Penrose Tilings: Patterns That Cover a Plane but Never Repeat

Take a bathroom floor tiled with squares. Slide the whole pattern over by one tile in any direction, and it looks exactly the same — the pattern repeats. Every tiling anyone had ever made, for thousands of years, worked this way: triangles, squares, hexagons, and every combination mathematicians could dream up eventually settled into a repeating grid. In 1974, a British mathematical physicist named Roger Penrose found a way to cover an infinite flat plane with simple tiles that never repeat, no matter how far you tile in any direction — and, weirder still, the pattern has a kind of five-fold symmetry that textbooks said was mathematically forbidden. Decades later, that "impossible" pattern turned out to describe real atoms in real metal, earned someone else a Nobel Prize, and got Penrose into a legal fight with a toilet paper company.

The Concept

A tiling is periodic if you can slide it sideways — without rotating it — and have it land back on itself, like wallpaper. Every periodic tiling of a plane using regular shapes is restricted to a short list of symmetries: you can tile the plane with equilateral triangles, squares, or hexagons, which each have 3-, 4-, or 6-fold rotational symmetry. Pentagons, with their 5-fold symmetry, famously don't tile the plane on their own — gaps and overlaps always show up. Mathematicians long assumed that any tiling using shapes with 5-fold symmetry couldn't cover a plane in an orderly way at all, periodic or otherwise. This wasn't idle speculation; it's a real, provable restriction called the crystallographic restriction theorem.

Penrose broke the rule in a different way. Instead of trying to tile the plane periodically with pentagons, he built tilings that are highly ordered — every finite patch of the tiling shows up infinitely often elsewhere in it, and the whole thing displays approximate 5-fold and 10-fold symmetry — but never periodic. No amount of sliding, in any direction, ever maps the pattern back onto itself. This property is called aperiodicity, and a tiling that manages it using only a small, finite set of tile shapes is an aperiodic tiling.

Penrose wasn't the first to look for such a thing. The question of whether an aperiodic tile set could even exist was posed by the mathematician Hao Wang in 1961, who initially conjectured that any set of tiles that could tile the plane could also do so periodically. His student Robert Berger disproved that conjecture in 1966, but his proof required an unwieldy set of 20,426 distinct tile shapes. Over the following years, other mathematicians — notably Raphael Robinson, who got the count down to six tiles in 1971 — chipped away at the number. Then in 1974, Penrose found a set of just two tiles that force aperiodicity: a "kite" and a "dart," two quadrilaterals built from golden-ratio proportions that slot together only in specific ways.

The catch is that the kite and dart can't be placed edge-to-edge however you like — arbitrary placements let them tile periodically. Penrose added matching rules, essentially notches and bumps (usually drawn as colored arcs that must line up) that force every legal arrangement into the aperiodic pattern. Get the rules right, and the tiles have no choice but to spiral out into an endless, never-repeating mosaic. Penrose later found an equivalent two-tile aperiodic set made of rhombi — one "fat" rhombus with 72-degree corners and one "thin" rhombus with 36-degree corners — which is often the version people picture today, since the pattern of fat and thin diamonds fanning out in pinwheels and starbursts is visually striking.

Why It Matters

For a decade after Penrose published his tiling, it lived almost entirely in recreational mathematics — a beautiful curiosity written up by Martin Gardner in Scientific American in 1977, the kind of thing puzzle enthusiasts and mathematicians admired without expecting it to matter to the physical world. Then, in 1982, Israeli materials scientist Dan Shechtman was examining an aluminum-manganese alloy under an electron microscope and saw something that shouldn't have been possible: a diffraction pattern with sharp, clear 10-fold symmetry. Crystals, according to a century of accepted crystallography, could only have 2-, 3-, 4-, or 6-fold symmetry — the same crystallographic restriction that rules out regular pentagon tilings. Shechtman's result was so far outside the accepted framework that colleagues told him to go back and re-read the textbook; the head of his research group reportedly asked him to leave the team over it.

What Shechtman had found was a quasicrystal — a material with atoms arranged in a pattern that is highly ordered but never repeats, the atomic three-dimensional analog of a Penrose tiling. Crystallographer Alan Mackay had already shown in 1982 that a Penrose tiling produces exactly this kind of 10-fold diffraction pattern mathematically, which gave Shechtman's skeptical peers a framework to eventually accept what he'd actually observed. It took years for the physics community to come around, but by 1991 the International Union of Crystallography had formally redefined "crystal" to include aperiodic, quasi-periodic structures — a rewrite of a definition that had stood since the 1800s. In 2011, Shechtman won the Nobel Prize in Chemistry, solely, for the discovery.

