How Islamic geometric patterns are constructed.
No pattern in this tradition was ever drawn freehand. Each one is the output of a construction - a sequence of compass-and-straightedge moves that any craftsman could repeat exactly. Here is the method, from underlying grid to finished strapwork, including the exact star depths the tradition favours and why only certain symmetries can tile a wall.
Islamic geometric patterns are constructed in three moves: lay an underlying polygonal tessellation, draw strap lines through the midpoint of every polygon edge at a fixed angle, then erase the polygons - the strapwork that remains is the pattern. This is the polygons-in-contact method, documented by E. H. Hankin from Mughal buildings in the early 20th century and consistent with the 15th-century construction drawings of the Topkapi Scroll. Everything else - interlace, colour, two-level patterns - is elaboration on that skeleton.
Move one: the underlying tessellation
Every pattern begins with a tiling the viewer will never see: a grid of triangles, squares, hexagons or richer combinations (octagon-and-square, kagome), chosen for the symmetry the finished work should carry. The choice is constrained by a hard mathematical fact - the crystallographic restriction: only 2-, 3-, 4- and 6-fold rotations can repeat periodically. Six-, eight- and twelve-fold stars therefore sit comfortably on repeating lattices, while true five- and ten-fold symmetry cannot repeat at all - the tradition's quasi-periodic girih arrangements are its ingenious way around that limit.
Move two: rays at the edges
Through the midpoint of every edge of the hidden tessellation, draw a pair of lines at a fixed angle to the edge. Because the angle is the same everywhere, the lines launched from neighbouring edges meet cleanly inside each polygon: at polygon corners they resolve into stars, along runs of edges into straight straps, and at meeting points into knots. Extend every segment until it hits its neighbours, erase the scaffolding, and the pattern is complete as pure linework.
Move three: depth - the one number that changes everything
The angle of those edge rays - equivalently, how deeply the star points cut toward each centre - is the construction's single free parameter. Any angle produces a pattern; the tradition settled on the specific values that make strap segments run dead straight from figure to figure, so the eye reads continuous lines across the whole wall. Stated as modern constants:
| Symmetry | Depth | Value | Where you have seen it |
|---|---|---|---|
| 6-fold | 1/√3 | 0.5774 | Hexagram lattices, jaali screens |
| 8-fold | √2 − 1 | 0.4142 | Star-and-cross, khatam fields |
| 10-fold | 1/φ² | 0.3820 | Decagonal girih, Darb-i Imam |
| 12-fold | 2 − √3 | 0.2679 | Great Seljuk and Mamluk panels |
None of these numbers is arbitrary: each is the tangent of the angle that aligns the straps for that symmetry family - the ten-fold value even involving the golden ratio, which is intrinsic to the decagon. The craftsmen never wrote the algebra; their compasses enforced it. The studio's engine carries these as exact constants and calls them harmonic depths; you can drag the depth away from them and watch the alignment dissolve, which teaches the rule faster than any equation.
Elaborations: interlace, colour, two levels
- Interlace widens each line into a ribbon and weaves the crossings alternately over and under. The weave is not an artistic choice at each crossing: alternation is forced by the geometry, the way a woven thread must alternate.
- Colour follows zellige practice: faces are grouped by congruence class, one colour per class, with no two adjacent faces sharing a colour - see What is zellige? for where that discipline comes from.
- Two-level patterns repeat the whole method at two scales: a large-scale construction whose straps are themselves filled with a small-scale pattern. The Topkapi Scroll records several; they remain the tradition's high-wire act.
Watch it happen
Reading the method is one thing; watching it perform is another. The derivation animation on our home page shows a khatam field drawing itself stroke by stroke in construction order, then taking colour one congruence class at a time. And the studio runs the full construction live: choose symmetry, mode and depth, and the same three moves resolve under your hands - then export as SVG, seamless repeat or a fabrication cut schedule.
Common questions
What tools did the original craftsmen use?
Compass, straightedge and templates. Every construction reduces to circles and straight lines - the classical instruments of geometry - and workshops kept pattern knowledge as construction drawings and physical templates rather than as written theory. The Topkapi Scroll, a 15th-century Persian roll of construction drawings, is the most complete surviving record of this practice.
Do you need mathematics to construct these patterns?
You need geometry, not calculation. The historical method is entirely constructive: draw circles, divide them evenly, connect intersections. The craftsmen who built the great pattern walls worked from constructions and templates, not equations. The equations - star depths like √2 − 1 - are the modern way of stating exactly what the compass was already doing.
Can a star with any number of points tile the plane?
No. Only 2-, 3-, 4- and 6-fold rotational symmetry can repeat periodically - the crystallographic restriction. That is why 6-, 8- and 12-fold patterns dominate: they sit compatibly on triangular and square lattices. True 5- and 10-fold symmetry cannot repeat periodically, and the girih tradition's answer to that problem - quasi-periodic arrangements like the one on the Darb-i Imam shrine - anticipated the mathematics of quasicrystals by five centuries.
Run the construction yourself.
Symmetry, mode, depth - the same three moves, live in your browser, free to explore.
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