The mathematical principles underlying Islamic geometric star patterns—specifically those that can tessellate infinitely without ever repeating—represent one of the most fascinating intersections of art and advanced mathematics.
Until the 1970s, Western mathematicians believed that creating an infinite, non-repeating (aperiodic) tiling with complex symmetries was a modern mathematical invention. However, in 2007, physicists Peter J. Lu and Paul J. Steinhardt discovered that Islamic artisans had been using advanced geometric concepts to create highly complex aperiodic tilings as early as the 13th century.
Here is a detailed explanation of the mathematical principles that allow these patterns to exist.
1. The Concept of Aperiodic Tessellation
To understand these patterns, one must first understand tessellation. A tessellation is a tiling of a flat plane using geometric shapes with no overlaps and no gaps. * Periodic Tessellation: Think of a checkerboard or a honeycomb. If you shift the pattern (translational symmetry), it perfectly aligns with itself. * Aperiodic Tessellation: The pattern covers an infinite plane without ever repeating the exact same arrangement of tiles. It lacks translational symmetry, yet it contains high levels of localized rotational symmetry.
2. The Five "Girih" Tiles
The mathematical genius of these Islamic patterns lies in an underlying toolkit known as Girih tiles. Instead of calculating complex math for every single line, Islamic artisans used a set of five specific polygonal tiles.
The five Girih tiles are: 1. A regular decagon (10 sides) 2. An elongated hexagon (6 sides) 3. A "bowtie" (a non-convex hexagon) 4. A rhombus (4 sides) 5. A regular pentagon (5 sides)
The Mathematical Constraints of the Tiles: * Equal Edge Lengths: Every side of every tile has the exact same length. * Specific Angles: All internal angles of these tiles are multiples of 36° ($\pi/5$ radians). * Decorated Lines: Instead of displaying the edges of the tiles, artisans drew continuous lines (strapwork) inside the tiles. These lines intersect the midpoint of every tile edge at exactly 54°. When the tiles are placed edge-to-edge, the internal lines match perfectly, creating the continuous, interlacing star patterns visible on the buildings. The outline of the tile itself vanishes.
3. Five-Fold and Ten-Fold Symmetry
Standard periodic tilings can only possess 2-, 3-, 4-, or 6-fold rotational symmetry. According to the Crystallographic Restriction Theorem, it is mathematically impossible to tile a plane periodically using 5-fold (pentagons) or 10-fold (decagons) symmetry.
Because Islamic art heavily favored 5-fold and 10-fold star patterns, the artisans were forced into a unique geometric space. By attempting to pack decagons and pentagons together tightly without gaps, they organically discovered the rules of aperiodic math.
4. Matching Rules and Penrose Tilings
In the 1970s, mathematician Roger Penrose discovered Penrose tilings—a set of two basic shapes (kites and darts, or thick and thin rhombi) that can tile a plane infinitely without repeating.
Mathematically, aperiodic tilings are governed by matching rules. You cannot simply place any tile next to any other tile; they must lock together based on specific edge conditions. * In Penrose tilings, these are usually defined by notches on the edges. * In Islamic architecture, the matching rules were enforced by the strapwork. The artisans had to place the tiles so that the decorative lines inside them continued seamlessly without dead ends.
Lu and Steinhardt proved that the five Girih tiles can be subdivided into the exact "kites and darts" formulated by Penrose. Therefore, by following the visual rules of the Girih strapwork, Islamic artisans were successfully generating mathematically rigorous Penrose tilings centuries before Penrose was born.
5. Self-Similarity (Inflation and Deflation)
The final mathematical principle that allows these patterns to tile infinitely without repetition is self-similarity, also known as scale symmetry.
A mathematically true aperiodic tiling can be scaled up or down infinitely. If you take a group of small Girih tiles arranged in a specific way, you can draw a boundary around them that forms a larger version of a single Girih tile. * Deflation: You can take a large Girih tile and subdivide it into smaller Girih tiles. * Inflation: You can group smaller tiles to act as a macro-tile.
This hierarchical fractal nature means the pattern can grow forever. Because the larger "macro-tiles" follow the exact same matching rules as the smaller tiles, the artisans could lay out a massive wall pattern by starting with a giant, simple template, and mathematically subdividing it into smaller and smaller interlocking star patterns.
Summary
The infinite, non-repeating star patterns found in structures like the Darb-e Imam shrine in Isfahan (1453 CE) are visually stunning but mathematically profound. By using a standardized set of five polygons with equal edge lengths and intersecting internal lines (Girih tiles), Islamic artisans created a physical algorithm. By ensuring the internal lines connected continuously (matching rules), they overcame the crystallographic restriction theorem to utilize 5- and 10-fold symmetries, resulting in an infinite, aperiodic fractal geometry long before modern mathematics codified the concepts.