Penrose Tiling
Infinite complexity, zero repetition.
What you're looking at
Two shapes, no repeats
The pattern above is built from just two tile shapes — "thick" and "thin" Robinson triangles. Together they cover the plane completely with no gaps and no overlaps. But unlike square tiles or hexagons, the pattern never repeats. No matter how far you extend it, you'll never find a translational period. It's ordered but not periodic — a strange middle ground between crystal and chaos.
The tiles
Both tiles are isoceles triangles related to the golden ratio φ = (1 + √5) / 2. The thick triangle has angles 36°-72°-72° and the thin triangle has angles 108°-36°-36°. The magic is in how they subdivide: split each at a golden-ratio point along one edge, and every thick triangle becomes one thick plus one thin, and every thin triangle does the same. This recursive subdivision is what produces the tiling — each level adds roughly φ² ≈ 2.618 times more tiles.
Five-fold symmetry
Look at the Sun starting pattern — ten triangles radiating from a center, with five-fold rotational symmetry. You'll see pentagons, decagons, and five-pointed stars everywhere in the pattern. This five-fold symmetry is exactly what makes it non-periodic. Crystallographers had long proved that periodic tilings can only have 2-, 3-, 4-, or 6-fold symmetry. Five is forbidden. Penrose found a way to have five-fold symmetry anyway, by giving up periodicity.
Quasicrystals
In 1982 Dan Shechtman shot an electron beam through a rapidly cooled aluminum-manganese alloy and saw a diffraction pattern with ten-fold symmetry — sharp Bragg peaks arranged in a pattern that was supposed to be impossible. The crystallography establishment resisted hard. Linus Pauling reportedly said "there is no such thing as quasicrystals, only quasi-scientists." But Shechtman was right. The atoms were arranged in a 3D analog of Penrose tiling — ordered enough to diffract sharply, but non-periodic. He got the Nobel Prize in 2011.
Self-similarity
Every time you click Subdivide, each triangle splits into smaller copies of the same two shapes. That means the pattern is self-similar — zoom in and you see the same structure at every scale. This is also what makes it mathematically interesting: the ratio of thick to thin triangles converges to φ as you subdivide. The golden ratio isn't just in the geometry of the tiles — it's in their relative abundance too.