The Girih Code
The Ulugh Beg Madrasah in Bukhara, constructed in 1417, displays geometric tilework known as girih. The term "girih" is Persian for "knot," and these patterns were traditionally thought to be made by artisans using only a straightedge and compass. Research published in 2007 by physicists Peter J. Lu and Paul J. Steinhardt showed a more sophisticated system. They showed that by the 15th century, artisans were using a specific set of five tile shapes—a decagon, a pentagon, a hexagon, a rhombus, and a "bow tie"—to create these designs.
These five "girih tiles" allowed for the creation of complex patterns that never repeat. Analysis of the patterns on buildings like the Darb-i Imam shrine (1453) in Iran shows they are nearly perfect quasi-crystalline tilings. This discovery suggests that medieval architects had developed a geometric toolkit. The five tile shapes have sides of equal length, and their interior angles are all multiples of 36 degrees (π/5 radians), which allows them to be combined in complex, non-repeating arrangements. This method enabled the creation of vast, complex patterns that appear ordered and lack the translational symmetry of simpler designs.
Aperiodic Tiling and Modern Physics
The patterns created with girih tiles are aperiodic, meaning they fill a plane without ever repeating exactly. This concept was explored in the 1970s by mathematician Roger Penrose with his Penrose tilings. The geometric principles behind girih tiles predate Penrose's work by five centuries. These 2D patterns are the geometric equivalent of a 3D structure first identified in a metallic alloy by Dan Shechtman on April 8, 1982. His discovery, published in 1984, identified a material that was ordered and not periodic.
This class of matter was termed a "quasicrystal." Shechtman's findings were initially met with skepticism and ridicule from the scientific community because they violated the established laws of crystallography, which stated that crystals must have repeating, periodic structures. The discovery of five-fold rotational symmetry in his electron diffraction patterns was considered impossible. Shechtman's persistence was eventually validated, and he was awarded the Nobel Prize in Chemistry in 2011 for the discovery. The artisans of 15th-century Bukhara, while unaware of the underlying atomic physics, had mastered the mathematical principles of quasicrystals, creating representations of a fundamental structure of matter 500 years before its formal scientific discovery.