The seventeen symmetries.

Every pattern that repeats across a plane, whatever it depicts, belongs to one of exactly seventeen types of symmetry. The claim that the Alhambra contains all seventeen is repeated in every popular account and is still argued over by the people who have counted. What the groups are, where the classical constructions sit among them, and why the answer changes the moment you add colour.

There are exactly seventeen wallpaper groups: seventeen distinct ways a flat pattern can be symmetric under translations, rotations, reflections and glide reflections. The Russian crystallographer Evgraf Fedorov proved the count in 1891, George Pólya rediscovered it in 1924, and M. C. Escher learned the seventeen from Pólya's paper and from the Alhambra, which he visited twice and drew for weeks. The theorem is complete: there is no eighteenth, and every repeating pattern ever made, from a brick bond to a Nasrid dado, falls into one of the seventeen.

What a symmetry group is

Take a pattern that continues forever in every direction and list every rigid motion that carries it exactly onto itself. Translations, since it repeats. Perhaps rotations about certain points. Perhaps reflections in certain lines, and perhaps glides, a reflection followed by a slide along the mirror, the symmetry of footprints. The list is the pattern's group, and two patterns have the same group when their lists have the same structure, whatever they are pictures of. The crystallographic restriction limits the rotations to two-, three-, four- and six-fold, which gives five kinds of lattice; combining each lattice with its possible mirrors and glides gives the seventeen. Their names are short codes: p6m, p4m, p3, cmm, and so on, where the number is the highest rotation, m means mirrors and g means glides.

Where the classical constructions sit

The simplest way to see the seventeen at work is to ask, of each construction the studio derives, what its group is.

• The six-fold star lattice, the hexagram field of the Fatehpur Sikri screens, is p6m: six-fold turns about every star, mirrors through every star in six directions. It is the most symmetric pattern a plane can carry, and it is the one every six-fold Islamic field belongs to as long as the colouring respects it.
• The eight-fold seal on a square net, the star-and-cross of Fez and the Comares Hall, is p4m. Each star is eight-fold about its own centre, but the field only turns by quarters, because a wallpaper cannot turn by an eighth. The mirrors run along and across the grid and along both diagonals.
• The twelve-fold field on its triangular lattice is p6m again: twelve-fold stars, a six-fold wallpaper. The extra six turns of each star are local, not global.
• The ten-fold field on its 72-degree rhombic lattice has no rotation above two-fold at all. Its mirrors lie along the lattice diagonals, which puts it in cmm, the group of the centred rectangle. Ten-pointed stars, a two-fold wallpaper: this is the restriction at its most visible.
• The crescent tiling of the Alhambra is the studio's only chiral construction: three-fold turns about the small stars and hexagons and no mirror anywhere, the group p3. Every crescent hooks the same way, and its mirror image is a different pattern.

Colour is a symmetry too

Everything above assumes the pattern is a line drawing. Colour it and the group can only stay the same or shrink, because a motion that carries the lines onto themselves may carry blue tiles onto gold ones. The classical zellige rule, that no two touching faces share a colour, usually forces a two-colour alternation across congruent faces, and that alternation halves the symmetry: the coloured pattern's translation is twice the line pattern's, and half the rotations and mirrors are lost. The studio's engine states this exactly. When alternation is on, the reported repeat tile doubles along both axes, because the true period of the coloured field is twice that of the geometry underneath. A generator that ignores colour will hand you a tile whose colours do not match at the seam.

Mathematicians handle this with colour symmetry, in which a motion may be allowed to permute the colours. Under that convention a pattern with a consistent two-colour swap keeps its whole group. Under the strict convention it loses half. Most of the arguments about historical patterns are, underneath, arguments about which convention to use.

The Alhambra claim

It is often said that the Alhambra is the one building in the world where all seventeen groups can be found. The claim has a history worth knowing before repeating it. Edith Müller's 1944 thesis, the first systematic count, found eleven. Branko Grünbaum, Zdenka Grünbaum and Geoffrey Shephard, surveying the palace in 1986, found thirteen and doubted the rest. Rafael Pérez-Gómez answered in 1987 with a paper claiming all seventeen, and that paper is the source of the popular version. Grünbaum returned to the question in 2006 and maintained that several of the missing groups are present only if colour is disregarded, or only in later restorations, or only in a single doubtful example.

The honest summary is that thirteen or fourteen are beyond dispute and the remaining three or four depend on what you count: whether a two-coloured pattern is classified by its lines or its colours, and whether nineteenth-century restoration work counts as Nasrid. That is not a small quarrel; it is precisely the colour question above. The Alhambra is astonishing either way, and the tradition as a whole, across Cairo, Isfahan, Fez and Delhi, certainly exhausts the seventeen. The single-building claim should be made with the asterisk attached.

Why the studio cares

A symmetry group is a checkable claim. Because every pattern here is derived from a lattice rather than painted to look like one, the engine knows the pattern's translations exactly and can verify them, vertex by vertex, before it hands you a repeat tile. It knows when colour has doubled the period and says so. And it knows when a construction has no translations at all, the medallion, the arched panel, the aperiodic field, and refuses to invent a tile for it. The seventeen groups are not a classification imposed afterwards on a picture; they are the structure the construction is built from, and the structure a fabricator will rely on when the tiles meet at the corner of a room.