In 1974, British mathematical physicist Sir Roger Penrose discovered a set of just two shapes that tile an infinite flat plane without ever repeating — a pattern now known as Penrose tiling. Until then, mathematicians largely assumed that any set of shapes covering the plane with no gaps or overlaps must eventually settle into a repeating pattern, the way ordinary floor tiles do. Penrose's tiles proved that assumption wrong, and the discovery earned him a share of a Wolf Prize and later a knighthood, decades before he won the 2020 Nobel Prize in Physics for unrelated work on black holes.
The best-known version uses two rhombi, or equivalently a kite and a dart shape, both built from the golden ratio — the same proportion found in sunflower seed spirals and nautilus shells. Despite never repeating exactly, the pattern is far from random: every finite region of the tiling, however large, reappears infinitely often elsewhere in it, and the whole tiling shows a striking five-fold and ten-fold rotational symmetry that ordinary repeating (periodic) tilings can never achieve — regular tilings are limited to two, three, four or six-fold symmetry.
The discovery turned out to have a striking real-world counterpart: in 1982, materials scientist Dan Shechtman found a metallic alloy whose atoms were arranged in a Penrose-like pattern, an arrangement long thought impossible in nature. The finding was so contrary to accepted crystallography that it was initially ridiculed, before it earned Shechtman the 2011 Nobel Prize in Chemistry and gave the world a new class of materials: quasicrystals.