What is girih? The Persian geometric tradition

Girih is the Persian Islamic tradition of complex interlaced geometric patterns, distinguished from the Maghreb zellij tradition by three things: a strong preference for decagonal (10-fold) symmetry rather than the octagonal (8-fold) symmetry that dominates Moroccan and Andalusian work, a method of construction using a small set of underlying polygonal tiles (the "girih tiles"), and a regional concentration in Iran, Central Asia, and the Persian cultural sphere more broadly. The word girih itself is Persian, meaning "knot," and refers to the interlaced structure that defines the visual character of the tradition. This is also the tradition that produced what the 2007 Lu and Steinhardt paper in Science identified as quasi-crystalline tilings — non-repeating patterns with mathematical properties Western science wouldn't formalize until the 1970s.

This post covers what girih is, how it differs from zellij, where to see it, and why it matters in the history of both Islamic art and mathematics.

What girih means

The word girih is used in modern scholarship in two related but distinct senses.

In the broader sense, girih refers to the entire Persian geometric pattern tradition — the work that fills the great Persian and Central Asian monuments from roughly the 11th century onward, in tile, brick, stucco, wood, and manuscript form. In this broad usage, girih is the Iranian counterpart to zellij in the Maghreb.

In a narrower technical sense, established by the Lu and Steinhardt 2007 paper, "girih tiles" refers specifically to a set of five polygonal shapes (a decagon, a pentagon, a hexagon, a rhombus, and a bowtie) that medieval Persian craftsmen used as underlying construction templates. The visible decorative lines were drawn over these polygons; the polygons themselves remained invisible in the finished work. This polygonal technique is what Lu and Steinhardt argue underlies the quasi-crystalline patterns at the Darb-e Imam shrine in Isfahan.

Both usages are current in modern scholarship. The narrower technical sense is more useful for understanding the mathematical structure; the broader cultural sense is more useful for understanding the tradition as a whole.

Where girih comes from

The Persian geometric tradition emerges in the 11th century, contemporary with the Sunni Revival described in the Sunni Revival post. The earliest surviving examples appear in Seljuk monuments in Iran, especially in carved brick decoration on tomb towers and mosques. By the 12th century, the tradition is recognizable as distinctly Persian, with the decagonal symmetry that would become its signature already in evidence.

The mature period of girih runs from roughly 1200 to 1500, encompassing:

• The Seljuks (11th–12th c.) in carved brick and stucco
• The Ilkhanids (Mongol dynasty, 13th–14th c.) — the patrons of the Soltaniyeh mausoleum among others
• The Timurids (14th–15th c.) — the patrons of the Samarkand monuments
• The early Safavids (16th c.) — who carried the tradition into its polychromatic peak at Isfahan

After 1500, Persian decoration shifts increasingly toward floral arabesque and away from pure geometric patterns. The Safavid mosques of Isfahan combine geometric and arabesque elements, with the floral often dominating. But the underlying geometric infrastructure remains present.

How girih differs from zellij

The contrast is sharper than most generalist accounts make it.

Symmetry. Zellij is built around the octagonal family — 8-pointed stars (the khatem) and related polygons. Girih is built around the decagonal family — 10-pointed stars and related polygons. The difference matters mathematically: octagonal patterns conform cleanly to the crystallographic restriction and produce periodic tilings; decagonal patterns can produce non-periodic or quasi-crystalline tilings, which is what Lu and Steinhardt identified.

Technique. Zellij is cut tile mosaic — individual tiles hand-cut and set in plaster. Persian work uses cut tile mosaic (called kashi-kari) but also extensively uses square tiles with patterns painted across multiple tiles (called haftrang), and brick patterning, and stucco. The Persian tradition is more varied in its physical execution.

Color palette. Zellij is dominated by earth tones, deep blue, green, white, and black. Persian work is famous for its bright turquoise blue, deep cobalt, gold, and white. The Imam Mosque in Isfahan is the canonical example of the Persian palette at full saturation.

Pattern logic. Zellij patterns tend toward strong central anchors (the dominant rosette in the centre of a panel). Girih patterns tend toward more uniform field patterns where the same motif repeats or quasi-repeats across the entire surface. This produces a different visual effect — zellij reads as composed; girih reads as woven.

For zellij specifically, see what is zellij — a guide to Moroccan tile work.

Famous girih monuments

The canonical examples:

The Friday Mosque of Isfahan (built and expanded from the 8th c. onward). The single richest collection of Persian Islamic geometric work in any one building. The mosque includes Seljuk brickwork from the 11th–12th centuries, Ilkhanid stucco from the 14th century, Safavid additions from the 17th century, and many other phases. Walking through the building is a tour through the history of Persian decoration.

The Darb-e Imam shrine in Isfahan (1453). The site that gave Lu and Steinhardt their 2007 paper. The tile work on the shrine's facade is mathematically equivalent to quasi-crystalline tiling, produced by 15th-century craftsmen working without formal mathematical theory. One of the most extraordinary single examples of Islamic geometric art anywhere.

The Imam Mosque (formerly Shah Mosque), Isfahan (1611–1629). The Safavid masterpiece. The interior and exterior are entirely covered in polychromatic tile — predominantly painted square tile rather than cut mosaic — in patterns combining girih geometry with floral arabesque.

