Girih tiles are the Islamic geometric tiling system that produced quasi-crystalline patterns centuries before Western mathematics. Learn their history and significance.
In 2007, a physicist named Peter Lu was walking through a mosque in Uzbekistan when he noticed something wrong with his understanding of the patterns on the wall. The conventional view held that the geometric star-and-polygon patterns in medieval Islamic architecture were drafted directly with straightedge and compass, line by line. But Lu realized the patterns were too complex and too consistent for that method. He went home, analyzed the patterns mathematically, and discovered that Islamic craftsmen had been using a system of interlocking polygon tiles to generate patterns that were mathematically equivalent to quasi-crystalline Penrose tilings, a structure Western science did not formally describe until 1974. The craftsmen had been doing this since the 12th century.
Girih tiles are a set of five equilateral polygon shapes decorated with line patterns that, when tessellated, produce the complex geometric star-and-polygon designs found in medieval Islamic architecture. The word "girih" is Persian for "knot" and refers to the interlacing strapwork that characterizes Islamic geometric ornament. The system was identified and named by Peter Lu and Paul Steinhardt in a 2007 paper published in Science magazine, which showed that by 1200 CE, Islamic designers had reconceived geometric patterns not as networks of individually drafted lines but as tessellations of decorated polygons. Read the paper at Science magazine.
This entry covers what girih tiles are, how they work, where to see them, and why their discovery changed the understanding of Islamic art and the history of mathematics.
The girih tile system consists of five polygon shapes: a decagon, a hexagon, a bowtie, a rhombus, and an elongated hexagon. Each tile is decorated with line segments that cross the tile's interior. When the tiles are tessellated, the line segments on adjacent tiles connect to form continuous patterns of stars and polygons that cover the surface. The tiles themselves are invisible in the finished work. What the viewer sees is the line pattern, not the underlying polygonal grid.
This is the key insight that Lu and Steinhardt identified. The conventional view was that Islamic geometric patterns were constructed by drawing zigzagging lines directly on the surface with compass and straightedge. This method works for simple patterns but becomes impractical for complex designs with 10-fold or higher symmetry. The girih tile system solves this problem: instead of drawing the final pattern directly, the designer lays out the polygon tiles, and the line pattern emerges automatically from the decorations on each tile. This allows the construction of far more complex patterns with far less effort.
The system also enables self-similar subdivision. Each girih tile can be subdivided into smaller versions of the same set of tiles, allowing patterns to be scaled to any level of detail. This is the property that makes quasi-crystalline patterns possible. A 2008 study from Radboud University Nijmegen demonstrated that a subset of three girih tiles (decagon, bowtie, elongated hexagon) allows various inflation rules with different scaling factors, producing genuine quasicrystal tilings. Read the study at Radboud University.
Before Lu and Steinhardt's 2007 paper, the standard understanding was that Islamic geometric patterns were constructed using compass and straightedge, with lines drafted directly on the surface. This method produces patterns based on circles and radii, and it works well for patterns with 4-fold, 6-fold, and 8-fold symmetry. It struggles with 10-fold symmetry, which requires dividing a circle into ten equal parts, a construction that is not possible with compass and straightedge alone in a simple, repeatable way.
Lu and Steinhardt showed that by 1200 CE, a conceptual breakthrough occurred in which girih patterns were reconceived as tessellations of decorated polygons. The earliest known example of a girih tile pattern is on the Gonbad-e Qabud, a tomb tower in Maragha, Iran, dated to approximately 1197 CE. The tower is decagonal in plan, and its walls are covered in geometric patterns that Lu and Steinhardt identified as girih tile constructions.
A 2008 study published in the journal Nexus Network Journal by Bonner and others traces the mathematical thinking behind these patterns to the intellectual culture of 11th and 12th century Islamic mathematics, which included geometry, spherical trigonometry, and conic sections. The study notes that the Persian word "girih" is not textually documented in sources until the 19th century, though the craft tradition is much older. Read the study at Springer.
