Here is a detailed explanation of the profound mathematical connections between medieval Islamic geometric art and modern crystallography.
Introduction: A Convergence of Art and Science
For centuries, the intricate geometric patterns adorning mosques, madrasas, and palaces across the Islamic world were viewed primarily as masterpieces of aesthetic decoration. From the Alhambra in Spain to the Darb-e Imam shrine in Iran, these designs were appreciated for their spiritual symbolism—representing the infinite and the unity of creation.
However, in recent decades, physicists and mathematicians have discovered that these patterns are not just artistic doodles. They encode sophisticated mathematical rules that predate their "discovery" in Western science by over 500 years. Specifically, certain Islamic patterns demonstrate aperiodic tiling, a mathematical structure identical to quasicrystals, a form of matter that was thought impossible until the 1980s.
1. The Mathematical Foundation: Tessellation and Symmetry
To understand the breakthrough, one must first understand the basics of tiling, or tessellation.
- Periodic Tiling: Standard wallpaper or bathroom tiles are "periodic." You can take a section, shift it up, down, left, or right, and it will perfectly overlap with the pattern next to it. Mathematically, these patterns are limited. You can only tile a flat surface perfectly using triangles, squares, or hexagons (3-fold, 4-fold, and 6-fold symmetry).
- The Forbidden Symmetry: For centuries, mathematicians believed it was impossible to tile a continuous flat surface using 5-fold symmetry (pentagons) or 10-fold symmetry (decagons) without leaving gaps. Try to pave a floor with only regular pentagons, and you will inevitably find empty spaces.
The Islamic Solution: Medieval Islamic artists wanted to express infinite complexity. They were unsatisfied with simple repeating squares or hexagons. They developed a modular system to bypass the limits of Euclidean geometry, creating patterns that utilized the "forbidden" 5-fold and 10-fold symmetries.
2. The Secret Code: The Girih Tiles
For a long time, historians believed artisans drew these complex patterns using a compass and straightedge for every single star and polygon—a laborious and error-prone process.
In 2007, physicists Peter J. Lu (Harvard) and Paul J. Steinhardt (Princeton) published a groundbreaking paper in Science. They discovered that artisans had developed a set of five template tiles, known as Girih tiles (Persian for "knot").
These five tiles are: 1. A regular decagon (10 sides). 2. An elongated hexagon (irregular). 3. A bow tie shape. 4. A rhombus. 5. A regular pentagon.
How it works: Every edge of these tiles has the same length. Decorating the tiles are specific lines. When the tiles are laid edge-to-edge, the internal lines connect to form a continuous, interlacing strapwork pattern (the visible art). The artisans laid down the tiles (the hidden math) to generate the pattern (the visible art).
Crucially, these tiles allow for the creation of patterns with 5-fold and 10-fold rotational symmetry that cover an infinite plane without gaps.
3. Quasicrystals: The Modern Discovery
Fast forward to 1982. Materials scientist Dan Shechtman looked at an alloy of aluminum and manganese under an electron microscope. He saw a diffraction pattern (the way atoms scatter X-rays) that showed 10-fold symmetry.
According to the laws of crystallography at the time, this was impossible. Crystals (like salt or diamond) are periodic—they repeat perfectly. Shechtman had found a structure that was ordered but aperiodic. * Ordered: It followed a strict mathematical rule. * Aperiodic: The pattern never repeated itself exactly. If you shifted the pattern over, it would never match the section next to it.
This new form of matter was named Quasicrystals. (Shechtman eventually won the Nobel Prize in Chemistry in 2011 for this discovery).
Mathematically, the structure of quasicrystals is often described using Penrose Tiling, a system invented by British mathematician Roger Penrose in the 1970s. Penrose Tiling uses two specific shapes (a "kite" and a "dart") to create an infinite, non-repeating pattern with 5-fold symmetry.
4. The Connection: The Darb-e Imam Shrine
The revelation provided by Peter Lu and Paul Steinhardt was that Islamic architects had intuitively created Penrose tiling 500 years before Roger Penrose.
The most stunning example is found at the Darb-e Imam shrine in Isfahan, Iran, built in 1453.
The spandrel of the shrine features a massive, complex geometric pattern. When Lu and Steinhardt analyzed it, they found: 1. Self-Similarity: The pattern is fractal. You can zoom in on a large decagon in the pattern and find it is filled with smaller versions of the same pattern. This scaling capability is a hallmark of quasicrystalline math. 2. Near-Perfect Quasicrystalline Structure: The arrangement of the Girih tiles on the shrine follows the same mathematical rules as Penrose tiling. It maps almost perfectly onto the atomic structure of quasicrystals.
The artisans had figured out how to project a slice of a higher-dimensional lattice (mathematically, quasicrystals can be viewed as projections of 6-dimensional hypercubes) onto a 2-dimensional surface.
Summary of the Phenomenon
| Concept | Traditional Crystallography | Islamic Art & Quasicrystals |
|---|---|---|
| Repetition | Periodic (repeats perfectly) | Aperiodic (never repeats exactly) |
| Symmetry | 2, 3, 4, 6-fold | 5, 10-fold (The "Forbidden" Symmetries) |
| Structure | Simple Grid | Complex, Self-Similar (Fractal) |
| Discovery | Ancient | 1453 (Art) / 1982 (Matter) |
Conclusion
The connection between Islamic tiling and quasicrystals changes our understanding of the history of science. It suggests that medieval Islamic mathematicians and artisans possessed a sophisticated, algorithmic understanding of geometry that the West would not unlock for half a millennium.
While they likely did not understand the atomic theory of matter, they understood the logic of the structure. They sought to represent the infinite nature of God through geometry, and in doing so, they constructed patterns that mirror the very building blocks of matter that 20th-century science deemed impossible.