Eight shapes.
Square, half-square, rhombus, half-rhombus, almond, jug, large biped, small biped. Every muqarnas ceiling ever built — from tenth-century Iran to the Alhambra — assembles from these eight. The angles between them are multiples of 45°. Al-Kashi wrote it down in 1427: "a right angle, or half a right angle, or their sum, or another combination of these two."
That's the entire vocabulary. Not a simplification. The actual constraint. Eight shapes and one angular rule produce ceilings that look like frozen cascades, stone stalactites, geometry dreaming about itself. The combinatorial space is infinite. The alphabet is not.
The compass can't open.
Abu'l-Wafa, Baghdad, sometime after 990. He writes A Book on Those Geometric Constructions Which Are Necessary for a Craftsman. Thirteen chapters. Regular pentagons, octagons, decagons — all constructed with a straightedge and a compass whose radius is fixed. The compass is "rusty." It doesn't adjust.
The reason is precision. A compass that opens and closes introduces error at every adjustment. A fixed compass eliminates that class of mistake entirely. You lose flexibility. You gain reliability. The constructions are harder to find but easier to trust.
Abu'l-Wafa documents the mistakes artisans make: approximations that work at small scale but compound at architectural scale. He documents how artisans mask slight imperfections. The mathematician sees the error. The artisan sees the building. They need each other. In Baghdad, they met regularly — mathematicians and craftsmen in the same room, working on the same surfaces.
The surfaces have wages.
Al-Kashi didn't measure muqarnas for the beauty of measurement. Craftspeople were paid per cubit. A muqarnas ceiling has a surface area vastly larger than its footprint — all those facets, all those cells stepping down in tiers, each one a small plane tilted at a precise angle.
Measuring it wrong means paying wrong. Al-Kashi developed formulas for computing the surface area of each cell type and summing them across tiers. The module — the base of the largest facet — serves as the unit. Everything scales from it. The mathematics existed because someone needed to write a check.
Nothing is left over.
The eight cells tile. Every cell's plan projection fits against every other cell's plan projection without gaps. The 2D plan maps directly to the 3D form because nothing overlaps in projection. This is what makes muqarnas buildable: you draw the plan, and the ceiling follows. No translation step. No interpretation. The constraint that limits the vocabulary is the same constraint that guarantees the assembly.
The plan is the building. The building is the plan. The compass is rusty and the ceiling is exact.