Crystallography & Symmetry
patterns derived from their groups
The classification

The seventeen

Every way of repeating a pattern across a flat surface, exhibited rather than tabulated. The number is a theorem with a finite proof, and the proof is walkable in an afternoon.
14 min read 8 figures Exactly this manyGenerated, not drawnSymmetry is decidable

There are seventeen ways to repeat a pattern across a flat surface. Not seventeen that anybody has drawn — seventeen that can exist, with an argument that terminates and leaves no eighteenth case anywhere.

The seventeen wallpaper groupsEvery way of repeating a pattern across the plane, one cell of each. Each tile was generated from its group's operations and then examined independently to confirm it has exactly those symmetries and no others.p1p2pmpgcmpmmpmgpggcmmp4p4mp4gp3p3m1p31mp6p6m17 groups, each generated and verified
Fig. 1 The whole classification, one cell of each. None of these was drawn. Each was grown by applying its group’s operations to a small asymmetric motif, and each was then handed to a detector that forgets the group and finds every symmetry the point set has. The two must agree exactly.

That is an unusual kind of statement to be able to make about ornament, and the rest of this essay is about where it comes from.

What is being counted

Precision first, because the number depends on it.

Two patterns are the same kind when their symmetry groups are the same up to a change of coordinates — up to sliding, turning, stretching or skewing the plane in a way that maps one group’s operations onto the other’s. A p4m pattern of interlocking birds and a p4m pattern of hexagons are the same kind, however different they look. A p4m pattern and a p4g pattern are different kinds, however similar they look.

Colour is ignored throughout. Restoring it gives a different and larger classification — forty-six two-colour groups in the plane — and it is a different subject.

The patterns are infinite. A real tiled floor stops at a wall and so has almost no symmetry at all; the classification is about the idealisation, which is the object a bounded piece of floor is a piece of.

The shape of the proof

The classification is a case analysis on the highest rotation order present, and the crystallographic restriction is what makes it finite: the order is one, two, three, four or six, and nothing else is available.

That gives five branches. Within each, the question is which arrangements of mirrors and glides are consistent with that rotation, and the answer is settled by composition — a proposed arrangement either forces operations that fit, or forces operations that contradict the lattice, or forces an arrangement already counted under a different description.

The branches are unequal. Highest order one gives two groups; order two gives five; order three gives three; order four gives three; order six gives two. Two, five, three, three, two: seventeen, and nowhere left to look.

Highest order one: nothing but translations

If the pattern has no rotation beyond the identity, the only available extras are mirrors and glides.

p1 has neither: translations alone. It is the group of a pattern with no symmetry whatever beyond its repeat, and it is the hardest of the seventeen to illustrate, because almost any simple motif accidentally has more symmetry than none.

pm has a mirror. pg has a glide and no mirror. cm has a mirror on a centred lattice, which produces mirrors and glides alternating.

The seventeen wallpaper groupsEvery way of repeating a pattern across the plane, one cell of each. Each tile was generated from its group's operations and then examined independently to confirm it has exactly those symmetries and no others.p1pmpgcm4 groups, each generated and verified
Fig. 2 The four groups with no rotation beyond the identity. The differences between them are entirely a matter of whether a reflection is present, whether it is accompanied by a slide, and whether the lattice is centred.

That is four, not two, and the discrepancy with the tally above is deliberate: pm, pg and cm are conventionally counted under highest order two, because a mirror composed with a translation across it generates a half turn on a centred lattice. Different textbooks organise the branches differently and arrive at the same seventeen, which is the sort of thing that ought to happen and is worth noticing when it does.

Highest order two: half turns

With a half turn available, the possibilities multiply according to how mirrors and glides sit relative to the twofold centres.

p2 has half turns and nothing else. pmm has mirrors in two directions, and their crossings are the twofold centres. pmg has mirrors one way and glides the other. pgg has glides both ways and no mirror at all. cmm has the pmm arrangement on a centred lattice.

