Crystallography & Symmetry
patterns derived from their groups
Lattices

The lattice underneath

Strip a pattern of everything but its repeats and a grid of points is left. That grid is not decoration — it is the object that decides which symmetries the pattern is permitted to have.
14 min read 6 figures The lattice forbidsSymmetry is decidableGenerated, not drawn

Take any pattern that repeats. Ignore the motif entirely — its shape, its colour, its charm. Keep only the answer to one question: which slides leave the pattern exactly as it was?

The answer is a set of vectors, and that set is remarkably rigid.

The hexagonal latticeEvery 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.hexagonalequal lengths, angle 120°the arrows are the basis; the shaded region is one unit cell
Fig. 1 A lattice: the set of all translations that map a pattern onto itself, drawn as the places a chosen point is sent. Nothing about the motif survives this operation. What remains is the pattern’s skeleton, and it is the skeleton that constrains everything else.

It contains the zero vector, since doing nothing is a slide. It is closed under addition, since two slides that each work compose to a third that works. It contains the negative of anything in it, since a slide that works can be undone. So the translations of a pattern are a group, entirely on their own, before any rotation or mirror is considered — and being a group is what makes them so constraining.

What the set looks like

For a pattern that repeats in two independent directions, the translation group turns out to be generated by two vectors. Every translation in it is a whole-number combination of those two, and nothing else is in it.

Written down, the lattice is the set

Λ={ ma1+na2:m,n∈Z }\Lambda = \{\, m\mathbf{a}_1 + n\mathbf{a}_2 : m, n \in \mathbb{Z} \,\}

for two vectors a1\mathbf{a}_1 and a2\mathbf{a}_2 that are not parallel. The vectors are called a basis; the parallelogram they span is a unit cell; and neither the basis nor the cell is unique, which is a source of confusion with its own essay.

The word “lattice” is used for the set of points and, loosely, for the group of translations that produces them. The two are the same object seen from opposite ends: pick an origin and the group generates the points; take the points and the differences between them are the group.

Why the whole numbers matter

The restriction to whole-number combinations is not a formality. It is the source of every finiteness result in the subject.

A set closed under addition with real coefficients would be the whole plane. A set closed under addition with whole-number coefficients is discrete — its points are separated, there is a shortest non-zero vector in it, and that shortest vector is the lever every impossibility proof pulls on.

The pattern of those proofs is always the same. Assume a pattern has some symmetry. Use the symmetry to construct a lattice vector shorter than the shortest one. Contradiction. This is how five-fold rotation is ruled out, and it works because a lattice has a shortest vector at all — which is exactly what discreteness buys.

Every other operation must respect it

Here is the constraint that gives this essay its title.

Suppose a pattern has a rotation RR as a symmetry, and a translation t\mathbf{t} as a symmetry. Then the composition — rotate, slide, rotate back — is also a symmetry, and it is a translation by RtR\mathbf{t}. So the rotated version of every lattice vector must itself be a lattice vector.

That is a severe demand, and it is worth stating in the form that makes its severity obvious: a rotation cannot merely be compatible with the motif. It must map the entire infinite lattice onto itself. The same goes for every mirror and every glide.

The square latticeEvery 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.squareequal lengths, angle 90°the arrows are the basis; the shaded region is one unit cell
Fig. 2 A square lattice, with the basis vectors marked. Any symmetry of a pattern on this lattice must send this grid of points to itself — not approximately, and not just near the origin, but exactly and everywhere.

The set of rotations and reflections that map a given lattice onto itself has a name: the lattice’s holohedry, or point symmetry. It is finite, it is small, and computing it is the first step in every classification argument. An oblique lattice has a holohedry of order two — the identity and a half turn. A hexagonal lattice has order twelve. Those numbers are the ceiling on what a pattern built on that lattice can have.

The lattice has more symmetry than the pattern

A point that trips up most readers on first contact: the lattice’s symmetry is an upper bound, not a description.

A pattern on a square lattice need not have fourfold symmetry. It needs only symmetries that are available on a square lattice. A pattern in p2 sitting on a square lattice has only half turns, and the lattice’s own quarter turns are not symmetries of the pattern because they do not respect the motif. The lattice permits; the motif disposes.

This is why the classification is not simply a list of the five lattices. Each lattice supports several groups — the square lattice supports p4, p4m and p4g, and also, if the motif declines the opportunity, everything the rectangular lattice supports. The seventeen come from pairing lattices with the point groups they can carry and then asking, for each pairing, where the elements can sit.

The pattern and its lattice, side by side

It helps to see the two objects in the same picture, because the relationship between them is easy to describe and easy to get backwards.