The Sheikh Lotfollah Mosque, Isfahan (1602–1619). Smaller than the Imam Mosque but with the highest tile quality. The dome's interior is among the most photographed surfaces in Islamic architecture.

The Soltaniyeh Mausoleum (1302–1312). The Ilkhanid tomb of Öljeitü, the third-largest masonry dome in the world after Hagia Sophia and the Florence cathedral. Geometric brick patterning at monumental scale.

The Registan, Samarkand. Three madrasahs from the 15th and 17th centuries, decorated with Timurid and post-Timurid tile work. Among the most photographed Islamic monuments anywhere.

The Tomb of Itimad-ud-Daula, Agra, India (1622–1628). The "Baby Taj," a Mughal monument that imports Persian decorative methods directly into India. Inlaid marble pietra dura execution of girih patterns.

For the wider Persian tradition, see Persian Islamic art — Isfahan, Yazd, and the Safavid flowering. For the Persian tile tradition specifically, see what is kashi.

The Lu and Steinhardt connection in detail

The 2007 paper deserves its own treatment. The argument:

The Darb-e Imam shrine's tile patterns appear to be standard decagonal Islamic geometric work — beautiful but not exotic. Lu and Steinhardt looked at the patterns more carefully and noticed two things. First, the patterns at the shrine don't repeat — they have well-defined rotational symmetry but no translational symmetry, which is the defining feature of quasi-crystals. Second, the patterns can be decomposed into a set of five specific polygonal tiles that, combined in specific ways, generate the full pattern. These five tiles are mathematically equivalent to Penrose's aperiodic tiling system.

Roger Penrose published his aperiodic tilings in 1974. The Darb-e Imam tile work dates to 1453. The medieval Persian craftsmen had figured out a mathematical theory five centuries before Western mathematics formalized it.

The Lu and Steinhardt paper provoked some scholarly debate when it was published. Some specialists argued that the patterns aren't strictly quasi-crystalline in the mathematical sense; others have defended the original analysis. The current consensus, twenty years on, is that the Persian craftsmen were definitely working with a polygonal technique that could and sometimes did produce mathematically interesting non-periodic patterns, even if every specific example doesn't meet the strictest definition of a quasi-crystal.

For more on the math, see the math behind Islamic geometric patterns and the Topkapı Scroll.

The contemporary relevance

For practitioners today, including myself, girih matters for two reasons.

The polygonal technique is the most flexible Islamic pattern construction method. Once you understand the five girih tiles and the rules for combining them, you can produce a very wide variety of patterns — both traditional ones and new ones — within the same systematic framework. Eric Broug's books teach this approach in accessible form. My own SVG Stack Studio implements the polygonal technique as one of its core design modes.

The mathematical interest of the tradition continues to draw new attention. Researchers in mathematics, computer science, and physics regularly engage with girih patterns as a real-world example of quasi-crystalline structure. This produces a steady stream of scholarship and a wide audience for the tradition that goes beyond conventional art history.

FAQ

What is girih?

Girih is the Persian Islamic tradition of complex interlaced geometric patterns, distinguished by a preference for decagonal (10-fold) symmetry and a construction method using a small set of underlying polygonal tiles. The word girih means "knot" in Persian, referring to the interlaced structure of the patterns.

How is girih different from zellij?

Several ways. Symmetry: girih favors decagonal (10-fold), zellij favors octagonal (8-fold). Technique: girih uses multiple methods including painted square tile, cut mosaic, and brick patterning; zellij is primarily cut tile mosaic. Color: girih palette includes turquoise, cobalt, and gold; zellij palette is more earth-toned. Region: girih is Persian and Central Asian; zellij is Moroccan and Andalusian.

Did medieval Persian craftsmen really invent quasi-crystals?

In a mathematical sense, effectively yes. The 2007 Lu and Steinhardt paper in Science demonstrated that 15th-century craftsmen working on the Darb-e Imam shrine in Isfahan used a polygonal construction technique mathematically equivalent to Penrose's aperiodic tilings, published in 1974. The medieval craftsmen weren't working from formal mathematical theory; they were using craft methods that happened to produce mathematically sophisticated patterns.

Where can I see the best girih in person?

Isfahan in Iran is the canonical destination — the Friday Mosque, the Darb-e Imam shrine, the Imam Mosque, and the Sheikh Lotfollah Mosque are all within a short walk of each other. Samarkand in Uzbekistan is the other major site, with the Registan complex and Shakh-i-Zindeh. For Western visitors who can't travel to Iran or Uzbekistan, the V&A and the Met have significant Persian tile collections; the David Collection in Copenhagen has one of the strongest concentrations of Islamic art outside the Islamic world.

Sources

• Wichmann, Brian, and David Wade. Islamic Design: A Mathematical Approach. Springer, 2017. Chapter 12 (Decagonal Patterns), Chapter 15 (Two-Level Patterns), Section 6.3 (Iran).
• Lu, Peter J., and Paul J. Steinhardt. "Decagonal and Quasi-Crystalline Tilings in Medieval Islamic Architecture." Science 315 (2007).
• Necipoğlu, Gülru. The Topkapı Scroll. Getty Center, 1995.
• Bonner, Jay. Islamic Geometric Patterns: Their Historical Development and Traditional Methods of Construction. Springer, 2017.
• Broug, Eric. Islamic Geometric Patterns. Thames & Hudson, 2008.