Two tiles, one golden rule
Penrose's tiling needs only two diamonds: a fat one (angles 72° and 108°) and a thin one (36° and 144°). Their sides are all the same length, and their diagonals stand in the golden ratio, φ ≈ 1.618 — the number that quietly governs the whole construction.
Ordinary diamonds would just repeat. These two, forced to obey a matching rule along their edges, cannot.
Grow it by splitting
You don't lay a Penrose tiling by hand — you grow it. Every tile is a rule that says how to cut itself into smaller tiles. Apply the cut everywhere, again and again, and an ever-finer, perfectly legal tiling unfolds. This process — inflation — is what guarantees the pattern can never settle into a repeat.
Drag the slider to deflate, step by step.
Five-fold, and forbidden
Look at the centre: a perfect five-fold star. Ordinary crystals are flatly forbidden this symmetry — you cannot tile a plane periodically with five-fold rotation. Penrose tilings get away with it precisely because they never repeat. When real materials were found doing the same in 1982, it won a Nobel Prize and a new word: quasicrystal.
Toggle the axes of symmetry through the central sun.
Patterns inside patterns
Inflation leaves a fingerprint: the tiling is self-similar. Bundle the small tiles back up and they reassemble into the very same two shapes, only larger — and those into larger still, forever. Colour each tile by the big "super-tile" it belongs to and the hidden hierarchy surfaces.
Step the grouping up a level and watch the regions merge.
The Penrose field
The whole idea, turning gently. Two diamonds, one golden rule, infinite non-repeating order. Push the depth, recolour it, and look as long as you like — somewhere out past the edge, the pattern is still going, and still never repeating.