Universal Turing Machine In Penrose Tile Cellular Automata

New submitter submeta writes "Katsunobu Imai at Hiroshima University has figured out a way to construct a universal Turing machine using cellular automata in a Penrose tile universe. 'Tiles in the first state act as wires that transmit signals between the logic gates, with the signal itself consisting of either a 'front' or 'back' state. Four other states manage the redirecting of the signal within the logic gates, while the final state is simply an unused background to keep the various states separate.' He was not aware of the recent development of the Penrose glider, so he developed this alternative approach."

Universal Turing Machines

[2.7182]

There is a reason there are 50 different definitions of computable function - they are not hard to come by. As a professional mathematician/theoretical computer scientist I find it totally unsurprising to find one in the Penrose tiles. If it is useful for something, that's different. But you build almost any sufficiently rich mathematical structure and you can interpret a subset of them as Turing machines.

Are there Penrose buckyballs?

[G3ckoG33k]

Are there Penrose "buckyballs", i.e. a version of the buckyball using the Penrose tiling?

I am not sure if they exist mathematically and have never seen them discussed anywhere.

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[Hentes]

Penrose tilings are flat, hence they can't cover a ball. It's impossible [wikipedia.org].

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[NoNonAlphaCharsHere]

Soccer balls are not regular polyhedrons. Nor are they spheres.

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[Mikkeles]

True, but it be interesting to know if a sphere could be "triangulated" with Penrose tiles.

Re:Are there Penrose buckyballs?

[Anonymous Coward]

No, you can't make a sphere with Penrose tiling. As has already been mentioned, a flat tile can't be used to cover a sphere. But more importantly, there isn't a generalization that will work either. The thing that makes Penrose tiling interesting is that it is aperiodic. No aperiodic pattern can work on a sphere since you necessarily are periodic when you make one complete revolution around any greater circle on a sphere.

Permutation City

[Shad0w99]

Somehow Greg Egan's book "Permutation City" came to my mind when reading this. With his Autoverse representation on cellular automats.

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[blackpaw]

Or "diaspora" where I belive he had naturally occurring penrose tiles in an alien biology performing turing calculations

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[mrsurb]

His short story "Wang's Carpets" - which then became a chapter in "Diaspora"

8 states?

[Hentes]

That's not very impressive, especially since he basically just copied the four-state WireWorld rule.

There you go

[dargaud]

Previous /. story [slashdot.org]: "Before we can totally discount the theory that space-time is comprised of Planck-scale pixels, [...]". There you go, you can have Penrose-tiled planck pixels and still move in straight lines. Where do I pick my Nobel ?

Thanks for posting this story!

[EnergyScholar]

I wish to offer a general apology for the terrible quality of comments on this story. Obviously, most readers failed to even understand what the story was about. I thought the best comment was the snarky 'Is this NKS?' at the end, as this story obviously does tie into NKS. Anyway, thanks, submeta, for posting a fine old skool slashdot story. Let's hope our readership is less ignorant and juvenile next time around.