The mathematics behind every line.
Every pattern on this site is built from a stated geometric construction, not sampled from a picture. That one fact is the whole product: it is why the pattern on your screen is the pattern in your cut file, to the millimetre, and it is the one thing no image generator can copy. Here is what it means, and where to read the constructions in full.
Constructed, not sampled
An image model learns what Islamic patterns look like and paints something plausible. This engine does the opposite: it computes what the pattern is. Every vertex is the exact intersection of two lines whose positions follow from the polygons-in-contact method the tradition itself used, and every proportion is an exact irrational number rather than a tuned decimal, the way the eight-fold star carries √2−1 and the ten-fold star carries 1/φ². No number is nudged to make a picture look right, because there is no picture, only a construction.
A generated pattern can only ever be plausible. A derived one is correct, and it can prove it.
The difference you can bank on
Because a real construction sits underneath every pattern, three things follow that a generator cannot offer. It is reproducible: the same construction at any size, so what you preview is what your maker cuts, exactly. It is fabricable: minimum piece sizes and tool limits are read off the geometry and checked before an export saves. And it can be wrong, and refuse: when a construction fails its own test, the engine declines rather than fake it, which is precisely the move an image model has no way to make. There is nothing to be wrong about in a picture.
The deepest sign that these are the tradition's own constructions, not a modern imitation, is how often the tradition's own answers appear unbidden. Left to derive its own proportions, the geometry keeps arriving at the same quiet number the medieval workshops did, centuries before anyone wrote the equations down. That story is the first piece below.
Four constructions, one idea each.
The substance, told the way the journal tells things: one construction at a time, no equations required, every figure drawn live by the engine.
The number that keeps appearing
Nowhere is anything set to three. Yet across four separate constructions, the trisection keeps arriving as the answer.
Patterns inside patterns
The Fez manner: rosettes seated inside the cells of bigger rosettes, at a scale the geometry fixes, never chooses.
The seal that weaves itself
Over-and-under is the easiest thing to fake and the hardest to prove. One symmetry rule settles every crossing at once.
The bones of the Alhambra
Curved cut-tile work with a genuine compass provenance: every arc scribed from a single setting, no asterisk.
Common questions
How is this different from AI pattern generation?
An image model samples what patterns look like; this engine computes what they are. Every vertex is the intersection of lines that follow from a stated construction, and every proportion is an exact irrational, so the pattern can be checked, reproduced at any size, and cut to millimetre-exact fabrication files. A generator makes a picture; this makes geometry. Derived, not generated.
Do I need to understand the mathematics to use the studio?
No. Every symmetry opens at its canonically correct proportions and every construction ships with a short plain-language note inside the studio. This page and its reading series are for anyone curious about why the patterns hold together, never a prerequisite for making them.
The same constructions, under your hands.
Everything here runs live in the studio - free to explore, at every symmetry.
Open the studio