Look up at the ceiling of almost any historic mosque from Morocco to Central Asia and you will see stars: ten-pointed, twelve-pointed, sixteen-pointed, wrapped in lines that weave over and under each other and seem to run on forever. For centuries, visitors assumed each of these patterns was drawn line by line with a compass and a straightedge. In 2007, a graduate student in physics in Cambridge, Massachusetts, proposed a simpler and more surprising answer, and it tied a centuries-old craft to one of the strangest discoveries in modern chemistry.
This is the story of that idea, what scholars still argue about, what geometry meant to the Muslim artisans who used it, and where you can stand in front of one of these patterns this week without leaving Massachusetts.
Three kinds of decoration, and a love of order
The Metropolitan Museum of Art describes geometric patterns as one of three main types of non-figural decoration in Islamic art, alongside calligraphy and vegetal (plant-like) designs. The Met notes that the basic shapes were not invented from scratch: they already existed in late antiquity among the Greeks, Romans and Sasanians in Iran. Muslim artists took those elements, then "complicated and elaborated upon them" to create a new style that stressed "the importance of unity and order." The museum adds that the work of Muslim mathematicians, astronomers and scientists was essential to that style.
The tools were humble. The Met's teaching guide explains that a compass and a straightedge, which make circles and straight lines, are enough to build rosettes, hexagons and eight-pointed stars, and that the grids made this way can repeat without end.
Five tiles that do the work
In a paper published in the journal Science in February 2007, Peter J. Lu, then a physics graduate student at Harvard University, and Paul J. Steinhardt of Princeton University in Princeton, New Jersey, argued that by around 1200, designers had made a conceptual leap. Instead of drafting every zigzag line by hand, they could lay down a small set of tiles, each already decorated with lines, and let the pattern form itself where the tiles met.
Lu and Steinhardt called them "girih tiles," from the Persian word for "knot." There are five:
- a regular decagon (ten sides)
- a regular pentagon
- an elongated hexagon
- a "bowtie" hexagon
- a rhombus (diamond)
Every tile has sides of the same length, and every angle is a multiple of 36 degrees, a tenth of a full turn. Lay them edge to edge and the decorative lines join up across the borders into the familiar stars and polygons. The Harvard Gazette reported that girih tiles may have been used to create patterns on buildings including mosques in Isfahan, Iran, and Bursa, Turkey; madrasas in Baghdad, Iraq; and shrines in Herat, Afghanistan, and Agra, India. He also matched the tiles to drawings in 15th-century Persian design scrolls.
"Straightedges and compasses work fine for the recurring symmetries of the simplest patterns we see," Lu told the Gazette, "but it probably required far more powerful tools to fully explain the elaborate tilings with decagonal symmetry."
The pattern that does not repeat
Most wallpaper and floor tiles are periodic: slide the design over by a set distance and it lines up exactly with itself. A pattern built on true tenfold symmetry cannot do that; as Lu and Steinhardt put it, that symmetry is "incompatible with periodicity." In the 1970s, the British mathematician Roger Penrose described tilings that are orderly but never repeat, now called Penrose tilings.
Lu and Steinhardt pointed to a large tile panel at the Darb-i Imam shrine in Isfahan, Iran. They found two sizes of girih tiles there, with the large tiles subdivided into small ones of the same shapes. That kind of rule, they wrote, is enough in principle to extend the pattern into an infinite, nearly perfect quasicrystalline tiling, a pattern with order but no repetition.
Then the chemistry. On April 8, 1982, the Israeli scientist Dan Shechtman looked at a rapidly cooled aluminum-manganese alloy under an electron microscope and saw a tenfold pattern that, by the textbooks of the day, could not exist in a crystal. According to the Royal Swedish Academy of Sciences, the discovery was "extremely controversial," and while defending it he was asked to leave his research group. In 2011 he received the Nobel Prize in Chemistry "for the discovery of quasicrystals." The Nobel press release made the link directly: "Aperiodic mosaics, such as those found in the medieval Islamic mosaics of the Alhambra Palace in Spain and the Darb-i Imam Shrine in Iran, have helped scientists understand what quasicrystals look like at the atomic level."
