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A Penrose tiling of a plane (note this not the same as a Penrose diagram) is a tiling that is "non-periodic", without "translational symmetry", meaning that you can't simply slide the pattern one direction or another and have the same pattern. This implies that if you are standing on the plane and can sense its structure you can figure out where you are (to an arbitrary accuracy) by studying the local structure.

I don't know if it's mathematically possible, but suppose one had a "Penrose space", one which exhibited the characteristics of a Penrose tiled plane, only 3-dimensional. Now suppose this Penrose space is actually the Universe.

What would be the implications if one could somehow sense the structure of this space and determine ones position within the Universe? (Clearly, this would give you something other than an inertial frame of reference -- though existing laws of physics still exist so inertial reference is still possible.)

Is this obviously ridiculous and impossible on the face of it, or is there some potential that such a space is mathematically possible and such a structure could actually be present in the Universe?

Hot Licks
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  • There are some studies of cellular automata on penrose tiling lattices, which might have something of the flavor of what you're asking for. If I get a chance I'll see if I can dig up some papers. – N. Virgo Jan 15 '16 at 06:51
  • "This implies that if you are standing on the plane and can sense its structure you can figure out where you are (to an arbitrary accuracy) by studying the local structure." can you give a link that substantiates this statement? In any case the answer is it will have the problems of all discrete topologies suggested as models of the underlying structure . – anna v Jan 15 '16 at 07:02
  • What do you mean, "but suppose one had a "Penrose space", one which exhibited the characteristics of a Penrose tiled plane, only 3-dimensional"? Are you asking what happens when you don't have translational symmetry? (momentum isn't conserved then, for one) – ACuriousMind Jan 15 '16 at 13:56
  • @ACuriousMind - I mean that the "background" is such a space -- an "ether", if using that term is not too prejudicial. At the gross level the laws of physics still apply, but in addition one can determine one's position in the space, without, eg, the use of a navigational gyro. (Whether the ability to determine this information would only be possible for intelligent beings or the information would be conceptually available to the quantum mechanical equations controlling elementary particles is up in the air. If subatomic particles had access I suspect it would have major ramifications.) – Hot Licks Jan 15 '16 at 14:08
  • I do not believe that the statement I quoted in my previous comment is true. Please prove or give a link that proves it. – anna v Jan 15 '16 at 14:17
  • @annav - It's based on my understanding of Penrose tiling. The tiling appears regular at a very "local" level, but does not repeat, and so by examining the structure at wider and wider radii one can determine where in the larger structure one is. One would have to examine the entire "universe" to determine position absolutely, but a smaller radius examination would presumably allow one to determine relative position within an arbitrarily large scope. But admittedly my understanding of Penrose tiles is based mostly on what I read in Scientific American 40-odd years ago, so I may be mistaken. – Hot Licks Jan 15 '16 at 14:22
  • Do you mean there is a unique geometrical pattern? Only then one could make out distances and coordinates from examining local geometry.,similar to a map – anna v Jan 15 '16 at 14:37
  • @annav - Yes, as I understand it, it's a unique geometric pattern. – Hot Licks Jan 15 '16 at 18:14
  • well, it is then inconsistent with quantum mechanics, which has an inherent uncertainty so no unique pattern – anna v Jan 15 '16 at 18:43
  • @annav - The uncertainty is within a small delta. The pattern could have a granularity larger than ol' Werner would suggest. – Hot Licks Jan 16 '16 at 13:46
  • you would not have a Big Bang model, as in the inflation period uncertainties are enormous. 2) think of chaos: small changes have big effects, and introducing a Heisenberg uncertainty certainly would act to turn the system to chaotic
    • – anna v Jan 16 '16 at 15:29
  • @annav - If inflation occurs on a fixed Penrose "background" then no additional uncertainties would be introduced. – Hot Licks Jan 16 '16 at 18:33
  • They would, due to the probabilistic nature of quantum mechanics. You cannot have quantum mechanics and this . You are just complicating the problem of quantization of gravity, except that your tiles and qauntum mechanics are not compatible, imo. – anna v Jan 16 '16 at 18:51