Since their discovery in 1982, exotic materials known as quasicrystals have bedeviled physicists and chemists. Their atoms arrange themselves into chains of pentagons, decagons, and other shapes to form patterns that never quite repeat. How can atoms possibly “know” how to form elaborate nonrepeating arrangements without an advanced understanding of mathematics? This finding helps explain how and why quasicrystals form. The tiles form pentagonal arrangements, yet pentagons can’t fit snugly side by side to tile the plane.