Muqarnas

A muqarnas is the stalactite vault of Islamic architecture: tiers of small carved cells, corbelled ring over ring, that turn a dome's underside into stepped concavities of light. This piece raises one — the vault implied by a thirteenth-century construction drawing — and lets you watch every step of the raising, because the subject of the piece is precisely that the raising is not in the drawing.

What is constructed, and what is read

No vertex is traced. Every element is generated from an alphabet of eight — two seeds, the square and the 45° rhombus, and their divisions and remainders — placed on the coordinate module (ℤ + ℤ·(√2/2))², the integer representation Harmsen builds for exactly this geometry, and computed in exact arithmetic over ℚ(√2), so the tiling closes with zero tolerance rather than within one.

The plate's arrangement is a different kind of fact, and it is not invented here. Which letter sits where, at what orientation, is digitised from the vector line work of Harmsen's fig. 5.17(a) by a tool kept in this repository: it snaps each coordinate to the lattice, splits the segments at their T-junctions, extracts the faces of the resulting subdivision, and classifies every face as one of the eight letters. What it records is a kind and an isometry — never a raw vertex — so the library still constructs each outline itself. Said plainly, because the distinction matters: the alphabet and all of its geometry are generated; the plate's arrangement is digitised from a published drawing and regularised onto the lattice.

Every measurement descends from one module, al-Kāshī's miqyās: the profile curve, the tier heights, the coefficient the plaster is billed in — all of it is that one length, repeated with knowledge.

The drawing

The canonical example is the thirteenth-century gypsum plate found at Takht-i Sulaymān, published and read in Muqarnas 22: a quarter-vault plan that this digitisation — its lines regularised and restored — resolves into 157 elements. That count is this reconstruction's reading, not a figure the publication states. No vault matching the plate survives at the site; the drawing may record a design never built — so everything raised here is reconstruction, not restoration. Turned four times about the centre and stitched with four seam-straddling diamonds — elements no quarter representation can hold — it spells the full sky with 4 × 157 + 4 = 632 letters, using only four of the eight kinds. The digitisation here surfaced a quiet fact: the regular field span is 7 + 3.5√2 ≈ 11.9497, but the printed plate stretches one band to make the span an even 12 — a deformation of exactly the remainder, 5 − 3.5√2 — about 1.8 mm on the plate, and the subject of a short note setting out the measurement and its limits.

The method

The profile every cell wears is al-Kāshī's "method of the masons" (Miftāḥ al-Ḥisāb, IV.9): strike a 30° oblique, divide it in five, fold two fifths onto the vertical — which forces the facet height 2 − (3/5)√3, nobody chose it — then one sixth of a circle of radius four fifths, and a ramp. Measured whole-and-half along its length — the facet whole, the curved part half — the construction's surface coefficient comes to 1;43,33,48,40 module-lengths: multiply a cell's summed facet-bases by it and you have the cell's surface. The manuscript tables 1;43,33,45,41; the discrepancy enters through al-Kāshī's value for the vertical GH, isolated by Dold-Samplonius five centuries later — and she thought a plain miscalculation unlikely, reading it rather as a mason's working value. The piece shows both numbers, because the record's texture is part of the record.

The ambiguity

The plan does not determine the vault. A solver enforces the buildability rules — every ring standing on the roofs of the ring below, heights propagating without divergence — and returns all the valid readings of its family, not the first. This site works in the plate's corner-starting family, the reading Harb proposed, where the first tier stands two cells to a corner: within it the solver finds completions rising through seventeen tiers to the crown ring in one reading and eighteen in another. (Other families are documented — Yaghan's and Dold-Samplonius's twelve-tier reading rests on removing nodes this alphabet keeps.) The climax of the piece descends one building into the flat drawing and raises a different building out of it, because at height zero both are the drawing. The master's knowledge was never fully in the plan.

The light and the colour

A muqarnas hangs over an opening, so light never falls on it from the sky: it bounces up warm from the court and rakes in low through the opening. The whole lighting of the piece is three fixed states of that language, and its eighth scene moves nothing but the sun. The cells' curved canopy wears a fired turquoise wash that sinks toward cobalt in its depths. This polychromy is an interpretation, not a record: what survives of Takht-i Sulaymān's decoration is glazed and lustre-painted tilework elsewhere in the palace, and how this vault's cells were finished — if the vault was built at all — is not documented. The glaze here borrows the palace's palette, and is colour and gloss only: no metal, no glow that the light did not put there.

Sources

• Yvonne Dold-Samplonius, “Practical Arabic Mathematics: Measuring the Muqarnas by al-Kāshī,” Centaurus 35 (1992).
• Silvia Harmsen, Algorithmic Computer Reconstructions of Stalactite Vaults — Muqarnas — in Islamic Architecture, dissertation, Heidelberg (2006).
• The Takht-i Sulaymān plate and its reading, Muqarnas 22 (2005).
• Ghiyāth al-Dīn Jamshīd al-Kāshī, Miftāḥ al-Ḥisāb (1427), book IV, chapter 9.

Colophon

TypeScript throughout; exact rational arithmetic over ℚ(√2) with bigint fractions; the tier solver and the lift are pure geometry with 184 tests; rendering is three.js WebGPU with TSL node materials, per-vertex occlusion baked against the real geometry, and GSAP ScrollTrigger under a Lenis scroll. The sister piece is From the Point — this site ends where that one begins.