Collage of AFM images of algorithmic crystals. Also known as the magic carpet.
Credit and copyright: Damien Woods

Fifth Bit

Computer Engineer: You know, someone wrote a book about that once, what's his name, you know, "The algorithmic beauty of plants."
Philosopher: I know the book. It's a wonderful thing. He shows that very simple algorithms can describe how plants grow, and branch, and flower. The computer graphics can be very realistic -- especially when the algorithms sometimes make random choices.
Farmer: I am well aware that every flower, every weed, every stalk of corn is different -- yet they share a certain... je ne sais pas... spirit, soul, character. And it's the same with these little self-assembled patterns the DNA computer grew. (sneezes) Someone should write a book, "The algorithmic beauty of crystals."
Mathematician: Someone did, almost. It's called "Tilings and patterns." It has all the most beautiful geometries for how tiles can fit together. It has a whole chapter on Wang tiles, which were invented in the 1960s to establish a connection between mathematical logic, automatic theorem proving, and pattern formation. But it's really about algorithms, because Wang's deepest discovery was that for some tile sets, the only way that they can fit together (using all tile types at least once) exactly mimics the execution of a Turing machine -- the simplest, yet universal, form of algorithm.
Poet: What's that got to do with crystals?
Mathematician: You remember the sugar crystals with the P21 symmetry group? Just like crystals, geometrical tiles can fit together to make periodic patterns, and those patterns can be characterized by their symmetries. There are exactly 17 different ways to make periodic arrangements in two dimensions, and 230 different ways in three dimensions. So, tilings -- of polygons in 2D and of polyhedra in 3D -- are one way of studying crystalline order.
Philosopher: But if tilings can describe geometrical order that goes beyond periodic order -- I mean, Turing machine algorithms can have arbitrarily complex behaviors -- then you could say that it's a more general way to talk about how matter can be organized. I'm thinking about the aperiodic Penrose tilings and how years later quasicrystals were discovered that had similar "impossible" 5-fold symmetries. So Wang's Turing tilings... if molecules are arranged with that sort of... "symmetry"... what would you call it?
Curator: I'm told the builders of this DNA computer were thinking along the same lines, actually.

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