Four stars, from Baghdad to Samarkand.

Four star patterns, recorded at a madrasa in Baghdad, a street of tombs in Samarkand, a madrasa in Cairo and a sultan's mausoleum at Soltaniyeh, are now exact constructions in the studio, each rebuilt from a drawing. Most of what fixes them can be seen in the drawings themselves; at Baghdad, two proportions are a draughtsman's choice. Here is which is which, and what the drawings leave unsaid.

Read from a drawing

Each pattern begins as a drawing, redrawn line for line from a photograph or a published plate and titled with the building where it was recorded. The studio does not trace it. It finds the grid the pattern repeats on, and the few lines from which every other line follows by reflection. What is left is a handful of open distances, and the work is to find what settles them.

Three things can. Lines can meet, a corner landing exactly where the drawing shows it. A star can be regular, every point alike. Or two lengths can be made equal. The first two can be seen in the drawing, and the rest follows by proof. The third, when nothing else enforces it, may be the draughtsman's own decision, and the studio calls it a construction choice wherever it uses one. Nothing in these four is measured, and each construction, laid over its drawing, falls on it line for line.

Baghdad: a pair of choices

The Mustansiriya in Baghdad, a madrasa where all four Sunni schools of law were taught, was founded by the Abbasid caliph al-Mustansir in 625 AH (1227-28) and opened in 631 AH (1234). The sources differ on the year of foundation; these dates follow Archnet. The drawing gives the pattern in the madrasa's open brickwork, after a photograph in Henri Stierlin's Islam: Early Architecture from Baghdad to Cordoba (1996, vol. 1, p. 223), and in a palace in the Baghdad citadel, after Jay Bonner's Islamic Geometric Patterns (2017, fig. 333b). It does not say where in the madrasa the brickwork is.

Twelve-pointed stars alternate with octagons on a square grid. Two distances are open, the sizes of star and octagon, and the usual way to settle them fails here: nowhere in the drawing do three lines meet at one point. What the drawing does show is that its edges come in whole multiples of one length: the star's edge, a run of strap and the octagon's edge stand as 2 : 3 : 4. Two of those equalities fix the whole pattern.

Those two are construction choices. They fit the drawing far better than chance would explain, but a good fit is still not proof, and nothing else in the pattern forces them. Whoever drew it gave its edges these lengths. The studio uses the equalities in place of measurements, and calls them what they are.

The drawing gives each shape of tile its own colour, six in all, in bright diagram tones. The preset softens them into the colours of baked brick, terracotta, sand and sage on a cream strap; they make no claim about the brickwork itself.

Samarkand: two regular stars

The Shah-i Zinda in Samarkand is a street of tiled mausolea leading to the shrine of Qusam ibn Abbas, most of them built in the fourteenth and fifteenth centuries, many for Timur's family and court. The drawing is titled with its name, and its source cites a plate in Hrbas and Knobloch's The Art of Central Asia (1965, pl. 62, dated 1360) and a photograph by David Wade at the site. Neither names a mausoleum. The same source gives the pattern at eleven other monuments, from Cairo to Thatta, Soltaniyeh among them, so the title is a way of finding the drawing rather than a claim about one building. The only dated reference falls before Timur's rise, and the studio calls the pattern Timurid and earlier.

The pattern shares the Mustansiriya's square grid but is fixed another way. Twelve-pointed stars alternate with eight-pointed ones, and each small star is drawn by the lines of its four large neighbours. On a square grid the large star need not be regular, since the grid ties its twelve lines together only in three separate sets. Making both stars regular leaves just one possible size. Both are plain in the drawing, and the source calls both stars regular. Nothing here is a choice.

The colours are the drawing's. With no mausoleum named, there is no glaze to compare them with.

Cairo: a floor in ninths

The madrasa and khanqah of Sultan al-Zahir Barquq, the first Circassian Mamluk sultan, was built in Bayn al-Qasrayn, Cairo, in 786-788 AH (1384-86). The drawing's source places the pattern on the floor before the mihrab in the qibla iwan, citing a Thesaurus Islamicus photograph, with Bernard O'Kane's The Mosques of Egypt (2016); it also gives the pattern at a mosque in Hama. The place on the floor rests on that photograph's caption alone. The photograph has not been seen, no description of the floor has been found, and the medium is not stated.

Here the grid is triangular. Each six-pointed star has a hexagon at its heart, and its edges, carried on past the points, close a rhombus between neighbouring stars. Two conditions fix it. The small triangle in each gap between three stars lands its corner exactly on the blunt corner of the rhombus, so that four tiles meet at one point; and the rhombus takes the star's edge, so that every short edge is one length. Each star's points then reach exactly a third of the way to the next. Nobody set that third; it falls out of the two conditions, as it does elsewhere in the tradition.

Unlike the Baghdad pair, this equal length does not stand alone: every simple relation among the drawing's lengths gives the same answer, and every corner of the drawing falls on a grid of ninths. The preset softens the drawing's grey, orange and tan to slate, rose and cream, the colours of a Mamluk marble floor; the floor's own stones are unsourced.

Soltaniyeh: everything in eighths

The mausoleum of the Ilkhanid sultan Oljaytu at Soltaniyeh, built in 1302-12 by UNESCO's dating, is an octagonal brick tomb under a dome about 50 m high, faced outside in turquoise glazed tile. The drawing is after a photograph by Aslam Qureshi, and its source does not say which part of the building the photograph shows. The one date the source attaches to it, 1308, is the date the same correspondent gives a hospital portal at Amasya, and it is not used here.

The pattern is six-fold, like Barquq's, and more austere, its lines running in only three directions. Three conditions fix it, one of each kind: hexagons touch corner to corner between the stars, the chevrons round each star have equal sides, and the hexagons are regular, as the source states. Every short edge is then an eighth of the distance from star to star, and every corner falls on a grid of eighths, as every corner of the drawing does.

The drawing has no border, and the studio carries the pattern on past its edge; whether the work at Soltaniyeh is a finite panel, the drawing cannot say. The white and slate blue are the drawing's colours, and the glaze is unsourced.

What the drawings do not say

The geometry of the four is settled; their history is not. The Mustansiriya drawing does not place the brickwork in the madrasa; the Shah-i Zinda drawing names no mausoleum; the Barquq floor rests on the caption of a photograph not yet seen; the Soltaniyeh drawing names no part of the building and carries another building's date. The studio gives the place each drawing gives, with its source, and claims no more.

For readers who want it, each preset's card in the studio gives the full construction, and the mathematics page sets out the proofs.