Why there is no five-fold wallpaper.

A pattern that repeats across a wall can turn by a half, a third, a quarter or a sixth of a circle and land on itself. It cannot turn by a fifth or a tenth. That is a theorem, not a limitation of any craftsman, and the tradition that loved the ten-pointed star above all others had to be cleverer than it.

Only two-, three-, four- and six-fold rotations are compatible with a repeating pattern. This is the crystallographic restriction, and it is one of the few theorems in mathematics that can be checked with a sheet of graph paper. It governs every wallpaper, every tiled floor and every crystal, and it puts the ten-pointed star, the most celebrated figure in Islamic geometry, in a peculiar position: the star itself is ten-fold, but no field of them can be.

The theorem, in one paragraph

A repeating pattern has a lattice: a set of points, all alike, that the pattern slides onto itself between. Pick two lattice points as close together as any two are, call the distance between them one, and suppose the pattern also turns onto itself by a fifth of a circle about the first point. Then the second point, turned by 72 degrees, is also a lattice point; and turned by 72 degrees about the second point, the first is a lattice point too. Those two new points are closer to each other than one, because 2 sin 36° ≈ 1.18 is the distance between them after the turns and the geometry of the two turns brings them inside the original spacing. But no two lattice points can be closer than the closest pair. So the fifth of a turn is impossible. The same argument, worked for a general angle, shows that 2 cos θ must be a whole number, and the only angles that oblige are 360, 180, 120, 90 and 60 degrees: the identity and the two-, three-, four- and six-fold turns.

The consequence for ornament is exact. A wall of six-pointed stars can be six-fold all the way down: each star is six-fold and the whole field turns onto itself by sixty degrees about any star. A wall of ten-pointed stars cannot. Each star is ten-fold about its own centre, and the field, taken as a whole, has some smaller symmetry: two-fold, or a mirror, or none. The eye is fooled because it sees the stars, not the field.

The first answer: an oblique lattice

The tradition's usual way of setting decagons is to put them in contact along a rhombic lattice whose corner is 72 degrees, so that every decagon touches its neighbours along two directions and a six-sided bowtie fills the gap between each pair. This is the layout the studio's engine derives for ten-fold work, and it is periodic: slide the field by one lattice step and it lands on itself. What it is not is rectangular. The two lattice directions meet at 72 degrees, and because cos 72° = (√5 − 1)/4 is irrational, no whole-number combination of the two steps ever lines up with an axis. A rectangular repeat tile for a ten-fold field does not exist, and the studio says so rather than approximating one: ask it for the seamless tile of a ten-fold pattern and it declines, with the reason. The six-, eight- and twelve-fold fields all have rectangular repeats; the ten-fold one has only the rhombus.

The second answer: the sebka

The Almohad and Nasrid builders of Seville and Granada took the same 72-degree rhombus and made it the whole point. In the sebka, the net of interlacing lobes that covers the Giralda and the walls of the Alhambra's Partal, ten-pointed stars sit at the vertices of the rhombic lattice and elongated hexagonal cells bridge them along both directions, so the field reads as a diagonal net rather than a grid. It repeats by the rhombus and only by the rhombus. The engine draws it on exactly that lattice, sets the decagon at the golden proportion of the lattice step at which the bridges meet cleanly, and reports its period honestly as a rhombic cell rather than pretending to a rectangle.

The third answer: stop repeating

The most remarkable answer is to give up the lattice altogether. A tiling of two rhombi, one thick and one thin, can cover the plane with perfect long-range five-fold order and no translation that carries it onto itself anywhere. It is quasi-periodic: ordered without repeating. The studio builds it the way de Bruijn showed such tilings can always be built, from five sets of parallel lines laid at 72 degrees to one another, with a rhombus at every crossing. Peter Lu and Paul Steinhardt argued in 2007 that the girih of the Darb-i Imam shrine in Isfahan, finished in 1453, is a fragment of exactly this kind of tiling, five centuries before Roger Penrose drew one and quasicrystals were found in metal.

Whether the craftsmen understood it that way is genuinely contested. Emil Makovicky had made a related claim for a Maragha tomb tower in the 1990s; Peter Cromwell and others have since argued that the same patterns follow from a decagonal construction method the workshops demonstrably used, and that intent to make a non-repeating pattern is not shown by the surviving evidence. What is not contested is the geometry: the tiling exists, the shrine carries a portion of it, and the studio can derive it. It draws the aperiodic field, refuses to give it a repeat tile, and lets the mathematics rather than the caption make the claim.

What this means for the studio

Every one of these facts is enforced rather than described. The engine computes the period of a pattern from its lattice and verifies it by translating every interior vertex by that period and requiring each one to land on another; a construction with no such period returns no tile. So a six-fold field exports as a seamless repeat, a ten-fold field exports as a rhombic one, the sebka as its rhombus, and the aperiodic tiling as a bounded panel with an honest note. A generator that paints ten-pointed stars into a square tile is not breaking the theorem; the theorem cannot be broken. It is fudging the stars, and a fabricator will find the fudge at the seam.