Here is a detailed explanation of the fascinating intersection between medieval Islamic art and modern crystallography: the discovery of "impossible" quasicrystalline geometry in 500-year-old mosaics.
1. The Core Mystery: What is a Quasicrystal?
To understand why this discovery was so shocking, one must first understand the rules of tiling (tessellation). For centuries, mathematicians and crystallographers believed there were strict rules governing how shapes could fit together to cover a surface without gaps or overlaps.
- Periodic Tiling: Standard crystals (like salt or diamonds) and standard tiles (like a bathroom floor) are periodic. This means they are constructed from a single unit shape (like a square or hexagon) that repeats endlessly in a regular pattern. You can shift the pattern over, and it looks exactly the same.
- The Forbidden Symmetry: Mathematically, you can tile a floor perfectly with 3-sided, 4-sided, or 6-sided shapes. However, it was mathematically proven that you cannot tile a floor using 5-fold symmetry (pentagons) or 10-fold symmetry without leaving gaps.
The Quasicrystal Revolution: In the 1970s, mathematician Roger Penrose discovered a set of two tile shapes (darts and kites) that could cover a surface in a pattern that never repeated. This is called "aperiodic tiling." In 1982, Dan Shechtman discovered this structure in actual matter (metal alloys), earning him the Nobel Prize. These structures, which possessed the "forbidden" 5-fold and 10-fold symmetries but never repeated, were named quasicrystals.
2. The Discovery: The Lu and Steinhardt Findings
In 2007, physicists Peter J. Lu (Harvard University) and Paul J. Steinhardt (Princeton University) published a groundbreaking paper in the journal Science.
Lu, fascinated by the geometric complexity of Islamic architecture during a trip to Uzbekistan, began analyzing the tile patterns known as girih (Persian for "knot"). When he examined the patterns on the Darb-i Imam shrine in Isfahan, Iran (built in 1453), he realized he was looking at something that shouldn't exist in the 15th century.
The patterns were not just pretty stars and polygons; they were nearly perfect Penrose tilings—quasicrystalline patterns created five centuries before the West "discovered" the math behind them.
3. How Did They Do It? The "Girih Tiles" Method
For a long time, historians believed Islamic artisans created these complex patterns using a straightedge and a compass, drawing the lines directly onto the plaster. However, Lu and Steinhardt argued that this method would have been incredibly difficult for such massive, error-free patterns.
Instead, they proposed that the artisans used a modular system of five specific tiles, now known as Girih tiles:
- A regular decagon (10 sides)
- An elongated hexagon (irregular convex hexagon)
- A bow tie shape
- A rhombus
- A regular pentagon
Each of these tiles was decorated with specific strapwork lines. When the tiles were laid edge-to-edge, the lines on them connected perfectly to form the complex, interlacing "knot" patterns visible on the walls.
The Significance of the Method: By focusing on the shapes of the tiles rather than the lines themselves, the artisans could create patterns with decagonal (10-fold) symmetry. The arrangement of these tiles at the Darb-i Imam shrine creates a pattern that does not repeat—essentially a medieval version of a Penrose tiling.
4. The "Impossible" Mathematics
The artisans of the Seljuk and Timurid eras had evidently developed a sophisticated geometric intuition that allowed them to bypass the "rules" of crystallography.
- Self-Similarity: The patterns at the Darb-i Imam shrine exhibit "self-similarity." This is a fractal concept where the pattern looks similar at different scales. The shrine features large girih tiles that are essentially filled with smaller versions of themselves.
- Aperiodic Infinite Extension: While the wall of a shrine is finite, the mathematical logic used to create the pattern implies that it could be extended infinitely without ever repeating exactly—the definition of a quasicrystal.
5. Implications and Legacy
This discovery forced a rewriting of the history of mathematics and art.
- Mathematics vs. Art: In the West, the discovery of quasicrystals was a triumph of abstract mathematics and materials science. In the Islamic world, it was a triumph of aesthetics and theology. The artisans were likely motivated by the desire to reflect the infinite nature of God through complex, non-repeating geometry, stumbling upon advanced mathematical truths in pursuit of beauty.
- Lost Knowledge: It appears this advanced understanding was not theoretical but practical. The artisans likely used pattern books (scrolls have been found, such as the Topkapi Scroll) to guide these constructions. However, the deep mathematical understanding of why these tiles worked seems to have remained within the guild traditions and was eventually lost or overtaken by changing artistic tastes.
Summary
The mosaics of the Darb-i Imam shrine represent a "technological anachronism." Islamic artisans, equipped with only compasses, rulers, and a set of five geometric tile templates, constructed patterns of such profound complexity that Western science would not be able to describe them mathematically for another 500 years. They successfully visualized the infinite and the "impossible" through the medium of glazed clay.