just one billiard ball and a complicated table shape, which is very different from the prior system of colliding multiple billiard balls.

Show threadEva MirandaDec 29
@philg @johncarlosbaez Just one ball in this article, several balls in the work of Fredkin and Toffoli