Derived from a plate.

In 1856 Owen Jones printed what he had measured at the Alhambra. Two of his Moresque plates are now constructions in this studio: a field of strapwork rings and Seals of Solomon on a hexagonal grid, and a field of eight-pointed roundels that somehow sits on a six-fold net. This is what it takes to turn a chromolithograph back into geometry - and what the printed page gives up when you do.

A page, not a wall

Owen Jones was an architect before he was a theorist of ornament. He went to Granada in 1834 with the French draughtsman Jules Goury, who died of cholera there that summer, and returned in 1837 to finish measuring the Alhambra. The results came out as Plans, Elevations, Sections and Details of the Alhambra, and then, in 1856, as the Moresque chapter of The Grammar of Ornament, the folio of colour plates that made his name. The plates were printed in chromolithography, one stone per colour, so what survives on the page is Jones's printed reading of the walls: his registration, his inks, his judgement about which pattern deserved a plate.

That matters for what follows. A plate is not a wall. It is a drawing that has already passed through one pair of eyes and a printing press, and when this studio rebuilds it the aim is not to trace the print but to find the rule the print is a picture of. Jones would have approved. His eighth proposition reads: "All ornament should be based upon a geometrical construction." The two plates below are the test of whether that is true of his own.

One lattice, two motifs

The first field looks busy and is simple. Its scaffold is the ordinary hexagonal tiling, and the only decision is a colouring: the hexagonal tiling can be coloured with three colours so that no two neighbours match, and that colouring is unique up to relabelling. Give one colour class the ring - a hexagon of straps turned through thirty degrees, so its sides cross the cell edges at their midpoints - and give the other two classes the Seal of Solomon, the six-pointed star whose two triangles also cross the cell edges at their midpoints. That is the whole construction.

Because both motifs cross every edge at the same point, at the same distance from the cell centre, the straps of a ring run straight on into the straps of the seal next door. The continuity is not drawn; it is a consequence. Every edge midpoint launches exactly four strap ends, two into each cell, and the faces that the straps cut out are only three kinds: triangles at the seal tips, pentagons between ring and seal, and the hexagonal cores. The interlace is settled by two rules, both already in the engine: the ordinary over-under at every ring-to-seal edge, and the three-family rule at seal-to-seal edges, where three strap directions meet.

The plate adds one more thing the lattice can carry. Jones gives his star cores two different fillings - a daisy of six small hexagons around a centre, and a star nested twice inside itself - and alternates them row by row. Read against the grid, the alternation is a stripe of the lattice, so the engine assigns it by arithmetic rather than by hand, and the repeat it declares honestly accounts for it: the field repeats every six cells along both axes, not every three.

Eight on six

The second plate is the harder one, and it is the one that explains why this studio bothers with lattice rules at all. It is a field of eight-pointed stars, each inside an octagonal band - and the field is six-fold. The roundels sit three to every seal on a hexagonal grid, with a small ochre tile tucked into every gap between two roundels on the seal's side. Eight-fold figures on a six-fold net ought not to close. They do, and the reason is that the eight-fold geometry is local: each star and its band live inside one cell and never have to meet another eight-fold figure edge to edge.

Here is the rule. On the hexagonal tiling, take every cell whose two lattice coordinates are both even - one cell in four - and keep it for the Seal of Solomon. Give every other cell an octagonal band, drawn at about two thirds of the cell's inradius and turned so that a corner points at each of the cell's two seal neighbours. Then carry the octagon's sides straight on through its corners, and the whole field follows. Through the two corners on the seal axis, each side runs on until it meets the extension of the side two along, at the apex of a 45-degree triangle, and there turns parallel to the axis and crosses the seal cell. Through the two corners perpendicular to the axis, each side runs on to the middle of the gap it enters, where the same side of the roundel across the gap arrives from the other direction, and the two stop in a sharp V. That is all the drawing there is.