What scholars still debate
Good science comes with caveats, and this story has several.
Lu and Steinhardt themselves were careful. In a follow-up letter in Science, they wrote that they did not claim the designers "recognized the meaning of quasi-crystals" or knew how to extend such a pattern forever. Other researchers, including Emil Makovicky, argued that the Darb-i Imam pattern is periodic, and concluded that the artisans were satisfied with creating a large repeating unit "without being concerned with a mathematical notion of infinitely expandable quasiperiodic patterns."
The date is also disputed. The 2007 paper placed the panel in 1453. A 2018 article in the Nexus Network Journal argued that it actually dates to 1715 to 1717, late in the Safavid period, which would make the gap with modern Western mathematics smaller than the headlines suggested.
What nobody disputes is the craft itself: a tradition of building astonishingly complex, nearly flawless patterns from a few simple shapes, carried across many centuries and many lands.
Geometry and faith
Why did geometry flourish so richly in Muslim lands? Art historians point to several reasons. The Met notes that geometric patterns are often associated with Islamic art "largely due to their aniconic quality": images of people and animals are traditionally kept out of mosques and Qur'an manuscripts, and geometry, calligraphy and arabesque filled that space.
For many believers, the patterns also speak to something deeper. The Qur'an describes God as the One who "created each thing and determined it with [precise] determination" (Surah Al-Furqan 25:2), and invites the reader to look at the sky: "You do not see in the creation of the Most Merciful any inconsistency. So return [your] vision [to the sky]; do you see any breaks?" (Surah Al-Mulk 67:3). Verses like these do not describe tiling theory, and Muslims do not need them to. They describe an attitude toward the world: that creation is ordered and measured, and that studying that order is a form of reflection.
The label beside a 16th- or 17th-century carved cedar ceiling panel from Morocco at the Harvard Art Museums puts it this way: for viewers glancing upward, the star-and-polygon pattern "may have seemed celestial, alluding to a divinely ordered universe."
Harvard's Gülru Necipoğlu, the Aga Khan Professor of Islamic Art and Architecture, made a similar point at book length. Her 1996 study The Topkapi Scroll analyzes a design scroll from the late 15th or early 16th century, preserved in the Topkapi Palace Museum Library in Istanbul, Türkiye, that contains 114 geometric patterns. A Publishers Weekly review summarized her argument that these patterns were not "mere decoration" but a "sign system" reflecting religious currents, mathematical and scientific advances, and mystical beliefs. Lu and Steinhardt wrote that their own approach was founded on the Topkapi scroll as Necipoğlu had published it, and that she "guided us."
See it in Massachusetts
Harvard Art Museums, 32 Quincy Street, Cambridge, Massachusetts. That Moroccan ceiling panel (object 1981.10) is listed on Level 2, Room 2550, in the gallery "Art from Islamic Lands: The Middle East and North Africa." The gallery text says this interlacing geometric style developed intensely around the year 1000 in Baghdad, Iraq, then spread east and west, finding particular favor across North Africa. The museums are open Tuesday through Sunday, 10 a.m. to 5 p.m., and closed Mondays. According to the museums' visit page, admission is free to all visitors every day, and advance registration is encouraged on weekends. Galleries change, so check the object page before you go.
A related talk tomorrow. On Monday, Oct. 26, at 6 p.m., MIT's Aga Khan Program for Islamic Architecture hosts artist Laurelie Rae for a lecture titled "Cities of Islamic Art" in Room 3-133 in Cambridge, Massachusetts; the program lists it as open to all. We covered it in this week's events roundup.
Try it at home. You need only a compass, a pencil, a ruler and paper. The Met's educators' booklet Islamic Art and Geometric Design: Activities for Learning starts with a single circle, then shows how overlapping circles of the same size divide it into six equal parts. Join the points and you have a hexagon and a six-pointed star, and from there rosettes and repeating grids. It makes a good weekend project with children, and a quiet one.
The next time you pray beneath a patterned ceiling or a star-and-polygon screen, take a second look. Five simple shapes, a compass and a great deal of patience produced work that artists admire for its beauty, believers read as a reminder of an ordered creation, and scientists in Cambridge, Massachusetts, found worth a paper in Science.