The wallpaper group p1A pattern with the symmetry of p1, generated by applying the group's 1 operations to an asymmetric motif and repeating across the lattice. The symmetries of the result were then found independently and match the group exactly.p1oblique lattice · 1 operations per cell
Fig. 3 A pattern in p1 — no symmetry at all beyond translation — with its unit cell drawn. The cell is a statement about the lattice; the marks inside it are a statement about the motif; and the two are independent choices that together determine the pattern.

The pattern determines the lattice: given the marks, the set of slides that map them onto themselves is fixed, with nothing left to decide. The lattice does not determine the pattern: any motif at all may be placed in the cell, and infinitely many do not disturb the lattice.

That asymmetry is the reason the classification counts groups rather than lattices. Five lattices support seventeen groups, because the question is not only which lattice but what else is compatible with it — and the “what else” is where the mirrors, the glides and the rotation centres live.

Coordinates along the lattice, and what that buys

Now the practical consequence that everything on this site depends on.

Ordinary page coordinates are the wrong coordinates for this subject. In them, a threefold rotation involves 3/2\sqrt{3}/2, a hexagonal lattice’s basis vectors are at 120°120° with irrational components, and every comparison between two operations becomes a comparison of floating-point numbers with a tolerance somebody had to choose.

Choose coordinates along a1\mathbf{a}_1 and a2\mathbf{a}_2 instead — so that a point is described by how many repeats along each axis it sits — and everything becomes exact. A lattice vector is a pair of whole numbers. A rotation that maps the lattice to itself is a matrix of whole numbers, because it maps basis vectors to lattice vectors and lattice vectors have integer coordinates. A translation attached to a glide is a fraction with a small denominator: a half, a third, a quarter.

In those coordinates the threefold rotation of a hexagonal lattice is the matrix

R3=(0−11−1)R_3 = \begin{pmatrix} 0 & -1 \\ 1 & -1 \end{pmatrix}

with four whole-number entries and no square roots anywhere. Cube it and the identity comes back exactly, not to fifteen decimal places.

Decidability, which is the unusual part

That exactness is why this site can round-trip every figure rather than merely check it.

Asking whether a pattern has a given symmetry becomes asking whether two lists of exact fractions are equal. There is no tolerance to choose and no residual to interpret. The detector enumerates the finitely many integer matrices in the lattice’s holohedry, determines each candidate translation exactly from a pair of pattern points, applies the operation, and compares point sets.

Very few subjects that draw pictures of computed things are in this position. Most are comparing a residual against a threshold, and choosing the threshold badly is a standing hazard — set it loose and wrong pictures pass, set it tight and right ones fail. Here the question has an answer rather than a confidence level.

The reciprocal lattice, mentioned early

There is a second lattice hiding in the first, and it is the one an experiment actually sees.

Given a lattice, there is a companion — the reciprocal lattice — whose vectors are perpendicular to the original’s planes and whose lengths are inversely proportional to the spacings. Long in one is short in the other. It sounds like an abstraction and it is not: it is literally where a crystal scatters. A diffraction pattern is a picture of the reciprocal lattice, weighted by what sits in the unit cell.

A lattice and its reciprocalThe reciprocal lattice is where a crystal scatters. A long axis in the crystal gives closely spaced spots and a short one gives widely spaced spots, so the diffraction pattern is the structure turned inside out.the crystal latticewhere it scatterslong in one is short in the otheraxis ratio 1.7
Fig. 4 A lattice and its reciprocal. The axis that is long on the left is short on the right, which is why a diffraction pattern always looks stretched the opposite way to the crystal that produced it.

That relationship is the subject of its own essay, and it matters here because it is the second independent route to the same information. A pattern figure asserts a group from the point set; a diffraction figure computes what the same point set would scatter. Two calculations sharing nothing but the atom positions.

Sublattices, superlattices and phase transitions

Lattices sit inside one another, and the relation is physically important rather than merely tidy.

Take every second point along one axis of a lattice and what remains is still a lattice — a sublattice of index two. Going the other way, adding points at the centres of the cells gives a superlattice containing the original. Both operations preserve the defining property, and both occur in real materials.

An alloy in which two metal species are randomly mixed has a small cell, because the species are indistinguishable to the lattice. Cool it and the species may order, alternating regularly; the true repeat is now twice as long in some direction, and the lattice has become a sublattice of what it was. The diffraction pattern responds by growing extra reflections — superlattice reflections — exactly halfway between the ones that were there before. Reciprocal space makes the halving visible as a doubling, which is one of the more satisfying consequences of long-becomes-short.

Counting points per cell

A small piece of bookkeeping that becomes important the moment centred cells appear.

Draw a unit cell on a lattice and count the lattice points it contains. A point at a corner is shared between four cells, so it counts a quarter; a point on an edge is shared between two and counts a half; a point strictly inside counts one. A primitive cell, by definition, contains exactly one lattice point when counted this way — four corners at a quarter each.