Everything the plate shows is a consequence. The six lines that cross a seal cell are parallel to the three roundel directions, and because the apex they start from sits half an octagon's circumradius off the axis, they cross on the gap axes at exactly that circumradius: the Seal of Solomon draws itself, its points reaching the same circle as the roundels' corners, its inner hexagon the ochre core. Between the Seal and each roundel the two line extensions and the two side extensions close a slate hexagon whose far corner is the roundel's own. The three V's that meet around every vertex where three roundels touch close the grey hexagon there. And the small ochre tile in every gap is not drawn at all: it is the hole left between the two bent straps, the two bands and the V, a pentagon whose base joins two strap corners and whose point sits in the V. The plate paints it; the studio paints it too, as ornament over the slate face, inset by half a strap. None of this is a Hankin ray family, and none of it needed to be: the engine's downstream machinery - the face graph, the interlace, the ribbons, the exports - asks only for a planar arrangement of straight straps, and this is one.

Inside each band the plate draws an eight-pointed star, and that too comes off the construction rather than off a ruler laid on the print. Join every third corner of the band's inner edge and the innermost crossings are the corners of a smaller octagon, at half the corner radius; on each of its sides stands an equilateral point, so the tips stop short of the band by exactly the margin the plate shows: the tips sit at (1 + √6 − √3)/2 of the inner corner radius, a little under 0.86, where the sheet reads 0.84. The four central triangles on the seal axis and across it are the darts. The colour law is the plate's, and it has a lattice rule of its own: the six roundels around one seal in four, the seals whose axial coordinates are both multiples of four, are vermilion with teal darts, and every other roundel is teal with vermilion darts, so the coloured field repeats only at twice the period of the strapwork. Seal cores and the small gap tiles ochre, the ground slate, every strap buff. Those are the plate's colours, so the palette in the studio is called jones and not Nasrid. It would be a small dishonesty to name a printer's inks after a dynasty.

Checking the drawing against the print

A construction that only resembles its source is a generated image with a footnote, so each field was checked the direct way. The engine's render was rotated through thirty degrees, scaled to the plate's measured cell pitch, and laid over a scan of the plate itself. In the first plate the ring's inradius, the seal's tip radius and the pentagon shapes coincide. The second plate needed a second look. The scan is coarse, seventy-one pixels to a cell pitch, so the plate was averaged over sixteen seals and the twelve symmetries of each, and the construction's lines were scored against the buff of that average: the bands, the side extensions, the seal lines and the V's all lie on Jones's straps for nine tenths of their length, where a line in the wrong place scores less than half. An earlier reading of this plate, which had the straps leaving the cell edges at their midpoints and the seal lines running across the roundel directions, scored like a wrong line, and was replaced by what is here.

Two honest residues remain. The engine's hexagonal cells stand on a point where Jones's stand on a side, so both fields read turned through thirty degrees against the plates; a rotation, not a difference of geometry. And the ornament inside the cores - the daisies, the nested stars, the octagrams - is paint laid over the strapwork, not part of the face graph, so a cut schedule treats each core as a single tile. A fabricator who wants the daisy cut as seven pieces should say so.

What repeats, and how often

Finally, the question every pattern in this studio has to answer before it may be exported as a seamless repeat: what is the smallest translation that carries the field onto itself? For the ring-and-seal field the three-colouring repeats every three cells, but the alternating star cores need twice that, so the honest period is 6√3 r across and 6 r down, where r is the cell's circumradius. For the roundel field the seal sublattice has index four, and the smallest rectangle that keeps both lattice coordinates even is 2√3 r across and 6 r down: a tall, narrow tile. That is the period of the strapwork; the colour classes follow one seal in four, so the coloured field repeats only at 4√3 r across and 12 r down, twice that tile on each axis, and the engine reports the longer period whenever the colours are on. Each was tested by shifting the render one full period along each axis and comparing pixel for pixel; both came back identical. A period the engine declares is a period it has proved.