Is the scale between the large and small rosettes a design choice?

No. The ratio is forced by the geometry. The small rosette's connectors must meet the host cell's corners at the small rosette's own edge length, and that single requirement is one equation with one admissible solution. For a square host cell the ratio is a specific irrational near 0.341; for a triangular host it is exactly one third. The scale is derived, never dialled in, which is why the nesting is correct and not merely plausible.

The mathematics · 30 August 2026

Patterns inside patterns.

Q: What is compound or hierarchical zellige?

Compound zellige is Islamic tilework built at more than one scale at once: a field of large rosettes whose in-between connecting cells are themselves subdivided into a small tiling that carries smaller rosettes. The effect, associated above all with the workshops of Fez, is a pattern you can read close up and far away, each distance showing a different order of the same geometry. It is distinct from a two-level pattern that simply re-runs a motif inside its own faces; compound work rebuilds the underlying tiling itself.

The richest Islamic tilework rewards two distances at once. Stand back and you read a field of large rosettes; step close and each of the cells between them turns out to hold a smaller rosette of its own. This is compound zellige, the manner of the Fez workshops, and the interesting thing is not that it nests but that the geometry, not the craftsman, decides how.

A plain pattern solves one tiling. A compound pattern solves a tiling, then goes back into the spaces left between its figures and solves a smaller tiling inside each of them, seating a subordinate rosette in the middle. Do it once and you have the two-scale plates of Fez and Marrakesh. Do it again on the new in-between cells and you have the dense, close-work surfaces that reward a magnifying glass.

FIG. 1 - A compound field, drawn live. The large rosettes carry the field; the smaller rosettes in the connecting cells are the second order, seated at a scale the geometry fixes.

A scale you don't get to choose

Here is the part that matters. To seat a smaller rosette inside a host cell, the short connectors that join the host's corners to the little rosette have to be the same length as the little rosette's own edges - the same rule of equals-meet-equals that governs the whole tradition. That single requirement is one equation, and it has one admissible answer. There is no slider for it.

When the host cell is a square, the ratio between the small rosette and the large one is a particular irrational number, close to 0.341, that falls out of a quadratic. Pick anything else and the small rosette pokes through the wall of its host.
When the host cell is a triangle, the equation collapses to something simpler and the answer is exactly one third - the small rosette lands flush, splitting each edge of the triangle into thirds. That is the same trisection that keeps appearing across the whole engine, showing up here first.

This is derived, not generated at its most concrete. An image model can paint something that looks nested, but it has no equation to satisfy, so its small rosettes are the right idea at the wrong size - close on inspection, never exact. Here the nesting is correct because the scale is the root of a real constraint. The pattern is not imitating hierarchical zellige; it is constructing it.

FIG. 2 - The same instinct, one step grander: whole framed medallions seated into a running field, each frame's radius chained from a single ratio so no two frames can ever touch.

COMPOUND ZELLIGE, IN ONE SENTENCE - solve a tiling, then solve a smaller tiling inside the gaps, at the one scale that lets the pieces meet as equals.

Nested, and still cuttable

Detail is cheap to draw and expensive to make, so the test of a compound pattern is whether it survives contact with a workshop. This one does. Every subordinate cell flows through exactly the same pipeline as the field around it: each piece is sorted into a congruence class, the cut schedule counts how many of each a maker needs, and the export reports the smallest piece at the deepest scale, so nobody commits to cutting a tile finer than their tools allow. The nesting adds orders of detail without adding a single exception. That is the difference between an ornament that photographs well and one you can put on a wall.

Read on

The seal that weaves itself: how Islamic strapwork decides over and under Interlace is the easiest thing in the tradition to fake and the hardest to prove. Here is how the geometry settles it.

Build a compound field yourself.

Nest the rosettes, recolour by class, and export the cut schedule - free to explore in the browser.

Open the studio

Common questions

What is compound or hierarchical zellige?

Islamic tilework built at more than one scale at once: a field of large rosettes whose in-between cells are subdivided into a small tiling carrying smaller rosettes. Associated above all with the workshops of Fez, it is a pattern you read close up and far away, each distance showing a different order of the same geometry.

Is the scale between the rosettes a design choice?

No. The small rosette's connectors must meet the host cell's corners at the small rosette's own edge length, and that single requirement is one equation with one admissible solution - a specific irrational near 0.341 for a square host, exactly one third for a triangular one. The scale is derived, never dialled in.

Does a compound pattern still export to fabrication files?

Yes. The subdivided cells flow through the same pipeline as a plain field: every piece is grouped by congruence class, the cut schedule counts them, and the export reports the smallest piece at the deepest scale. Nesting adds detail, not exceptions.