Quasi-Crystalline Pattern

Darb-i Imam shrine spandrel, Isfahan, Iran (1453). The tile pattern contains a near-perfect quasicrystalline structure — built five centuries before Western science could describe it. Wikimedia Commons

What Is a Quasicrystal?

A Quasicrystal is a structurally ordered pattern that never repeats at regular intervals — ordered without being periodic. In classical crystallography, dating from Von Laue's first x-ray diffraction experiment in 1912, the only rotational symmetries permitted in a repeating lattice are 1-, 2-, 3-, 4-, and 6-fold. Fivefold symmetry — the symmetry of a pentagon or ten-pointed star — was not merely undiscovered; it was mathematically forbidden in any real material. To fill a plane with fivefold tiles you are forced to leave gaps or create overlaps. Neither is permitted in a true tiling. The quasicrystal resolves this by abandoning periodicity while preserving long-range order: a pattern that looks regular, follows strict geometric rules, but never exactly repeats.

Dan Shechtman and the Forbidden Crystal

On April 8, 1982, Israeli materials scientist Dan Shechtman was alone in an electron microscope laboratory at the National Bureau of Standards in Washington, examining a newly created aluminum-manganese alloy. The diffraction pattern on his screen showed ten sharp spots arranged with perfect tenfold symmetry. He counted them once. He counted them again. It was impossible — and yet there it was.

He spent the rest of the day running experiments, searching for an alternate explanation — crystal twins, artifacts, instrument error. He found none. His peers were not sympathetic. The head of his laboratory placed a crystallography textbook on his desk and suggested he read it. When Shechtman persisted, he was asked to leave the research group. Nobel laureate Linus Pauling — two-time Nobel recipient and one of the most decorated scientists in American history — publicly dismissed the findings from stages across the country, and continued to do so until his death. Two years after his initial observation, Shechtman published in Physical Review Letters, and the community of believers slowly grew. The definition of a crystal was eventually broadened to include quasiperiodic materials. In 2011, Shechtman received the Nobel Prize in Chemistry — awarded solely for this one discovery.

Professor Dan Shechtman describes the morning of his discovery. Nobel Prize in Chemistry, 2011.

Roger Penrose and Aperiodic Tiling

A Penrose tiling (P3 rhombus variant) — two rhombus shapes, fivefold symmetry, never repeating. By Inductiveload, Wikimedia Commons (CC BY-SA 3.0).

In the 1970s, Oxford mathematician Roger Penrose described a class of aperiodic tilings — flat patterns assembled from two rhombus shapes that fill the plane without gaps or overlaps, yet never repeat exactly. They exhibit fivefold rotational symmetry and are self-similar at multiple scales: zoom out and the large-scale structure mirrors the small. When Shechtman's quasicrystals emerged, physicists recognized immediately that Penrose's geometry provided exactly the structural model needed to explain them. The Nobel committee cited Penrose tilings directly in its 2011 award citation. Western mathematics arrived at quasicrystalline order in the 1970s. Medieval Islamic craftsmen had been building it since at least the 13th century.

The spiral seed arrangement of a sunflower head follows Fibonacci proportions — the same mathematical ratios that underlie Penrose tilings and quasicrystals. Fivefold order appears throughout nature. Wikimedia Commons.

Girih Tiles: The Hidden Mathematical System

The conventional assumption had always been that the intricate star-and-polygon patterns covering medieval Islamic mosques, shrines, and madrasas — from Turkey through Iran and across to India — were drawn directly with a straightedge and compass, line by laborious line. Beautiful, certainly. But essentially craft rather than theory.

In 2007, Harvard physicist Peter J. Lu and Princeton physicist Paul J. Steinhardt published a paper in Science that changed that picture entirely. Studying thousands of photographs of medieval Islamic buildings alongside 15th-century Persian architectural scrolls — construction manuals of the era — they identified a set of five polygon shapes consistently used as underlying structural units: a decagon, pentagon, diamond, bow tie, and hexagon. All sides equal. All angles multiples of 36°. These are the girih tiles — girih being Persian for "knot."