Quasicrystals turned out to have genuinely useful physical properties: low friction, high hardness, poor heat conduction, and resistance to reacting with other materials. That combination made them attractive as coatings — a French company tried quasicrystal-coated non-stick frying pans in the 1990s (the line was eventually discontinued after the coating reacted with salt during cooking), and the Swedish steelmaker Sandvik still uses quasicrystalline particles to strengthen stainless steel used in razor blades and surgical and dental instruments, prized for being both very hard and non-corroding against skin.

The Details

The mechanism that forces aperiodicity is worth sitting with, because it's not a trick — it's structural. Any legal Penrose tiling can be broken down ("deflated") into a unique arrangement of larger tiles following the same shapes, scaled up by the golden ratio, φ = (1 + √5)/2 ≈ 1.618. Run the process in reverse ("inflation") and each tile splits into a fixed pattern of smaller tiles. Because φ is irrational, no finite patch of the tiling can ever line up with a shifted copy of itself — the self-similar scaling at the golden ratio is precisely what makes periodicity impossible while keeping the pattern locally predictable. It's the same irrational number that shows up in the Fibonacci sequence and in the spiral packing of sunflower seeds, here doing structural duty in geometry instead of biology.

Despite never repeating exactly, Penrose tilings have a strange consolation prize called the local isomorphism property: any finite region of any legal Penrose tiling, no matter how large, appears infinitely many times throughout the same tiling — and also appears somewhere in every other legal Penrose tiling. You could zoom into two entirely different Penrose tilings, find matching patches, and be unable to tell which infinite tiling either patch came from. There is, in a real sense, essentially only one Penrose tiling, viewed from infinitely many different starting points.

The pattern's real-world head start actually predates Penrose by five centuries. In 2007, Harvard physicist Peter Lu and Princeton's Paul Steinhardt published findings in Science showing that medieval Islamic architects had been using a set of five decorated polygons — called girih tiles — to design decagonal, quasi-periodic patterns on buildings across the Islamic world. The most striking example is the Darb-i Imam shrine in Isfahan, Iran, built in 1453, whose tilework shows girih patterns nested at two different size scales in a way mathematically equivalent to a Penrose subdivision — meaning the medieval builders had, empirically, worked out a scaling method that in principle could be extended to tile an arbitrarily large surface without the pattern ever repeating. There's no evidence the architects had (or needed) the underlying theory Penrose developed; they arrived at a working aperiodic construction through centuries of geometric craft tradition.

Penrose's tiling also had an unexpectedly petty legal afterlife. In 1997, Pentaplex Ltd — a company set up to license Penrose's patented tiling — discovered that Kimberly-Clark was embossing quilted Kleenex-brand toilet paper with a pattern matching the Penrose kite-and-dart design. Penrose, through Pentaplex, pursued the matter, reportedly quipping that it was "flattering" that a multinational company would want to use his work, but that "when it comes to the population of Great Britain being invited by a multinational to wipe their bottoms on what appears to be the work of a Knight of the Realm, then a last stand must be made." The dispute was settled out of court, and Kimberly-Clark discontinued that pattern.

And in 2023, mathematicians finally answered the question Penrose's work had left dangling: does a single tile shape exist that forces aperiodicity all on its own, with no partner tile needed? A team led by David Smith found "the hat" — and shortly after, a chiral version called "the spectre" — a single 13-sided polygon that tiles the plane only aperiodically, closing a fifty-year search for what's now called an "einstein" tile (from the German ein Stein, "one stone," not the physicist).

Takeaways

Aperiodic ≠ random. Penrose tilings are highly ordered — every finite patch recurs infinitely often — yet no shift or slide ever maps the whole pattern back onto itself.
Two simple tiles, endlessly combined, with strict matching rules, are enough to force an infinite, never-repeating pattern — a striking demonstration that complexity doesn't require complicated building blocks.
The golden ratio isn't decorative here — it's structural. The tiling's self-similar scaling behavior depends on φ being irrational.
Pure math predicted physical matter before anyone had seen it: Penrose's 1974 geometry gave physicists the framework to recognize Shechtman's 1982 quasicrystal instead of dismissing it as an error.
The same idea keeps getting rediscovered — by 15th-century Islamic tile artisans working by eye and tradition, and by mathematicians in 2023 finally finding a single tile that does the whole job alone.

Resources: - Peter Lu & Paul Steinhardt, "Decagonal and Quasi-Crystalline Tilings in Medieval Islamic Architecture," Science, 2007 - The Nobel Prize, "The Nobel Prize in Chemistry 2011" (Dan Shechtman)