The Strange Mathematics of Penrose Tilings: Aperiodic Order and the Crystals That Shouldn't Exist

The mathematics of tiling looks like it ought to be settled. Squares tile the plane. Triangles tile the plane. Hexagons tile the plane. Mathematicians worked out which shapes can tile the plane periodically — repeating their pattern in regular intervals — sometime in the 19th century, and the topic seemed like a closed chapter of geometry. The interesting question, which turned out to be much harder than anyone expected, was whether shapes can tile the plane aperiodically — covering the plane without gaps or overlaps but never repeating the same pattern.

The question opened in 1961 with a paper by the logician Hao Wang, who proved that if a finite set of shapes can tile the plane, it can tile the plane periodically. The conjecture seemed plausible — there was no obvious counterexample — but the proof depended on a hidden assumption about the relationship between the tiling problem and a class of decidable logical questions. Wang's student Robert Berger destroyed the conjecture in 1966 by constructing a set of 20,426 tiles that could only tile the plane aperiodically. The set was reduced to 104 tiles, then to 40, then to 6 by Raphael Robinson in 1971. Roger Penrose, working independently in 1974, found a set of just two tiles that did the same thing. The Penrose tilings are the most famous example, but the broader phenomenon — aperiodic order — is mathematically much stranger than two shapes that almost-but-never-quite repeat.

What the two Penrose tiles do

The simplest Penrose tiling uses two rhombi: a thin rhombus with 36° and 144° angles, and a thick rhombus with 72° and 108° angles. The angles are all multiples of 36°, which is one-tenth of a full circle and connected to the regular pentagon. The tiles must be assembled with matching rules — patterns on the edges or markings on the tiles that constrain how adjacent tiles can fit together. With the matching rules, the only legal tilings of the plane are aperiodic: they cover the plane perfectly but contain no fundamental repeating unit.

The properties of these tilings are remarkable. Local patches of any size repeat infinitely many times throughout the tiling, but the patches occur at unpredictable intervals — there is no shift that maps the entire tiling to itself except the trivial identity. The ratio of thick to thin rhombi in any sufficiently large region approaches the golden ratio φ ≈ 1.618. The tilings have five-fold rotational symmetry on average — any rotation by 72° produces a tiling that is locally identical to the original at every patch, but globally distinct.

This last property is the one that caused trouble in physics. The classical theorem of crystallography, proved in the 1840s, states that a periodic crystal in three dimensions can have only 2-fold, 3-fold, 4-fold, or 6-fold rotational symmetry. Five-fold symmetry is forbidden because it cannot tile the plane periodically. The Penrose tilings show that aperiodic structures can have five-fold symmetry. In 1974 this was a beautiful piece of mathematics with no obvious physical implication.

The 1982 experiment that should not have worked

On April 8, 1982, the Israeli materials scientist Dan Shechtman, working at NIST in Maryland on rapidly cooled aluminum-manganese alloys, observed a diffraction pattern that he could not explain. The pattern showed sharp peaks in a ten-fold symmetric arrangement — the unmistakable signature of a crystal with five-fold rotational symmetry, which the 1840s theorem said could not exist. He spent two years getting the result published. The Journal of Applied Physics rejected it. The chemistry community largely rejected it. Linus Pauling, the two-time Nobel laureate and most prominent figure in 20th-century structural chemistry, denounced Shechtman's results in increasingly public and increasingly personal terms, declaring famously: "There is no such thing as quasicrystals, only quasi-scientists."

The dispute was finally settled by the accumulation of additional examples — by 1990, dozens of materials had been confirmed to show the same impossible diffraction patterns — and by the recognition that Shechtman's "impossible" crystals were physical realizations of three-dimensional analogs of Penrose tilings. The objects were not periodic and therefore did not violate the crystallographic theorem; they were a new kind of order that the theorem had not anticipated. Shechtman won the Nobel Prize in Chemistry in 2011, twenty-nine years after the discovery and sixteen years after Pauling's death.