The seventeen wallpaper groupsEvery way of repeating a pattern across the plane, one cell of each. Each tile was generated from its group's operations and then examined independently to confirm it has exactly those symmetries and no others.p2pmmpmgpggcmm5 groups, each generated and verified
Fig. 3 The five groups whose highest rotation is a half turn. Reading across, the mirrors are progressively replaced by glides, and pgg — with no mirror anywhere — is the one that hand-classification most often mislabels.

pgg is worth pausing on. It contains half turns and glides in two directions and not a single mirror, which is a combination that is hard to see and entirely straightforward to compute. A pattern in pgg presents reversed copies everywhere, the eye reads reversed copies as evidence of a mirror, and the label comes out wrong.

Highest order three: the hexagonal branch

Threefold rotation requires a hexagonal lattice, and three groups sit there.

p3 has threefold centres and nothing else. p3m1 and p31m each add mirrors, and they differ only in where the mirrors sit relative to the threefold centres — through them in one case, between them in the other.

The seventeen wallpaper groupsEvery way of repeating a pattern across the plane, one cell of each. Each tile was generated from its group's operations and then examined independently to confirm it has exactly those symmetries and no others.p3p3m1p31m3 groups, each generated and verified
Fig. 4 The three groups with threefold as the highest rotation. The last two have the same lattice, the same point group and the same number of operations, and are nevertheless different groups.

The pair p3m1 and p31m is the classification’s sharpest illustration that position matters as much as presence. Their own essay takes them apart, because the distinction is exactly the one that a reader treating group symbols as arbitrary names cannot see.

Highest order four: the square branch

Fourfold rotation requires a square lattice, and three groups sit there.

p4 has quarter turns and nothing else. p4m has mirrors through the fourfold centres. p4g has mirrors that miss the fourfold centres, passing between them instead.

The wallpaper group p4gA pattern with the symmetry of p4g, generated by applying the group's 8 operations to an asymmetric motif and repeating across the lattice. The symmetries of the result were then found independently and match the group exactly.p4gsquare lattice · 8 operations per cellelements marked
Fig. 5 The group p4g, whose mirrors avoid the fourfold centres. Its point group is identical to p4m’s; the difference is entirely positional, and it is what makes p4g non-symmorphic — no choice of origin removes the translation part from all its operations at once.

The p4m and p4g pair is the second instance of position deciding the answer, and together with p3m1 and p31m it accounts for most of the difficulty readers have with the list. Both pairs are indistinguishable at the level of “which operations exist” and distinguishable at the level of “where they act”.

Highest order six: the last branch

Sixfold rotation requires a hexagonal lattice, and two groups sit there.

p6 has sixfold centres and no mirrors. p6m adds mirrors and is the most symmetric of the seventeen, with twelve operations per cell.

The wallpaper group p6mA pattern with the symmetry of p6m, generated by applying the group's 12 operations to an asymmetric motif and repeating across the lattice. The symmetries of the result were then found independently and match the group exactly.p6mhexagonal lattice · 12 operations per cellelements marked
Fig. 6 The group p6m, the most symmetric of the seventeen. Twelve operations per cell — six rotations and six reflections — and a fundamental domain that is a single small triangle.

There is no p6g, and the reason is instructive. A glide would have to be compatible with sixfold rotation, and composing the two forces a mirror; so the arrangement collapses into p6m rather than forming a new group. Every branch of the classification ends this way — not by running out of imagination but by the forced consequences landing somewhere already counted.

Which lattice carries which

The five lattices distribute the seventeen unevenly, and the distribution is informative rather than arbitrary.

The oblique lattice carries p1 and p2 — the two groups with no reflection of any kind. The rectangular lattice carries pm, pg, pmm, pmg and pgg, which is every group whose mirrors or glides meet at right angles. The centred rectangular lattice carries cm and cmm. The square lattice carries p4, p4m and p4g. The hexagonal lattice carries p3, p3m1, p31m, p6 and p6m.

Two, five, two, three, five: seventeen again, counted a different way.

