Crystallography & Symmetry
patterns derived from their groups
Operations

The fundamental domain

The smallest piece of a pattern from which the group rebuilds the rest. Drawing one is easy and drawing one correctly is not, because a region with a gap or an overlap looks exactly like a region without.
14 min read 6 figures Generated, not drawnSymmetry is decidableThe motif must be asymmetric

A pattern with the symmetry of p4m repeats eight times over inside every unit cell, so seven eighths of the drawing is redundant. The eighth that is not is the fundamental domain, and the whole pattern is what the group does to it.

A fundamental domain for p4mOne representative from every orbit of p4m, shaded, with the images that tile the rest of the cell. The domain was found by computing orbits rather than by drawing a region, and every sample's orbit was checked to meet it exactly once — so the region has neither a gap nor an overlap.p4m55 of 324 samples17.0% of the cell8 operations, so one part in 8grid 18×18
Fig. 1 A fundamental domain of p4m, shaded, with the images that fill the rest of the cell. The domain was not drawn: it is the set of points that come first in their own orbit, and every other point in the cell is an image of exactly one of them.

The idea is simple enough to state in a sentence and its correctness condition is not. A region qualifies when the group’s images of it cover the plane and no two images overlap — and the second half is where hand-drawn domains fail, silently, because an overlap and a gap both produce a picture that looks entirely reasonable.

The definition, stated so it can be checked

A fundamental domain for a group acting on the plane is a region DD such that every point of the plane lies in g(D)g(D) for some gg in the group, and no point lies in the interior of two different images.

Reading it as two conditions rather than one is what makes it testable.

Covering. Nothing is left out: applying the group to DD reaches everywhere. A domain that fails this leaves a strip of the pattern that no image of the domain accounts for, and rebuilding the pattern from the domain would leave a hole.

Non-overlap. Nothing is counted twice: no two images share more than a boundary. A domain that fails this describes some part of the pattern more than once, and a count made by multiplying the domain’s contents by the group order comes out too large.

Neither condition is visible. A region with a five per cent overlap looks like a region without one, and the way to tell them apart is to take a point, compute its whole orbit, and ask how many members of that orbit lie in the region. The answer must be exactly one.

How the domain here is found

Nothing on this site draws a fundamental domain. The domain is computed, by a procedure short enough to describe completely.

The cell is sampled on a grid of exact rational coordinates. Each sample is pushed through every operation of the group, giving its orbit. The orbit’s members are compared under a fixed ordering — bottom to top, then left to right — and the smallest is the orbit’s representative. The domain is the set of samples that are their own representative.

Since every orbit has exactly one smallest member, this construction cannot produce an overlap and cannot produce a gap. That is not a claim about the code; it follows from the definition of a minimum. The check that runs anyway is the one worth having: for every sample, count the members of its orbit that lie in the domain, and require the count to be one. It is a test of the orbit computation and the ordering against each other, and it caught a real error the first time it ran.

A fundamental domain for p3m1One representative from every orbit of p3m1, shaded, with the images that tile the rest of the cell. The domain was found by computing orbits rather than by drawing a region, and every sample's orbit was checked to meet it exactly once — so the region has neither a gap nor an overlap.p3m164 of 324 samples19.8% of the cell6 operations, so one part in 6grid 18×18
Fig. 2 The domain of p3m1, on the hexagonal lattice. The sample grid here has to be the plain grid of thirds and sixths rather than one offset by half a step, and the reason is the failure recorded below.

The failure the check caught

The first version of the sampler put its points at the centres of small squares — the natural choice, since each sample then stands for the square around it. It worked for twelve of the seventeen groups and failed for the five on the hexagonal lattice, with a message about an orbit that missed the domain entirely.

The cause is worth stating because it is a general trap. A grid offset by half a step is closed under the operations of a square lattice: negating an odd-numbered half-step gives another odd-numbered half-step, and swapping the coordinates keeps the offset. It is not closed under a threefold rotation, which sends the point (i,j)/N(i, j)/N to (−j, i−j)/N(-j,\, i - j)/N — and the difference of two odd numbers is even, so the image lands between the sample points rather than on one.

The symptom was a hole in the domain. The picture showed a shaded region with a bite out of it, which looked like an interesting feature of p3 rather than a bug in the sampler, and the only reason it was not shipped that way is that the partition assertion refused it before it was drawn. This is the site’s argument in miniature: the picture had no way of telling anybody it was wrong.

The area, and why it is not exactly one over the order

For a group with ∣G∣|G| operations per cell, the domain ought to be 1/∣G∣1/|G| of the cell — eight operations, an eighth of the area. The figures here report a fraction slightly larger than that, and the excess is real rather than an artefact.

