It Took 20,426 Tiles, Then Six, Then Two

Hao Wang, a logician at IBM in 1961, noticed that tiling problems could be reframed as logic problems. He proposed a question: is there a finite set of tiles that can cover the infinite plane, but only in patterns that never repeat? Wang's own conjecture was no — any tiling that worked at all could be made periodic.

His graduate student Robert Berger disproved him in 1964. Berger's PhD thesis at Harvard turned the Domino Problem into a halting-problem reduction; in proving the tiling question was undecidable, he constructed an aperiodic set as a side effect. The set he produced had 20,426 tiles. He later cut it to 104. Both numbers told you the answer existed and gave you no reason to look at the tiles.

Roger Penrose came at it sideways. In 1974, working with five-pointed symmetry that no periodic tiling can have, he found a set of six tiles that forced aperiodicity. He kept reducing. By the late 1970s he had two: a fat rhombus and a thin one, or equivalently a "kite" and a "dart," with edge-matching rules that made periodic patterns impossible. Two shapes. Pages of arrangements. None of them ever quite repeating, no matter how far you walked.

Then in 1982 Dan Shechtman, a materials scientist at the U.S. National Bureau of Standards, ran electron diffraction on a rapidly cooled aluminum-manganese alloy and saw a tenfold-symmetric pattern that was supposed to be impossible. He had found a quasicrystal: real atoms arranged on a Penrose-style lattice. The discovery was so unwelcome that Linus Pauling spent years saying it could not be true. It won Shechtman the 2011 Nobel Prize in Chemistry. Penrose's two tiles had been a preview of the chemistry.