The Aperiodiad

As you probably know, rectilinear and polar coordinates are the ones normally used when doing any sort of engineering or scientific calculations. However, given the finity of computers, these coordinates, while conceptually continuous, are necessarily discretized, often by utilizing so-called floating-point or fixed-point schemes. Coordinate axes or angular measures based upon these numbers utilize a finite set of rational numbers, thus defining a finite universe of possible locations. If some coordinate of your particle or wave should “actually” between two of the available numbers, it will be assigned one or the other.

These computerized coordinate systems exhibit periodicities. A simple picture is a checkerboard or Rubik’s cube, with rectilinear shapes tiling an area or a volume, extended spatially to the limit of the numbering scheme. With higher dimensions, we wave our hands and say “it’s like going from a square to a cube, just moreso and with special sauce”. Given all this, some of the squares or cubes (or parallelo-whatsises: right angles aren’t required) that occur in the imagination can’t actually be denoted in floating or fixed point, especially when you get to very small or very large numbers. One can of course sometimes use tricks such as mentally shifting calculations so that the tractable and dense populations of available coordinates is moved from around the origin to the regions of interest, but this can be complex and might not always work. Even with these tricks, the underlying periodicity of the finite numbering scheme being imposed on an infinite substratum suggests that considerable thought be given to anticipating and avoiding the kinds of artifacts that might ensue from using these ubiquitous approximations.

Now, imagine other tilings of the plane, such as Penrose’s aperiodic rhombs. Each tile is reminiscent of the squares that build up the universe of a traditional xy plot, and again most numbers can only be approximated in the usual computer numbering schemes. Unfortunately for Penrose tilings, there is no easy trick, as with periodically distributed tiles, to locating tiles in faraway or highly magnified regions. One has to mathematically pre-assemble the tiling of interest to determine the available coordinates, and as the tiling is aperiodic, the metaphor of tracing your patterns onto a transparent sheet and shifting it cannot possibly work. There are infinite numbers of different ways to assemble such tilings. Furthermore, given any finite subset of a Penrose tiling, there is, nearby, an exact copy of that finite subset that is part of the ultimate, unique, infinite tiling. The upshot is that even once you have determined a (finite) tiling (no matter how capacious or doughty your computer), you can’t even tell which of the infinite number of different tilings it belongs to, because it belongs to all of them an infinite number of times.

Let us now don our speculation-spectacles and loosen up our arm-waving muscles. Extend the above to n dimensions, such as 11 for string theory or 4 for general relativity, or what have you, and assume suitable aperiodic tilings exist in these dimensions. Further assume that such coordinate systems provide artifact-avoiding and performance advantages for physics calculations. Any time a re-casting of the activities of daily physics improves our abilities, we assume that our models are closer to reality. If aperiodic coordinates bring us closer to reality, but we can’t even distinguish between different specific instantiations of said coordinates, then distinguishing between determinism and indeterminism becomes questionable, potentially rendering the very question “not even moot”.