The intersection of medieval Islamic art and modern advanced mathematics is one of the most fascinating discoveries in the history of science and architecture. In 2007, physicists Peter J. Lu and Paul J. Steinhardt published a groundbreaking paper revealing that medieval Islamic artisans had unknowingly created near-perfect aperiodic quasicrystalline patterns centuries before Western mathematicians formally described the underlying principles.
Here is a detailed explanation of this phenomenon, exploring the math, the historical method, and how art unintentionally anticipated modern physics.
1. The Mathematical Context: What is an Aperiodic Quasicrystal?
To understand the significance of the discovery, one must first understand tiling. * Periodic Tiling: Think of a standard bathroom floor or a checkerboard. The pattern translates (shifts) and repeats perfectly at regular intervals. * Aperiodic Tiling: An aperiodic pattern completely fills a two-dimensional space without leaving gaps, but it never repeats exactly. Even though it doesn't repeat, it isn't random; it follows strict mathematical rules.
In the 1970s, British mathematician Roger Penrose famously discovered a set of two shapes (often called "kites" and "darts") that could tile a plane infinitely without ever repeating, creating what is known as Penrose tiling. This geometry exhibits "five-fold" or "ten-fold" rotational symmetry—something previously thought impossible in crystallography. When scientists later discovered physical materials structured this way at the atomic level, they named them quasicrystals (a discovery that won the 2011 Nobel Prize in Chemistry).
For decades, the scientific community believed that these complex, non-repeating geometric structures were purely a product of 20th-century advanced mathematics.
2. The Artisanal Tool: Girih Tiles
In Islamic architecture, depictions of humans and animals were traditionally avoided, leading to a profound focus on complex geometric ornamentation. By the 12th century, artisans were creating incredibly intricate star-and-polygon patterns.
Originally, these patterns were drafted using a compass and a straightedge. However, as the patterns became more complex, this method became mathematically cumbersome and prone to compounding errors. To solve this, artisans abstracted the geometry into a physical toolkit known as girih tiles.
There are five standard girih shapes: 1. A regular decagon (10-sided polygon) 2. An elongated hexagon 3. A bowtie shape 4. A rhombus 5. A regular pentagon
The secret of the girih tiles lies in their decoration. Each tile has decorative lines drawn across it. When the artisans laid the tiles edge-to-edge, the borders of the underlying tiles essentially disappeared, and the intersecting lines on top of the tiles connected to form a continuous, complex, overarching web.
3. The Discovery: Quasicrystals in Medieval Shrines
In 2007, Lu and Steinhardt analyzed photographs of Islamic architecture, most notably the Darb-e Imam shrine in Isfahan, Iran, built in 1453.
When they overlaid Penrose's mathematical models onto the walls of the shrine, they found a stunning correlation. By mapping the kites and darts of Penrose mathematics onto the girih tiles, they realized that the 15th-century artisans had created a near-perfect decagonal (10-fold) quasicrystalline pattern.
The pattern at Darb-e Imam is particularly astounding because it exists on two scales. There is a large, overarching macro-pattern, and within the shapes of that pattern is a micro-pattern following the exact same mathematical rules. This concept, known as "self-similarity" or fractal geometry, is a hallmark of true quasicrystals.
4. The "Unintentional" Brilliance
Did medieval Islamic artisans understand the algebra, group theory, and non-periodic mathematics behind their creations? The consensus among historians and physicists is no. The mathematical presence of quasicrystals was an unintentional byproduct of an aesthetic philosophy and a practical methodology.
- The Aesthetic Drive: Islamic artisans were driven by a desire to represent the infinite and the divine. A pattern that never repeats, yet maintains perfect harmony and order, was the ultimate theological and artistic achievement.
- The Practical Methodology: The artisans did not calculate quasicrystals algebraically. Instead, they discovered local "matching rules." By simply ensuring that the lines on one girih tile connected seamlessly to the lines on the next, they were unknowingly enforcing rules that generated global aperiodicity.
- Empirical Evolution: The transition from simple repeating patterns (in the 11th century) to true quasicrystalline patterns (by the 15th century) shows an evolutionary process of trial, error, and refinement. They developed a practical geometric algorithm that acted as a shortcut to a vastly complex mathematical truth.
Summary
The presence of aperiodic quasicrystals in medieval Islamic architecture is a spectacular example of art leading science. Driven by a cultural and religious desire to create infinite, non-repeating beauty, Islamic artisans invented the girih tile system. By strictly following the local geometric rules of these physical templates, they unintentionally solved a highly advanced spatial problem, physically rendering Penrose tilings over 500 years before Western mathematics formally discovered that such structures could even exist.