The most remarkable example of girih tile construction is on the Darb-i Imam shrine in Isfahan, Iran, built in 1453. The patterns on this shrine are not just complex. They are genuinely quasi-periodic, meaning they cannot tile the plane by simple repetition and have a mathematical structure equivalent to a two-dimensional section through a five-dimensional periodic lattice. This is the same structure that Roger Penrose described in 1974 and that was found in quasicrystals in the 1980s, earning Dan Shechtman the Nobel Prize in Chemistry in 2011.
The Darb-i Imam patterns were constructed by Islamic craftsmen working with compass and straightedge, using the self-similar subdivision property of girih tiles, approximately 500 years before Western mathematics formally described quasi-periodic structures. This is not a coincidence or a rough approximation. The patterns are mathematically equivalent to Penrose tilings.
The Topkapi Scroll, a 30-meter design document held at the Topkapi Palace Museum in Istanbul, provides direct evidence of how Islamic designers worked with girih tiles. Pattern number 28 of the scroll shows line ornaments on two different scale levels, with additional lines indicating how the pattern can be constructed from the set of five girih tiles. The scroll is essentially a design manual, showing the subdivision and inflation rules that generate complex patterns from simple tile sets.
This decagonal tomb tower is the earliest identified example of girih tile construction. Its walls are covered in decagonal geometric patterns that demonstrate the transition from direct line drafting to the polygonal tessellation method. The tower's decagonal plan is itself a reflection of the 10-fold symmetry that the girih system was designed to produce.
The Darb-i Imam shrine contains the most mathematically advanced girih patterns known. The quasi-periodic patterns on this shrine are equivalent to Penrose tilings and represent the highest achievement of the Islamic geometric tradition. The shrine is still standing and can be visited in Isfahan.
The Alhambra contains numerous geometric patterns, some of which have been analyzed as girih tile constructions. The patterns in the Alhambra cover walls, ceilings, and arches in a density of geometric ornament that is unmatched in Western architecture. While not all Alhambra patterns use the girih system, the most complex ones do.
The Topkapi Scroll is the primary documentary evidence for the girih tile system. It is held at the Topkapi Palace Museum and contains templates, construction lines, and completed patterns that served as reference material for architects and craftsmen.
Girih tiles are part of the broader tradition of Islamic geometric art and represent its most mathematically advanced expression. They connect to sacred geometry as the Islamic contribution to the global tradition of geometric patterns with spiritual significance. The arabesque is the vegetal counterpart to the geometric patterns produced by girih tiles, and the two are often combined on the same surface. The mathematical properties of girih tiles relate to the golden ratio, which appears in the proportions of decagonal patterns. Pattern as a formal element of art is the broader category that girih tiles exemplify.
The discovery of girih tiles matters for two reasons. First, it corrects a historical injustice. The conventional view assumed that Islamic craftsmen were skilled artisans but not mathematicians, producing complex patterns through patient, repetitive drafting rather than conceptual innovation. Lu and Steinhardt's work showed that these craftsmen had developed a sophisticated mathematical system that was, in some respects, ahead of Western mathematics by five centuries. The Islamic geometric tradition was not merely decorative. It was doing mathematics.
Second, girih tiles demonstrate that significant mathematical discoveries can be embedded in craft traditions rather than in written texts. The girih system was transmitted through workshop practice, design manuals like the Topkapi Scroll, and direct apprenticeship, not through mathematical treatises. This means that other craft traditions may contain mathematical knowledge that has not yet been recognized by modern science.
To see girih tile patterns in person, visit the Darb-i Imam shrine and the Jameh Mosque in Isfahan, the Gonbad-e Qabud in Maragha, or the Alhambra in Granada. The Topkapi Palace Museum in Istanbul holds the Topkapi Scroll. For more on the broader context, read our entries on sacred geometry, arabesque, and Islamic geometric art, or explore our guide on how to read a painting to learn how geometric structure underlies visual art across traditions.
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