Aperiodic Tiling

While I was designing our Christmas cards, I thought about doing something math-related, and the obvious choice was to use some variant of the Penrose tiling as the background. I took the card in a different direction, but the process gave me an opportunity to reflect on life with a new baby.

An aperiodic tiling, among which the Penrose tiling is the most famous, is a way of covering space with simple geometric shapes, with an interesting feature–you could go on forever in every direction, and there would never be a repetitive pattern. No region would ever look exactly like another (as long as you’re looking at regions that contain more than a few tiles). There are ways of doing this that use just two different tile shapes, which astonishes me. That’s not to say that different areas don’t look similar–that’s unavoidable–but they’re never exact copies. I see a similarity with an infant’s life. Although his repertoire has been growing, it used to be that Roman could only do about four things: eat, sleep, cry, and (rarely) be calmly awake. Every day is a mix of those activities that is sometimes frustratingly repetitive, but never exactly the same. We have to keep on our toes and adapt as best we can, which isn’t easy but is certainly preferable to the monotony of doing exactly the same thing over and over again.

There is a point to all of this rambling. How can we be sure that a tiling really is aperiodic? What if there’s some spot out in the distance, out close to infinity where nobody has charted yet, where everything looks just the same as it does right here? One way of proving that can’t happen is to show that a tiling has hierarchical structure–certain motifs that reappear at ever-increasing scales. If the pattern of tiles ever repeated, then there would be a largest motif size that it could contain, so if the maximum size of the motif is unlimited the tiles can never repeat. It’s similar to an argument that proves that there are infinitely many prime numbers. Down in the trenches, holding Roman while he’s going through another bout of inconsolable screaming, it’s hard to understand what’s going on. Every day we lay another few tiles that help to expose the larger structure, and hope that the actions we take will effect positive influence on the larger structures of personality and character.

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  1. Pingback: Blog Resolutions | The Number Field

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