The cause is the boundary. A sample sitting exactly on a mirror line has a stabiliser: the reflection fixes it, so its orbit is half the usual size, and it is its own representative more often than a generic point is. Those samples are counted whole, so the measured domain includes a little of every mirror and every rotation centre in the cell. The excess shrinks as the grid is refined, since the boundary is one-dimensional and the interior is two-dimensional, and the assertion accompanying each figure is an inequality with the slack stated rather than an equality that would have to be fudged.

That is the honest form of the claim and it points at something structural. A fundamental domain is only defined up to its boundary, and the boundary is where all the interesting points live: the mirror lines, the rotation centres, the positions an atom can occupy without breaking the symmetry. Any implementation has to decide what to do with them, and the decision shows up in the numbers.

The size of the effect is a good diagnostic in its own right. A group whose special positions are one-dimensional — the mirror groups, pm and cm and pmm and the rest — shows an excess that falls only as fast as the grid step, because the mirror lines have length and the sampling always catches a strip of them. A group whose special positions are isolated points shows an excess that falls as the square, because the points have neither length nor area. So the rate at which the measured fraction approaches one part in the group order says which kind of special position the group has, without anybody looking at the picture. That is a small thing and it is the sort of small thing worth noticing: a number that was collected to check one claim turns out to answer a different question.

A fundamental domain for pggOne representative from every orbit of pgg, shaded, with the images that tile the rest of the cell. The domain was found by computing orbits rather than by drawing a region, and every sample's orbit was checked to meet it exactly once — so the region has neither a gap nor an overlap.pgg101 of 400 samples25.3% of the cell4 operations, so one part in 4grid 20×20
Fig. 3 The domain of pgg, at a finer grid. Four operations per cell, so a quarter of the area — and here the measured fraction sits very close to the ideal, because pgg has no mirrors at all. Its only special positions are isolated twofold centres, and isolated points contribute almost nothing to an area.

Building the pattern back from the domain

The domain is only worth having if the pattern can be recovered from it, and recovering it is the same construction the rest of this site is built on.

Growing the p4 orbitOne motif, then more of the group's operations applied to it, until applying another produces nothing new. The pattern is the orbit; the drawing is only its shadow.1 of 43 of 44 of 4the pattern is grown from the group, never drawnp4
Fig. 4 The orbit of a motif, built one operation at a time. A fundamental domain is what remains when this construction is run backwards: the piece of the cell each of these copies came from.

Take the contents of the domain, apply every operation of the group, and take the union. Because the images tile the cell without overlap, nothing is drawn twice; because they cover it, nothing is missing. The result is the cell, and translating the cell gives the plane.

There is a subtlety in the word “contents” that is easy to skate over. What the domain holds is not a picture but a rule for a picture: a set of positions with whatever is at them. A crystallographer’s asymmetric unit holds atoms with their elements and displacement parameters; this site’s domains hold sample points. Applying the group to either produces the whole, and the two are the same operation with different payloads.

The reverse question is the useful one. Given a finished pattern, which piece of it is the domain? That cannot be answered without the group, and finding the group from the pattern is the detector’s job rather than the domain’s. The two constructions run in opposite directions and meet in the middle, which is what makes the round trip on this site possible at all.

A fundamental domain for p6mOne representative from every orbit of p6m, shaded, with the images that tile the rest of the cell. The domain was found by computing orbits rather than by drawing a region, and every sample's orbit was checked to meet it exactly once — so the region has neither a gap nor an overlap.p6m37 of 324 samples11.4% of the cell12 operations, so one part in 12grid 18×18
Fig. 5 The domain of p6m without the motif drawn over it. Twelve operations per cell, so a twelfth of the area — the smallest fundamental domain any wallpaper group has, and the reason a p6m pattern is the cheapest of the seventeen to specify.

The asymmetric unit, which is the same idea under a different name

Crystallography’s version of the fundamental domain is the asymmetric unit, and the shift in name comes with a shift in emphasis worth noting.

A structure determination reports the contents of the asymmetric unit and the space group, and nothing else. Everything else in the crystal is generated. A protein structure with four identical chains in the cell usually deposits one, and the other three are the group’s images of it — which is why a downloaded structure file often looks a quarter the size of the molecule it describes, and why software that ignores the symmetry operations displays a quarter of a complex.

The International Tables specify a standard asymmetric unit for each space group as an explicit region, given by inequalities on the fractional coordinates. Those inequalities are a hand-made version of exactly what the computation above produces, chosen once and for all so that two laboratories describing the same structure describe the same eighth of it.

The economy is not small. For a space group of order 96, storing the asymmetric unit rather than the cell is a factor of ninety-six, and the factor compounds across a database of two hundred thousand structures.

Where the exactness stops

Two limits, and the second is the one that matters when the idea leaves this site.

A fundamental domain is not unique. Any region meeting the two conditions qualifies, and there are infinitely many — the one computed here is the one a particular ordering picks out, and a different ordering gives a differently shaped region of the same area. Statements about the shape of a domain are therefore statements about a convention. Statements about its area are not, since the area is 1/∣G∣1/|G| of the cell for every valid choice.

