What is interlace or strapwork in Islamic geometry?

Interlace is the treatment that turns the lines of a geometric pattern into ribbons that weave over and under one another, so the design reads as an endless plaited band rather than a flat network. It is one of the signatures of Islamic ornament, seen in the strapwork of the Alhambra and the woven seal stars of Cairo and Isfahan. The lines themselves are the ordinary geometric construction; interlace is the rule for which ribbon passes over which at every crossing.

The mathematics · 30 August 2026

The seal that weaves itself.

Q: Is the over-and-under drawn by hand?

Not in Zakhraf Studio. The weave is decided by the geometry, not placed by eye. For the khatam seal, a single rule settles every one of its sixteen crossings, and the rule is proved never to contradict itself: because the star is mirror-symmetric, the two strands at any crossing always sit at positions that sum to an odd number, so one is always even and one always odd. The rule 'pass over at your even crossings' therefore never asks both strands to go over. Consistency comes from the symmetry, not from patching conflicts by hand.

Q: What is the khatam seal?

The khatam is the eight-pointed seal star, one of the most recognisable figures in Islamic geometric ornament, formed by joining the midpoints of an octagon's edges in a fixed skipping pattern. It appears across the tradition, from the marble dados of the Sultan Hassan complex in Cairo to Safavid tilework in Isfahan. Its interlaced form weaves its eight straps into a single continuous seal.

The thing that turns a flat geometric network into something that looks plaited from a single endless ribbon is interlace: at every crossing, one strand passes over and the other under. It looks like a matter of taste. It is not. It is the easiest thing in the whole tradition to fake, the hardest to get right everywhere at once, and in this engine it is decided entirely by the geometry.

FIG. 1 - The khatam seal, woven, drawn live. Follow any one ribbon and it passes over, then under, then over, without a single break across the whole field.

The easiest thing to fake

Drawing one crossing is trivial: pick a strand to go over, break the other behind it. The difficulty is consistency. A ribbon that goes over at one crossing must go under at the next, all the way along its length, and it shares every crossing with a second ribbon that has its own such demand. Satisfy them locally, crossing by crossing, and you can paint yourself into a corner three crossings later where a strand is forced to pass over itself, or two ribbons both claim to be on top. The weave quietly breaks, and the eye - which is very good at reading woven things - catches it at once.

This is exactly the kind of thing an image generator gets wrong. It produces something that looks like weaving locally, because it has seen a great deal of weaving, but it is not solving the global constraint, so across a real field the over-and-under drifts out of true. It has no ribbon to keep honest.

One rule, and a proof it can't fail

The engine never places a single crossing by hand. For the khatam seal it uses one rule - a strand passes over exactly at its even-numbered crossings - and then proves that rule can never contradict itself. The proof is a piece of pure symmetry, and it is genuinely lovely:

the seal is mirror-symmetric, so the two strands meeting at any crossing always sit at positions that add up to an odd number. If two whole numbers add to something odd, one of them is even and the other is odd - always. So at every crossing, exactly one strand is at an even position and exactly one is at an odd one. The rule "go over when you are even" therefore never asks both strands to go over, and never asks both to go under, anywhere in the pattern. Global consistency falls straight out of the star's symmetry, with nothing left to check by hand. Sixteen crossings on the seal, all settled by one sentence.

INTERLACE, IN ONE SENTENCE - the weave is not a drawing decision but a consequence of the pattern's symmetry, so it is correct at every crossing, at any size, in every export.

FIG. 2 - The same discipline on a twelve-fold field. The weave holds across the whole surface because it is computed, not decided.

The same reasoning generalises to the whole family of seal stars, so turning on interlace for a new one is a matter of applying the rule, not of hunting crossing by crossing for a consistent solution. And because the over-and-under is part of the geometry rather than a layer painted on top, it survives everywhere the pattern goes: the woven ribbon you see on screen is the woven ribbon in the print vector and the fabrication file, without a redraw. That is what it means for a pattern to be derived, not generated - even its weave has a reason.

Read on

The bones of the Alhambra: curved cut-tile work, still compass-and-straightedge Not every Islamic pattern is made of straight lines. The one that is all curves is still, provably, compass work.

Weave a seal yourself.

Turn on interlace and follow a single ribbon across the field - free to explore in the browser.

Open the studio

Common questions

What is interlace or strapwork?

Interlace is the treatment that turns the lines of a geometric pattern into ribbons that weave over and under one another, so the design reads as an endless plaited band. The lines themselves are the ordinary construction; interlace is the rule for which ribbon passes over which at every crossing.

Is the over-and-under drawn by hand?

Not here. The weave is decided by the geometry. For the khatam seal a single rule settles all sixteen crossings, and it is proved never to contradict itself: the star's mirror symmetry means the two strands at any crossing always sit at positions summing to an odd number, so one is always even. Consistency comes from the symmetry, not from patching conflicts by hand.

What is the khatam seal?

The eight-pointed seal star, one of the most recognisable figures in Islamic geometric ornament, formed by joining the midpoints of an octagon's edges in a fixed skipping pattern. It appears across the tradition, from the marble dados of the Sultan Hassan complex in Cairo to Safavid tilework in Isfahan.