Islamic Geometric Patterns: Compass, Straightedge and Girih Tiles

Everything starts with a circle

From roughly the 8th century onward, geometric ornament became one of the dominant decorative languages across the Islamic world, appearing on mosques, madrasas, palaces and manuscripts from Spain to Central Asia. The patterns are built almost entirely with compass and straightedge, starting from a circle divided into equally spaced points. Connecting those points in different ways — skipping one, skipping two, skipping three — generates the whole family of star shapes that anchor the design, before the pattern is extended by translation and rotation to tile an entire wall or ceiling.

Star polygons: the {n/k} construction

A star polygon is described by two numbers, written {n/k}: place n points evenly around a circle, then connect every k-th point in order until you return to the start. The most common Islamic star shapes are {8/3} (an 8-pointed star), {10/3} and {12/5}. Because k and n share no common factor, the connecting line visits every point exactly once, tracing a single continuous, self-intersecting path rather than several separate polygons.

for i in 0..n-1:
    angle_i = 2*pi * i / n
    point_i = (cos(angle_i), sin(angle_i))

connect point_i to point_(i+k mod n), for every i   // the {n/k} star

{8/3}: 8 points, skip 3 each time  → classic 8-pointed rosette
{10/3}: 10 points, skip 3 each time → decagonal star, very common
gcd(n,k) must equal 1, or the path splits into gcd(n,k) separate loops

From one star to an infinite pattern

A single star is only the seed. The craftsman surrounds it with a lattice of secondary points — often the vertices of the polygon connecting the gaps between star points — and repeats the same construction around each one, then trims every line segment to a fixed length so that stars and the polygons between them interlock without gaps. Because the underlying point lattice is periodic (typically based on a square, triangular or hexagonal grid), the whole pattern can, in principle, extend forever, repeating over one of the 17 wallpaper symmetry groups. Which symmetry group a pattern uses can be identified from its rotational and reflective symmetries alone, independent of the specific star shapes drawn within it.

Girih tiles: the workshop shortcut

Later craftsmen, from roughly the 12th century onward, developed a faster method that avoided redrawing the compass construction for every tile: a set of five edge-matching polygons — a regular decagon, pentagon, hexagon, elongated hexagon (bowtie) and rhombus — each pre-marked with straight guide lines called girih (Persian for "knot"). A craftsman could tile a wall with these five shapes edge to edge, matching the guide lines across every seam, and then draw only the guide lines in the final artwork, leaving the underlying tile outlines invisible. The result reads as a single continuous interlaced strapwork pattern, even though it was assembled from a small, reusable kit of parts.

A 2007 discovery: proto-quasicrystals

In 2007, physicists Peter Lu and Paul Steinhardt analysed girih patterns on buildings including the Darb-i Imam shrine in Isfahan (built 1453) and found that some tilings used subdivision rules matching the mathematics behind Penrose tiling — a non-repeating, quasi-periodic pattern not formally described in the West until the 1970s. It suggests that some medieval craftsmen had, through purely geometric trial and refinement, discovered principles of quasi-periodic tiling centuries before they were understood mathematically, though this quasi-periodic technique was one specialised strand within a broader tradition that remained overwhelmingly periodic.

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