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Jules Mikhael and his colleagues didn’t set out to make a material with a structure that had never been seen before, much less one that combines order and irregularity in a whole new way, one that Archimedes hinted at 2,000 years ago, one bound together by the Fibonacci sequence. They just wanted to understand a quasicrystal.
Even that wasn’t such a modest goal, because quasicrystals are pretty odd critters. Slice one in half, and there is a sort of mosaic with repeating shapes like tiles, much like a crystal. But here’s the bizarre part: Spin the resulting mosaic a fifth of a turn and often its tiles will line up exactly as they were before you spun it.
But that kind of five-fold symmetry is “forbidden,” because mathematicians have shown that no repeating flat pattern has it. That’s why you’ve never seen a bathroom tiled with regular pentagons—it’d be impossible to cover the whole surface with no gaps.