The discovery of complex aperiodic quasi-crystalline geometry within medieval Islamic architecture is one of the most fascinating intersections of art, history, and advanced mathematics. For centuries, the intricate geometric star-and-polygon patterns adorning mosques and shrines across the Islamic world were admired purely as masterful works of art. However, a groundbreaking 2007 study revealed that these medieval artisans had intuitively grasped a highly complex mathematical concept—quasi-crystalline geometry—nearly 500 years before Western mathematicians formally defined it.
Here is a detailed explanation of this discovery, the mathematics behind it, and its historical significance.
1. The Basics: What are Girih Patterns?
Girih (Persian for "knot") is an Islamic decorative art form consisting of geometric lines that create interwoven strapwork patterns. These patterns typically feature stars and polygons. Historically, historians and mathematicians believed that these intricate designs were created entirely using a "compass-and-straightedge" drafting method, drawn locally line-by-line.
While this method works well for simpler, repeating patterns, it becomes almost impossibly cumbersome to maintain accuracy over large surface areas (like the dome or wall of a mosque) without the lines drifting out of alignment.
2. The Math: What is Quasi-Crystalline Geometry?
To understand the discovery, one must understand the difference between periodic and aperiodic patterns: * Periodic Patterns: Like a standard chessboard or honeycomb, the pattern repeats uniformly in all directions. You can pick it up, shift it, and it will perfectly overlap itself. * Aperiodic Patterns: These patterns fill an infinite two-dimensional plane completely, without any gaps, but they never repeat the exact same way twice.
In the 1970s, British mathematician Sir Roger Penrose discovered a way to create an aperiodic tiling using just two distinct shapes. This became known as Penrose tiling. These tilings exhibit a "forbidden symmetry" (such as 5-fold or 10-fold decagonal symmetry) which was thought impossible in traditional crystallography. In 1982, scientist Dan Shechtman discovered molecular structures in nature that behaved this way, earning him the Nobel Prize in Chemistry for the discovery of "quasicrystals."
3. The Breakthrough: The 2007 Discovery
In 2007, Harvard physicist Peter J. Lu and Princeton physicist Paul J. Steinhardt published a paper in the journal Science. Lu had been traveling in Uzbekistan and noticed that the Islamic geometric patterns on the buildings looked remarkably similar to the Penrose tilings he studied in physics.
Lu and Steinhardt analyzed thousands of architectural photos and architectural scrolls. They discovered two major things: 1. The Girih Tile System: Artisans were not using compasses and straightedges for these complex patterns. Instead, they had developed a set of five master tiles (a regular decagon, an irregular pentagon, a hexagon, a bowtie shape, and a rhombus). 2. Quasi-Crystalline Execution: By the 15th century, the arrangement of these tiles had evolved from simple, repeating patterns into complex, non-repeating (aperiodic) quasicrystalline patterns.
4. How the "Girih Tiles" Work
The genius of the medieval artisans lay in the creation of the tiles themselves. The five Girih tiles were not the final visible artwork; they were the templates.
On each of the five tiles, the artisans drew continuous decorative lines. When the tiles were laid edge-to-edge according to specific matching rules, the lines on the tiles connected seamlessly to form the continuous, overlapping star-and-polygon Girih pattern. Once the design was complete, the outlines of the five base tiles were erased or hidden, leaving only the complex interwoven strapwork visible.
The Topkapi Scroll, a 15th-century Persian architectural manual held in Istanbul, provided the smoking gun. It clearly shows the faint outlines of these five Girih tiles drawn beneath the intricate strapwork, proving that this tile-based method was the standard operating procedure for master builders.
5. The Apex: The Darb-e Imam Shrine (1453)
The most profound example of this mathematical mastery was found at the Darb-e Imam shrine in Isfahan, Iran, built in 1453.
Lu and Steinhardt discovered that the patterns on this shrine possess two defining characteristics of quasicrystals: * Aperiodicity: The pattern on the wall is perfectly mapped using Girih tiles, but it does not repeat itself with strict regularity. * Self-Similarity (Fractal Geometry): The pattern exists on two different scales. If you look closely at the wall, you see a small star-and-polygon pattern. If you step back, you realize that those small patterns are grouped together to form the exact same shapes on a macro-level. Large decagons are constructed out of smaller decagons, bowties, and hexagons.
This self-similar subdivision is the exact mathematical property that allows Penrose tilings to stretch out to infinity without ever repeating perfectly.
Summary of Significance
The discovery that medieval Islamic artisans created quasicrystalline patterns forces a re-evaluation of the history of mathematics and art.
While there is no evidence that these 15th-century artisans understood the underlying algebraic equations or formal physics of quasicrystals, they possessed an incredibly sophisticated spatial intuition. By developing the Girih tile system, they successfully translated highly abstract, complex geometric principles into a practical, modular building tool. In doing so, they created perfect aperiodic geometries half a millennium before modern mathematicians even realized such patterns were possible.