The five girih tiles: decagon, pentagon, hexagon, bow tie, and rhombus. All sides equal; all angles multiples of 36°. The visible strapwork lines are inscribed on the tile surfaces — the tile boundaries themselves remain hidden beneath the finished pattern. Wikimedia Commons (CC BY-SA 3.0).

The key insight is that the finished surface patterns — the visible strapwork — are not the edges of the tiles themselves, but decorative lines inscribed on each tile. The girih tiles are hidden scaffolding. By around 1200 CE, Islamic artisans had made a conceptual leap: rather than constructing each pattern stroke by stroke from scratch, they were tessellating pre-designed polygon units — a system more efficient, more precise, and capable of generating far greater complexity than compass-and-straightedge drafting alone.

Darb-i Imam, Isfahan, 1453: Five Centuries Ahead

Spandrel tile detail, Darb-i Imam shrine, Isfahan, Iran (1453). Of approximately 3,700 tiles mapped by Lu and Steinhardt, only 11 deviated from a perfect Penrose pattern — each correctable by a simple rotation. Photo: Reuters / Peter J. Lu.

The most remarkable example Lu and Steinhardt documented was at the Darb-i Imam shrine in Isfahan, Iran, completed in 1453. Among approximately 3,700 tiles mapped on its portal spandrel, only 11 showed any deviation from a theoretically perfect quasicrystalline Penrose pattern — each correctable by a simple tile rotation, consistent with construction error or centuries of repair. The pattern is quasicrystalline by every geometric measure: it never repeats, it exhibits tenfold symmetry, and it is self-similar across two distinct scales — large girih tiles subdivided into the same five shapes at a smaller size. That self-similarity across scales is not decorative coincidence. It is the defining structural property of a Penrose tiling, and its presence in a 15th-century Iranian shrine places the conceptual discovery of quasicrystalline geometry roughly 500 years before Penrose described it and 530 years before Shechtman observed it at the atomic scale.

The same underlying logic appears across the medieval Islamic world: the Al-Mustansiriyya Madrasa in Baghdad (1227–34), the Gunbad-i Kabud tower in Maragha, Iran (1197), the Timurid Tuman Aqa Mausoleum in Samarkand, Uzbekistan (1405), the Friday Mosque of Isfahan (late 15th century), and the Hall of the Ambassadors at the Alhambra in Granada, Spain (1354). This was not a regional anomaly. It was an architectural language spoken across three centuries and thousands of miles.

Order Without Repetition

Western design has long privileged the grid — repeating, modular, predictable. The quasicrystal offers a different kind of order: coherent without being mechanical, complex without being chaotic, and infinite without being monotonous. It is the geometry of a system that follows strict rules but never exhausts its configurations. The pattern on the Darb-i Imam portal reads as complete and resolved at every scale of observation — without ever being the same thing twice.

Medieval Islamic craftsmen built this geometry into stone and tile with a precision Western science would not theoretically ground for 500 years — possibly as an expression of ideas about the inexhaustible nature of divine creation, possibly as pure geometric exploration, most likely both. Today, quasicrystalline materials appear in surgical-grade stainless steel, electric shaver components, and high-performance non-stick coatings. The geometry on that 15th-century Iranian shrine and the microstructure of a modern surgical blade share the same underlying mathematical object — expressed at radically different scales and separated by half a millennium of rediscovery.

Referenced Links:
Reuters: Islamic Tile Patterns and Quasicrystals (2007)
Lu & Steinhardt — "Decagonal and Quasi-Crystalline Tilings in Medieval Islamic Architecture," Science (2007)
Harvard Gazette: Medieval Islamic Architecture Presages 20th Century Mathematics
Peter J. Lu: Quasicrystalline Medieval Islamic Architectural Tilings
Wikipedia: Girih Tiles
Wikipedia: Quasicrystal
Wikipedia: Roger Penrose