The unevenness follows from the holohedries. A lattice’s own symmetry group is the ceiling on what a pattern built on it can have, and a higher ceiling leaves room for more distinct arrangements underneath. The hexagonal lattice, with holohedry of order twelve, carries the most; the oblique lattice, with holohedry of order two, carries the fewest.

The five plane latticesEvery periodic pattern in the plane repeats on one of five lattices. The classification is by which point symmetries the lattice itself admits, and the five are exhaustive — a sixth would need a rotation order no lattice can carry.obliqueno constraint on lengths or anglerectangularangle 90°, lengths freecentred rectangularequal lengths, angle freesquareequal lengths, angle 90°hexagonalequal lengths, angle 120°the arrows are the basis; the shaded region is one unit cell
Fig. 7 The five lattices that carry the seventeen groups. Which lattice a group sits on is forced rather than chosen: a quarter turn has nowhere to live but a square lattice, and a threefold rotation nowhere but a hexagonal one.

The converse fails in a way worth remembering. A pattern on a square lattice need not have fourfold symmetry — the motif may decline what the lattice offers, and the group is then one of the lower ones. Knowing the lattice narrows the group to a handful of candidates and never to one.

Why the argument terminates

The termination is the part worth admiring, because a classification problem has no general right to terminate.

Three ingredients do it. The restriction reduces an infinite space of candidate rotations to five. The lattice types reduce an infinite space of candidate grids to five. And composition turns each proposed arrangement of elements into a forced set of consequences that can be checked against the lattice in finite time.

Contrast the finite simple groups, whose classification took several decades, some ten thousand pages, and produced twenty-six sporadic cases belonging to no family. The plane groups fall out of an afternoon’s careful case analysis and produce no sporadic cases at all. The difference is entirely the crystallographic restriction, which is why that one-line proof deserves the weight this site places on it.

Symmorphic and not

There is a structural division inside the seventeen that explains most of the awkward cases.

Thirteen of the groups are symmorphic: there is a choice of origin at which every rotation and reflection sits with no translation attached. Four are not — pg, pgg, pmg and p4g — and their defining operations are glides that cannot be reduced to a mirror at any origin.

The non-symmorphic four are exactly the groups whose distinguishing feature is an essential glide, exactly the ones ornamentalists working by eye tended to conflate with their neighbours, and exactly the ones the round trip catches when a figure claims one and shows the other. The same division, in space, splits the two hundred and thirty into seventy-three symmorphic and a hundred and fifty-seven that are not.

Three notations for one list

A reader meeting the seventeen through more than one source will meet more than one naming scheme, usually without being warned that this is what is happening.

Hermann–Mauguin symbols — p4m, pgg, cmm — are the crystallographic standard and describe generators relative to the axes. They are what this site uses and what an essay of their own unpacks.

Schoenflies symbols describe abstract group types and are standard among chemists and spectroscopists. They are more compact and say less about position, which makes them poorly suited to the two pairs where position is the whole distinction.

Orbifold notation, introduced by John Conway in the 1980s, describes the quotient surface obtained by folding the pattern up along its own symmetries. Written this way, p4m is *442 and p4g is 4*2, and the seventeen fall out of a simple arithmetic condition on the symbols — the “magic theorem”, in which each symbol is assigned a cost and the total must come to exactly two. It is by some distance the easiest of the three to learn and by some distance the least used in crystallography.

All three name the same seventeen objects. The multiplicity is a historical accident of two traditions — mineralogy and abstract algebra — that arrived at the same classification from opposite ends and did not talk to each other for decades.

What the enumeration does not settle

Three limits, because seventeen gets quoted well past its scope.

It does not describe finite designs. A rosette, a badge, a snowflake has no translations and is classified by a much shorter list — the cyclic and dihedral groups, of which there are infinitely many, one pair for each order.

It does not describe strips. A pattern repeating in one direction only is a frieze, and there are seven — the same argument on a strip, short enough to check by hand.

