What is girih? The geometry of Islamic star patterns.

Girih - from the Persian gereh, "knot" - is the strapwork geometry of Islamic architecture: interlacing straight lines that weave stars, polygons and knots across walls, domes, doors and manuscripts. It is one of the most mathematically sophisticated ornament systems ever put on a building, and its method can be stated precisely.

Girih is the network of straight strapwork lines that forms the star-and-polygon patterns of Islamic geometric ornament. The word names both the lines themselves and the design system that produces them. A girih pattern is not drawn freehand and not assembled from a picture: it is constructed, line by line, from an underlying polygonal grid, which is why the same pattern can be reproduced exactly at any size - on a ceramic panel, a carved screen or a city gate.

Where does girih come from?

Geometric strapwork appears in Islamic architecture from around the 9th century - the window grilles of the Mosque of Ibn Tulun in Cairo (879 CE) are among the earliest surviving examples, each screen a different star lattice. The style matured from the 11th century across Persia, Anatolia, Syria and Central Asia, reaching extraordinary refinement under the Seljuks and Timurids. In the west, the same geometric instinct produced the zellige mosaics of Morocco and the Alhambra; in the east, the tile vaults of Isfahan and Samarkand.

One document above all preserves the working method: the Topkapi Scroll, a 29-metre roll of construction drawings from 15th-century Persia, now in the Topkapi Palace Museum in Istanbul. It contains no text explaining the patterns - only the constructions themselves, drawn the way a master would hand them to a workshop.

What are the five girih tiles?

The five girih tiles are a decagon, an elongated hexagon, a bowtie, a rhombus and a regular pentagon - each decorated with the same fixed strapwork lines. Craftsmen discovered that instead of constructing every star from scratch, they could tile a surface with these five shapes; the lines painted on the tiles then join automatically across every edge, and the strapwork appears as if by itself. It is a profound act of modularisation: the geometry is solved once, on the tiles, and any arrangement of the tiles is a correct pattern.

In 2007, physicists Peter Lu and Paul Steinhardt showed that by 1453 this method had produced something modern science only named in the 1970s: the girih pattern on the Darb-i Imam shrine in Isfahan is a quasi-periodic tiling - a pattern with perfect long-range order that never repeats, mathematically parallel to Penrose tilings and to the atomic structure of quasicrystals. Medieval craftsmen, working with templates and compasses, had reached geometry five centuries ahead of its formal discovery.

How is a girih pattern actually constructed?

The standard method is called polygons-in-contact: tile the surface with polygons, then draw a pair of strap lines through the midpoint of every polygon edge, at a fixed angle to that edge. Where the lines from neighbouring edges meet inside each polygon, they intersect cleanly - because the angle is the same everywhere - and the surface resolves into stars where the polygons had corners and knots where they met. The polygons themselves are then erased; only the strapwork remains. English engineer Ernest Hanbury Hankin documented the method in colonial-era India in the early 20th century, and it remains the clearest account of how the patterns were made.

The single most consequential choice in the construction is the angle the straps make with the polygon edges - equivalently, how deep the star points cut toward each star's centre. The tradition settled on values that make the strap segments align dead straight as they cross from figure to figure. In the studio's engine these are carried as exact constants: 1/√3 for six-fold work, √2−1 for eight-fold, 1/φ² for ten-fold and 2−√3 for twelve-fold. Read more in How Islamic geometric patterns are constructed.

Girih today

Girih never stopped being built - restoration workshops in Fès, Isfahan and Istanbul still cut it by hand - and it is enjoying a design revival in contemporary interiors, from screens and joinery to tiled feature walls. What has changed is the drafting. Zakhraf Studio's engine performs the polygons-in-contact construction live in the browser: choose a symmetry family, a girih mode and a depth, and the same geometry that Hankin documented resolves before your eyes - then exports as print-ready SVG or a fabrication cut schedule. Every figure in this article was drawn that way, on this page, as you loaded it.