Connectedness is not guaranteed. Nothing in the definition says the domain has to be in one piece, and for some groups the natural constructions produce a domain in several. The pictures on this page happen to come out connected, which is a property of the ordering used and not a theorem, and a reader inferring “domains are connected” from three examples would be inferring too much.

There is also the standing limit of everything on this site: the computation is a computation about a periodic pattern with a lattice. An aperiodic tiling has no fundamental domain in this sense, because it has no group of translations to quotient by, and the analogous object — a window in the space the tiling was projected from — belongs to a different construction entirely.

What the domain is good for

Three uses, in increasing order of surprise.

It bounds the work. Anything computed pointwise over a pattern — a density, an integral, a sum over atoms — need only be computed over the domain and multiplied. That is the reason the asymmetric unit is the unit of storage as well as of description.

It identifies the special positions. The boundary of the domain is where the stabilisers are non-trivial, and an atom placed there has its symmetry constrained. The list of special positions in the International Tables is the list of boundary strata, and placing a motif on one without noticing is the commonest way to produce a figure of the wrong group.

It names the group. Fold the plane up along the group’s own symmetries and the domain becomes a small surface with marked points — an orbifold — and its shape determines the group completely. That is a change of language rather than of content, and it is the shortest of the three notations in use for the seventeen.

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.p1p2pmmp4p4mp6m6 groups, each generated and verified
Fig. 6 Six of the seventeen, in increasing order of symmetry: one operation per cell, then two, four, four, eight and twelve. The fundamental domain shrinks in exact proportion, from the whole cell to a twelfth of it, and a pattern’s group can be read off the size of the piece it needs.

The surprising part

The size of the domain is the reciprocal of the size of the group, so the more symmetric a pattern is, the less of it has to be specified. p1 needs a whole cell; p6m needs a twelfth.

Push that far enough and it becomes a statement about information rather than geometry. A pattern’s symmetry group is a compression scheme, and the compression ratio is exactly the group order. What makes this more than a metaphor is that the scheme is lossless and the ratio is not approximate: given the domain and the group, the pattern is reconstructed exactly, with nothing left to specify.

The connection worth carrying is to the accidental-symmetry problem. If the compression ratio is the group order, then a figure drawn with a motif that is accidentally too symmetric has been compressed harder than intended — the domain is smaller than the author thinks, because the real group is larger than the caption. The failure that this site’s round trip exists to catch is, in this language, a compression ratio that came out better than expected. Which is exactly when a result should be looked at twice.

Who thought of it

The idea comes from the study of Riemann surfaces rather than from crystallography. Poincaré and Klein, working in the 1880s on functions invariant under groups acting on the hyperbolic plane, needed regions from which the group rebuilt the whole space, and the machinery of fundamental domains was built there and imported afterwards.

Dirichlet’s construction — take the points closer to a chosen origin than to any of its images — gives a canonical domain for any discrete group and is the same construction that produces the Voronoi cell of a lattice and the Wigner–Seitz cell of a crystal. Three names for one region, arrived at independently in number theory, in computational geometry and in solid-state physics, which is a fair indication that the object was going to be found whatever anybody called it.

Crystallography’s asymmetric unit arrived with the International Tables in 1935 as a matter of bookkeeping: somebody had to decide which eighth of the cell got printed.

Where the ladder goes next

The construction the domain is built out of is the orbit, and the reason its boundary matters is the stabiliser.

The language that turns a domain into a name is orbifold notation, and the classification it makes tractable is the seventeen themselves.

What the pictures here cannot show. The domain drawn on each figure is a set of small squares approximating a region, and the approximation is visible: the boundary is stepped rather than straight. The true domain has a boundary made of mirror lines and segments joining rotation centres, and no sampled construction produces that exactly. What the sampling does give — a partition with neither gap nor overlap — is checked, and it is a different claim from the shape of the outline being right.

Where this goes next

1 of 43 of 44 of 4the pattern is grown from the group, never drawnp4 The orbit is the pattern A wallpaper is not designed and then found to have symmetry. It is the set of places a group sends a single mark, and once that is taken literally the pattern can be grown, checked, and caught out. p4m5 classes: 1 + 1 + 2 + 2 + 22 × 4-fold rotationsquare lattice · 8 operations per cellone class marked The same symmetry, somewhere else Two mirrors in a pattern can be the same symmetry or two different ones, and looking will not settle it. Conjugation is the operation that decides, and it turns an intuition about sameness into arithmetic. centred rectangularequal lengths, angle freethe arrows are the basis; the shaded region is one unit cell The cell is a choice, the lattice is not Every lattice has infinitely many unit cells and infinitely many bases, and crystallography picks one by convention. Knowing which convention is in force is the difference between a symbol that means something and a symbol that means nothing.