The mathematics

The crescents of the Alhambra.

Q: What is the Alhambra crescent tiling?

It is a cut-tile mosaic (alicatado) from the Nasrid tradition of the Alhambra, built from slim crescent-shaped tiles that dominate a scatter of small six-pointed stars and small hexagons. The dominant tile is a hook: one flank is concave, bitten in by its neighbours, and the other bulges convex to fill the bites next door. Bump and bite pair off exactly, so the field tiles with no gaps and no leftover ground. Like the bone tiling, it is cut ceramic rather than overlaid strapwork: the curved tiles themselves are the pattern.

Q: What does it mean that the crescent tiling is chiral?

Chiral means handed: the pattern has rotational symmetry but no mirror symmetry, so its reflection is a genuinely different pattern that cannot be slid back onto the original. Every crescent hooks the same rotational way. Its symmetry group is p3, rotations only. This is rare in Islamic geometry, which is usually full of mirror axes, and it is what makes the crescent field the most distinctive tiling in the studio.

Q: Is the crescent tiling still compass-and-straightedge?

Yes. The scaffold is a rhombille - a 60-120 rhombus straddling every edge of a regular star lattice - and every straight rhombus edge is replaced by a single circular arc of the same radius, L over root 3, where L is the rhombus edge. One lattice, one radius: no curve is placed by eye. A pattern made entirely of curves is still, provably, ruler-and-compass work, so the line derived, not generated holds with no asterisk.

The bone tiling gave every tile two curves. The crescent has more nerve. It is a field of hook-shaped tiles - slim crescents crowding around a scatter of small stars and hexagons - and it is the one pattern in this studio with no mirror at all. Hold it up to a looking-glass and you get a pattern you cannot slide back onto the original. That is rare, and it is exact.

FIG. 1 - The crescent alicatado, drawn live. Slim crescents dominate; small six-pointed stars sit at the lattice vertices and small hexagons at the up-centroids. Every crescent hooks the same way round.

A tile shaped like a hook

Most of what this studio makes is strapwork: a network of lines laid over a grid. The crescent, like the bone tiling, is a different kind of object. It is alicatado - cut-tile mosaic, the craft of Moroccan zellige and the Alhambra dados - and there is no line network at all. The tiles themselves are the pattern. The dominant one is a slim crescent: a hook whose inner flank curves inward, as if bitten, and whose outer flank bulges out. Among the crescents sit small six-pointed stars and small hexagons, and the count is fixed - three crescents to every star and every hexagon.

A curved rhombille

The scaffold is a rhombille. Take a regular star lattice and lay one 60-120 rhombus across every edge of it, its two sharp corners on the stars and its two blunt corners on the cell centres either side. That is a grid of straight rhombi. Now replace each straight rhombus edge with a single deep circular arc - turned through 120 degrees, radius L / √3 where L is the rhombus edge, so each bulge stands out by exactly 1 / √3 of the edge it replaces.

One rule decides which way each arc bends, and it is the whole difference between a crescent and a fat lens. Call it flank-coherence: the entire run of a tile's boundary on one side curves inward, and the entire return run curves outward. A tile that alternates in-out-in-out around its edge reads as a lozenge; a tile that keeps each flank whole reads as a true slim hook. The Alhambra plate is unambiguous - it is a hook - so the rule is coherence, not alternation.

Why it tiles, and why it has no mirror

Two tiles share every edge, and they travel along it in opposite directions, so one tile's outward bulge is exactly the other tile's inward bite. Bumps and bites pair off, edge for edge, and the field closes with no gaps and no leftover ground. That much it has in common with the bone.

Here is what it does not share, the property worth a page on mathematics. Each six-way vertex of the lattice opens into a small hexagram star; each "up" cell centre opens into a small hexagon shared by three crescents; and the "down" centres stay bare points where three tails meet. Up and down are not the same, and that asymmetry destroys every possible mirror. What is left is rotation only - the tiling is chiral, symmetry group p3. Reflect it and you get its mirror image, a pattern that no amount of sliding and turning will bring back into register. Most Islamic geometry is dense with mirror axes; this one has none, and it is the more striking for it.

THE CRESCENT TILING, IN ONE SENTENCE - a chiral curved-rhombille mosaic whose every crescent's bulge is its neighbour's bite, so a field of pure hooks tiles the plane with no gap, no straight edge, and no mirror.

Coloured like hand-cut zellige

The glaze does not follow the geometry's order. A zellige cutter sets colour by hand, and the one law is that no two touching pieces share a glaze. The studio colours the crescent field the same way - by a proper five-colouring arranged on a colour torus, five glazes with no two adjacent tiles alike and no rule tying colour to orientation, the cream ground carrying the whole field. Turn the scatter off and the crescents fall into three colours by direction: the plainest possible picture of the chirality, one hook in three rotations.

FIG. 2 - The same tiles, coloured by direction instead of scattered. The crescents sort into three glazes - one per rotation - which is the three-fold chiral order laid bare. Stars and hexagons keep their own.

Curves a machine can cut

Because the arcs are exact, they stay exact all the way to the workshop. When the studio writes a fabrication file for a crescent field, the curved edges go out as true arcs - one radius, L / √3, struck for every one of them - not as a fan of little straight segments pretending to be a curve. A waterjet or router reads the real circle and cuts a clean edge, and the narrow waist of each crescent is reported in millimetres so a maker knows the slim tile will survive the cut. A pattern with no straight line anywhere leaves the screen as a single compass setting. That is the whole idea of this studio, on its most wilful pattern.

The series

Its straight-laced sibling: the bones of the Alhambra The other Alhambra cut-tile pattern made entirely of curves - and the one compass opening that scribes the whole wall.

Tile a crescent field yourself.

Build the Alhambra crescent alicatado, recolour it by hand or by direction, and export true-arc fabrication files - free to explore in the browser.

Open the studio

Common questions

What is the Alhambra crescent tiling?

A Nasrid cut-tile mosaic (alicatado) built from slim crescent-shaped tiles that dominate a scatter of small six-pointed stars and small hexagons. Each crescent is a hook: one flank concave, bitten in by its neighbours, the other convex, bulging to fill the bites next door. Bump and bite pair off exactly, so the field tiles with no gaps. The curved tiles themselves are the pattern; there is no overlaid line network.

What does it mean that the crescent tiling is chiral?

Chiral means handed: rotational symmetry but no mirror symmetry, so the reflection is a genuinely different pattern that cannot be slid back onto the original. Every crescent hooks the same rotational way, and the symmetry group is p3 - rotations only. That is rare in Islamic geometry, which is usually full of mirror axes.

Is the crescent tiling still compass-and-straightedge?

Yes. The scaffold is a rhombille on a regular star lattice, and every straight rhombus edge is replaced by a single circular arc of the same radius, L over root 3, where L is the rhombus edge. One lattice, one radius - no curve placed by eye. A pattern made entirely of curves is still, provably, ruler-and-compass work.