The deeper structure: cut-and-project

The mathematical relationship between Penrose tilings and quasicrystals turned out to be precise. Both can be described as projections of higher-dimensional periodic structures onto a lower-dimensional space at an irrational angle. A Penrose tiling can be constructed by taking a five-dimensional cubic lattice, slicing it with a two-dimensional plane oriented at angles related to the golden ratio, and projecting the lattice points near the plane down onto it. The construction is called "cut and project," and it generalizes to higher dimensions and other irrational angles.

The cut-and-project framework explains why Penrose tilings have the properties they do. The five-fold symmetry comes from the symmetry of the higher-dimensional lattice. The aperiodicity comes from the irrational projection angle. The local repetition comes from the periodicity of the higher-dimensional structure. The framework also generalizes to physical quasicrystals: a three-dimensional quasicrystal is the projection of a six-dimensional cubic lattice onto three-dimensional space at an angle related to a different irrational number.

The current status

Quasicrystals are now an established field of materials science. They occur naturally in some meteorites — the first natural quasicrystal, icosahedrite, was identified in a Russian meteorite in 2009 — and have been synthesized in a wide range of metallic alloys. Their physical properties (low thermal conductivity, low surface friction, hardness) make them useful for surface coatings on cookware and aerospace components.

The mathematics has continued to develop. The minimum number of tiles required for an aperiodic tiling has been pushed downward over the decades. Penrose's two-tile construction held the record from 1974 until 2023, when David Smith, an amateur mathematician from Yorkshire, discovered a single shape (the "hat" tile) that tiles the plane only aperiodically. Smith's discovery, refined and proved by collaborators including Craig Kaplan and Joseph Myers, settled the open question of whether a single aperiodic tile exists. The answer, after fifty years of searching, is yes, and the tile is shockingly simple — a 13-sided polygon with no special properties other than the fact that any tiling using it must be aperiodic.

The broader pattern

The Penrose-quasicrystal story is one of those cases where pure mathematics turned out to describe a physical phenomenon nobody had asked it to describe. Penrose was working on a question of geometric possibility — can two shapes tile the plane only aperiodically — that had no apparent connection to materials science. The answer turned out to be the structural theory of a class of materials that already existed and were being mistaken for crystallographic anomalies. The mathematics arrived first and was waiting when the physics caught up.

The pattern recurs in the history of science. Riemannian geometry was developed in the 19th century as a study of curved spaces with no physical motivation, and it became the mathematical language of general relativity sixty years later. Group representation theory was developed for purely algebraic reasons in the 1890s and became the language of quantum mechanics in the 1920s. The persistent observation that mathematicians keep stumbling onto the structural theories of phenomena that physicists have not yet discovered is what Eugene Wigner called "the unreasonable effectiveness of mathematics" — and the Penrose-quasicrystal story is one of the cleanest modern examples.

The aesthetic legacy

The Penrose tilings have also escaped into art and architecture. The tiling pattern adorns the floor of the Wadham College quadrangle in Oxford, the lobby of the Mitchell Institute at Texas A&M, and the entrance hall of Penrose's own institution at Oxford's Mathematical Institute. Islamic architecture from the 13th-15th centuries — particularly the Darb-i Imam shrine in Isfahan from 1453 — contains patterns that mathematicians Peter Lu and Paul Steinhardt argued in 2007 are mathematically equivalent to a quasiperiodic tiling structure that anticipates Penrose's discovery by five centuries. The Lu-Steinhardt interpretation is contested by some historians of Islamic art, but the geometric similarity is striking, and the broader question of whether the Islamic mathematicians had a theoretical understanding of aperiodicity or arrived at the patterns through artistic search is genuinely open.

The five-fold symmetry that crystallography forbade and quasicrystals revealed has been visible in human-made patterns for at least a thousand years. The mathematics required to describe it took until the 1970s. The physics required to recognize it took until the 1980s. The honest summary is that the world has been quietly full of aperiodic order all along, and we only recently developed the apparatus to notice it.