Forbidden symmetry. A crystal can only repeat itself with 2-, 3-, 4- or 6-fold symmetry. In 1982 Dan Shechtman found an alloy with 10-fold symmetry: a quasicrystal, perfectly ordered yet never repeating. Here 5 waves crossing at equal angles build that impossible order, and de Bruijn’s method turns it into a tiling of just a few kinds of rhombus. As you move, tiles flip one by one, the “phason” jumps that real quasicrystals make. Move to shift the waves · click for another symmetry.
Forbidden symmetry
Order that never repeats. A floor of tiles that could go on for ever without a single pattern coming back.
How it works
Waves that cross. Five sets of straight waves, turned by 36° from one another, add up to a pattern with ten-fold symmetry. No ordinary crystal can have it: you cannot tile a floor with pentagons. Yet in 1982 Dan Shechtman found an alloy whose atoms did exactly this. He was told it was impossible, and was awarded the Nobel Prize in Chemistry in 2011.
From waves to tiles. Nicolaas de Bruijn showed in 1981 how to turn those waves into a tiling. Each family of waves becomes a family of lines, and every crossing of two lines becomes one rhombus. With five families you get Roger Penrose’s famous tiling of fat and thin rhombi; with four, the eight-fold Ammann–Beenker tiling; with six, a twelve-fold one.
Phasons. Shifting the waves does not slide the tiling: it makes individual tiles flip, one at a time, while the pattern stays perfect. Real quasicrystals make these same jumps.