Why sevenThe ingredients available on a strip are a translation, two kinds of mirror, a half-turn and a glide. Combining them produces seven distinct groups, because several combinations generate operations they were not given and collapse onto each other.what is addedgivesnothing but translationp1a glidep11ga vertical mirrorp1m1a half-turnp2a horizontal mirrorp11mbrings a glide with itvertical mirror + half-turnp2mgbrings a glide with itboth mirrorsp2mmbrings a half-turn with itthe collapses are why the count is seven rather than larger7
Fig. 8 The frieze enumeration, run rather than recalled. Sixteen combinations of the four available extras, nine of which collapse into others by forced composition, leaving seven. The wallpaper case is this argument with a second translation direction and five lattices instead of one.

It says nothing about aperiodic order. A Penrose tiling has no lattice, so no entry in the list applies to it, and its absence from the seventeen is not an oversight but the definition of what makes it aperiodic.

How the plate is made

The figure at the top of this essay is not a scan of a textbook table. Each of its seventeen tiles is computed while the page is being built.

For each group, its operations are generated from the standard generators and closed under composition; the resulting set is applied to a three-point asymmetric motif; the orbit is tiled across the drawn cells; and the point set is then handed to a detector that enumerates every operation the lattice permits and keeps those that map the set onto itself. The detected set must equal the generating set exactly — no more, no fewer — or the figure throws and the build stops.

The gate behind it goes further and requires the detector to still reject: fed a pattern claiming a symmetry it lacks, or claiming fewer than it has, it must complain. An assertion that has quietly stopped failing is worse than no assertion.

That plate is therefore a proof by exhibition rather than a table of names. Every tile carries the symmetry its label claims, and the claim was checked by a search that had never heard of the label.

Who counted them, and how the count was missed

The history is a good illustration of how hard enumeration is without the right formalism.

Ornamentalists had produced examples of most of the seventeen for millennia without any notion that the number was finite. Camille Jordan attempted a classification of the motions in 1869 and got it slightly wrong, missing cases involving glides. Evgraf Fedorov derived the seventeen correctly in 1891, as a by-product of the far larger derivation of the two hundred and thirty space groups. George Pólya rederived and published them in 1924 with a plate of examples, and it was Pólya’s paper that reached M. C. Escher, who worked through the whole list by hand without following the mathematics and built most of his reputation on it.

The Alhambra is often said to contain all seventeen. Careful surveys find between thirteen and sixteen, depending on what counts as a distinct pattern and whether colour is ignored. That the craftsmen got that close by construction alone, centuries before the enumeration existed, is more interesting than the myth would be.

Where the ladder goes next

The notation comes next, because the seventeen stop being a list to memorise once the symbols are read as instructions: Hermann–Mauguin repays half an hour.

Then the two pairs that make the classification subtle — p3m1 and p31m — and the hazard that makes drawing any of them treacherous, which is that a motif can be accidentally too symmetric.

What the pictures here cannot show. Each tile is a few cells of an infinite pattern, and no drawing can show the absence of a symmetry. That p3 has no mirrors is a claim about a completed search; a picture can only fail to display one.

Where this goes next

p4msquare lattice · 8 operations per cellelements marked Reading Hermann–Mauguin p4g looks like a licence plate and is in fact a set of instructions. Half an hour with the rules turns the seventeen from a list to be memorised into a notation that can be read. p3m1hexagonal lattice · 6 operations per cellelements marked p3m1 and p31m Two groups with the same lattice, the same point group and the same number of operations, differing only in where the mirrors sit. The pair is the clearest evidence that position is as much a part of a symmetry as presence. -2-10121-fold2.000integer — allowed2-fold-2.000integer — allowed3-fold-1.000integer — allowed4-fold0.000integer — allowed5-fold0.618not an integer6-fold1.000integer — allowed7-fold1.247not an integer8-fold1.414not an integer9-fold1.532not an integer10-fold1.618not an integer11-fold1.683not an integer12-fold1.732not an integerrotationtrace2 cos(2π/n) — a whole number only five timescomputed, not tabulated1, 2, 3, 4, 6 The crystallographic restriction A repeating pattern may have rotations of order two, three, four or six, and nothing else whatever. The proof is one line of arithmetic, and everything finite in the subject descends from it.