The centred rectangular latticeEvery 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.centred rectangularequal lengths, angle freethe arrows are the basis; the shaded region is one unit cell
Fig. 5 The centred rectangular lattice, drawn with the cell crystallographers actually use. The conventional cell contains two lattice points rather than one, which looks like waste and is not: it is what makes the rectangular symmetry of the lattice visible in the choice of axes.

Why accept a cell with two points in it when a primitive one exists? Because the primitive cell of this lattice is a rhombus whose axes sit at an awkward angle to the lattice’s mirror lines, so a description in those axes conceals the symmetry it is supposed to exhibit. The centred cell has orthogonal axes aligned with the mirrors, at the cost of an extra lattice point. Crystallography chose clarity over economy, consistently, and the essay on cells takes up why.

The same trade-off, in three dimensions, is why Bravais’s fourteen lattices include face-centred and body-centred varieties rather than being reduced to primitive cells throughout. Reducing them is always possible and almost always a bad idea.

Real crystals, and where the idealisation frays

Everything above is about an infinite perfect lattice. Real materials are neither.

A real crystal is finite, so strictly it has no translational symmetry at all: every candidate translation eventually runs off the surface. It contains vacancies, substitutions, dislocations and grain boundaries, each of which locally destroys the repeat. It is at a temperature, so its atoms are not at lattice points but oscillating about them.

None of this makes the idealisation wrong; it makes it an idealisation, which is a different thing. A crystal a millimetre across contains some millions of repeats along each axis, and for any question about local structure the difference between millions and infinity is not detectable. Where it is detectable, the effects have names and a literature — finite-size broadening of diffraction peaks, diffuse scattering from defects, thermal factors that reduce peak intensities without moving them.

The right way to hold it is that the lattice describes the ideal the material approximates, and the quality of the approximation is measurable. A good single crystal of quartz approximates it extremely well. A lump of window glass does not approximate it at all, which is why glass is not a crystal.

The oblique latticeEvery 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 anglethe arrows are the basis; the shaded region is one unit cell
Fig. 6 An oblique lattice, the least constrained of the five: no equal lengths, no special angle. Every periodic pattern has a lattice at least this general, and most of the subject consists of asking what happens when it is more special than this.

What a lattice is not

Three confusions are common enough to be worth naming.

A lattice is not a drawing of lines. The grid lines in every figure here are an aid to the eye. The lattice is the set of points, or equivalently the set of translation vectors; the lines connecting them are notation.

A lattice is not the crystal structure. The structure is the lattice plus the contents of the unit cell. Sodium chloride and diamond can share a lattice type and be entirely different substances. Conflating the two is the commonest error in introductory accounts, and it produces the false statement that there are only fourteen kinds of crystal.

A lattice is not the same as a tiling. A tiling covers the plane with shapes; a lattice is a discrete set of points. Every periodic tiling determines a lattice, but a Penrose tiling determines none, because it has no translations at all.

Who worked it out

The lattice idea arrived from mineralogy rather than from mathematics, and it arrived as an explanation of something puzzling.

René Just Haüy, a French mineralogist, dropped a specimen of calcite in about 1781 and noticed that the fragments cleaved into shapes with the same angles as the original, at every scale he could examine. His inference, published in 1784, was that a crystal is built from identical small units stacked periodically — molécules intégrantes — and that the flat faces of a crystal are the planes along which such a stack can be cut.

That is the lattice hypothesis, proposed a century before there was any way to test it, and it explained the law of constant interfacial angles that Nicolas Steno had observed in 1669. Auguste Bravais put it on a proper footing in 1848 by enumerating the fourteen lattice types in three dimensions, and it stayed a hypothesis until 1912, when Max von Laue put a crystal in an X-ray beam and got the diffraction pattern that only a periodic array could produce.

Where the ladder goes next

The immediate question is how many lattices there are, and the answer in the plane is exactly five, for reasons that come straight out of the holohedry.

The immediate hazard is that a lattice does not determine its own description: the unit cell is a choice, several conventions are in use, and the centred cell that looks like a redundancy is doing real work.

And the immediate consequence is the one this whole field turns on. A lattice tolerates rotations of order one, two, three, four and six, and nothing else whatever — a proof of one line, and the constraint that makes seventeen a theorem rather than a tally.

What the pictures here cannot show. A lattice is infinite and every figure shows a patch. More importantly, no drawing can show that a set of translations is complete — the claim that a pattern has no shorter repeat than the one drawn is a claim about a search, and the search is arithmetic rather than visual.

Where this goes next

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 Five lattices, and no others A repeating grid can be oblique, rectangular, centred, square or hexagonal. That is the complete list for the plane, and the argument that closes it is a page long. 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. -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.