Three Anatolian nets.

Three patterns from Anatolia, at Malatya, Kayseri and Erzurum, are now presets in the studio, each rebuilt from a drawing after a figure in a published book, so that every line sits where it does for a stated reason. A fourth, from a hospital in Sivas, is drawn but held back: one of its proportions can only be measured off the page.

From a drawing to a net

Each pattern is read from a modern line drawing made after a figure in a published book: for Malatya, Oktay Aslanapa's Turkish Art and Architecture (1971, fig. 24); for Kayseri, Erzurum and Sivas, Jay Bonner's Islamic Geometric Patterns (2017, figs 146, 139 and 109d). The studio reads the drawing, not the building. The captions given for the four name no material; two do not say where in the building the pattern is, and a third names the drum of a dome without saying which. The colours in the figures are therefore the draughtsmen's, and make no claim about brick or glaze.

The patterns themselves are nets of straight straps running in only a few directions. The symmetry settles which way the lines run but not where they sit, and each position has to be pinned by something that can be checked: three lines through a single point, a star drawn regular, or two lengths made equal.

Each proportion then stands in one of three ways. It is proved when it follows from something the drawing plainly shows, and a choice when the drawing agrees with it but does not force it, a rule a draughtsman might have worked to. It is measured when nothing fixes it and the number is read off the page, and the studio keeps measured values out of its presets until they are proved or deliberately accepted.

Malatya: one length throughout

The Great Mosque of Malatya stands at Battalgazi, the old town of Malatya. It is usually dated to 621 AH (1224) under the Seljuk sultan Kayqubad I, from an inscription now in the Malatya Museum, though the Evkaf (pious foundation) records place the building under Kaykaus I, who reigned from 1211 to 1220. A gate inscription of 645 AH (1247) records later work. The brick iwan, the great vaulted hall open at one end, is framed in glazed tile mosaic and bears the signature of Ya’qub ibn Abi Bakr of Malatya. Aslanapa's caption places the pattern in the spandrel of the arch on the right-hand side of the iwan; the source does not say whether that spandrel is of 1224 or of 1247.

The stars sit on a triangular grid. Round each stand two hexagons, one reaching three-fifths and the other four-fifths of the way to the next star, and every line in the pattern is a side of one of them. Their overlaps cut chevrons round each star, with diamonds and small triangles between.

Two things fix those sizes, and the drawing shows both: halfway between two stars three lines pass through one point, and each chevron has equal sides. So the pattern is proved, and unusually tidy. Every edge has the same length, and every shape, from the stars to the diamonds, is built from small equilateral triangles.

The drawing is a flat mosaic in blue and tan, and so is the preset. It could not be woven in any case: a weave carries one strap over another, and where Malatya's lines meet there are three.

Kayseri: a star and two choices

The Çifte Medrese in Kayseri, the "twin madrasa", pairs a hospital with a madrasa beside it. The hospital was founded under the Seljuk sultan Kaykhusraw I at the bequest of his sister Gevher Nesibe, and an inscription naming her is dated 602 AH (1205-06). Bonner gives the pattern for the complex; the source does not say which part of the complex carries it.

Here crosses and eight-pointed stars alternate on a square grid, like the squares of a chessboard. Each star is two squares laid one over the other, their sides running on as straps, and eight more straps draw an octagon round it. Pentagons, hexagons, kites, triangles and small bowties fill the rest.

The star is regular, as the drawing shows, but two more things are needed to set its size, and both are choices. One makes each bowtie fit exactly inside a square. The other makes the strap from a cross's inner corner to a star's point four times the star's radius.

Nothing on the page forces either, as three lines meeting at a point would. They are proportions a draughtsman could have worked to, and of the simple rules that might have set the pattern they are the ones that fit, closely enough that luck is an unlikely explanation. That is good evidence, still short of proof, and if a better source disagreed, these are the two that would give way.

The drawing is woven and so is the preset, its over-and-under matching the drawing's at every crossing compared.

Erzurum: squares and octagons

The Great Mosque of Erzurum is the earliest of the three buildings by the dates given for them, and the only one that is not Seljuk. It was built for the Saltukid ruler Nasir al-Din Muhammad in 575 AH (1179-80), a date taken from a foundation inscription now lost. Repairs from 1639 to the 1960s rebuilt parts of it, and the Saltukid fabric survives mainly in the qibla wall, the wall that faces Mecca, and part of the central aisle. The source does not say which part of the mosque carries Bonner's pattern, so the pattern's own date is not established: it may belong to the first building or to any of the repairs.

The grid is square again, with centres of two kinds alternating, and every line runs level, upright or on a diagonal. The straps form closed loops, a regular octagon round every centre and a long octagon round every second one, and where the loops cross they frame small squares, some lined up with the grid and some turned.

Both things that fix the pattern are plain in the drawing: the small squares are all one size, and the octagon round each centre is regular. So it is proved, and built on very few measures. Every stretch of strap between crossings is one or two lengths of a single unit, or the side of a small square, and the rectangles between the loops have the proportions of a sheet of A4.

The drawing is woven, but the studio's weave would take every crossing the opposite way, so the preset lays the straps flat, in the drawing's light blue, dark blue and green.

Sivas: one number still measured

The hospital that the Seljuk sultan Kaykaus I founded in Sivas is dated by its portal inscription to 614 AH (1217-18), and the sultan lies in the tomb built into it. Bonner's caption places the pattern on the drum of the dome, the ring of wall the dome stands on; the source does not say which dome.

The pattern is six-fold. A small hexagon stands at the centre of each group of three stars and a larger one round each star, and every strap is a side of one of them. Where they cross they leave a twelve-pointed star with right-angled points at every centre.

The star is regular, as the drawing shows; the other condition is unusual. In the ideal pattern three straps would cross in a single point at the centre of each group of three stars, which cannot be woven, so the drawing moves each line out by half a strap's width and lets the straps' inner edges meet on that point instead. That makes the width of the strap part of the geometry: change it and the star changes too.

That width is the one number in the four patterns that nothing in the drawing settles, so it is measured, from more than six hundred readings of the straps' edges. It is almost exactly the width of the pen the straps were drawn with, which suggests it belongs to the drawing rather than to the pattern. Yet it cannot be dropped. Without it the pattern fits the drawing no better than chance; with it, the two agree line for line.

So the studio draws the Sivas pattern, on the Twelve Star button with the measurement stated in its hint and its note, but does not yet offer it as a preset. The preset waits for a condition that fixes the width, or for the measurement to be accepted, and the answer may have to come from the building, once a photograph of the drum can be read against the drawing.

Proved, chosen, measured

• Malatya, Double Hexagon. Every line follows from what the drawing shows. Proved.
• Kayseri, Star and Cross. A regular star and two choices the drawing agrees with. Exact, but chosen.
• Erzurum, Linked Octagons. Equal squares and a regular octagon, both shown. Proved.
• Sivas, Twelve Star. Exact except for the width of its strap, which is measured. Drawn, not yet a preset.

A rebuilt pattern says nothing about how a mason at Malatya or Erzurum set the design out, and the buildings themselves, their materials unstated and, at Erzurum, the pattern undated, remain the larger question. What it settles is what the studio draws: every corner placed by a stated rule, every proportion marked proved, chosen or measured. Derived, not generated, with the standing of each number stated.

For readers who want it, each preset's card in the studio gives the full construction, and the mathematics page explains how the studio works.