Discrete, or dense, and nothing between

Every count in this collection rests on a hypothesis that is never stated, because it is inside the word lattice: the translations of a pattern form a discrete subgroup of the plane.

Drop that and nothing becomes more general. The counts do not grow — they stop existing, because the object being counted is no longer a lattice and the machinery has nothing to act on. There is no eighteenth wallpaper group waiting outside the hypothesis; there is a different subject.

A subgroup of the plane under addition is one of five things, and the classification is short enough to be worth having whole.

The five #

A subgroup of the plane is:

• the origin alone;
• a lattice of rank one — a line of evenly spaced points, ℤv;
• a lattice of rank two — the case every essay here assumes;
• a dense subset of a line — points on one line, coming arbitrarily close together;
• dense in the plane.

Nothing else occurs. There is no subgroup that is spread out along a line but sparse, no subgroup that is discrete in one direction and dense in another while still being a group, and no exotic middle case.

The reason is one line. If a subgroup has points arbitrarily close to the origin, adding those points to themselves produces points arbitrarily close to every point of the line they lie on. So a subgroup that fails to be discrete does not merely fail — it fills something. Discreteness is an all-or-nothing property, and the classification is the list of what happens on each side of it.

The measurement that separates them #

Which case a subgroup is in is decided by a measurement, and the measurement is the more useful object.

Take the shortest non-zero vector produced by integer combinations with coefficients up to some bound, and watch it as the bound grows.

For a lattice it is constant. The shortest vector is the shortest vector; looking further finds nothing nearer, because a lattice has a minimum distance and it is attained.

For a subgroup that is not a lattice it falls, without limit.

Why it falls is the continued fraction. The convergents of √2 give integers p and q with |q√2 − p| less than 1/q, and that combination is an element of the group. As q grows the element approaches zero, so the group has elements as near the origin as one likes — which is exactly what a lattice cannot have.

Why the plane has no more cases than the line #

The five cases are the line’s three — trivial, a lattice, dense — arranged in two dimensions, and the reason there is nothing new is worth a paragraph.

Take a subgroup G of the plane. Look at the set of directions along which G has points arbitrarily close to the origin. If that set is empty, G is discrete, and a discrete subgroup is generated by at most two independent vectors, so it is a lattice of rank nought, one or two. If the set is a single direction, G contains a dense subgroup of that line and is otherwise discrete across it — the fourth case. If it is more than one direction, the sums fill the plane — the fifth.

So the classification’s content is that the middle case cannot be partial. A subgroup cannot be dense along a line in some places and not others, because it is a group: whatever happens near the origin happens near every one of its points, by translation. Homogeneity is what turns a local property into a global one, and it is the same reason a crystal’s symmetry near one atom is its symmetry everywhere.

Where this bites: the reciprocal object of a quasicrystal #

The classification is not a curiosity, because one of the non-lattice cases is a structure this collection spends a whole field on.

A ten-fold diffraction pattern has spots closed under addition — the sum of two scattering vectors is one — and closed under a five-fold rotation, since the pattern has that symmetry. So the spots form a subgroup of the plane containing five unit vectors at seventy-two degrees, and that subgroup is dense.

That is why the object is called a module and never a lattice. It needs four integers rather than two to index, its points are dense, and the sharp spots a photograph shows are the countably many with large amplitude rather than the whole set. Every one of the dense infinity of positions has some intensity; almost all of it is unmeasurably small.

Four, and not three, and the reason is worth having exactly — because three unit vectors at seventy-two degrees is the natural first guess and it is not enough. Write the turn as multiplication by a fifth root of unity ζ. The module generated by 1, ζ and ζ² is dense, and it is not carried onto itself by the turn: multiplying by ζ sends ζ² to ζ³, and ζ³ is not an integer combination of 1, ζ and ζ², because the only relation the fifth roots satisfy is that all five of them sum to zero and that relation involves ζ⁴. So the smallest module closed under the turn is spanned by four of them, and it is free of rank four. A pattern with five-fold symmetry needs four integers because its symmetry demands the fourth generator, not because the experimenter chose an awkward basis.

That is also the cleanest available statement of why five-fold and periodicity are incompatible in the plane. A lattice’s translations are a free group of rank two; a module closed under a five-fold turn is free of rank four; and rank four does not fit inside rank two. The crystallographic restriction, proved elsewhere in this collection from the trace of an integer matrix, is the same obstruction counted a different way.

So a quasicrystal’s diffraction is not a lattice with extra spots. It is a genuinely different object, and the classification above says exactly how different: the fourth case rather than the third.

Why the hypothesis is not automatic #

Two examples show that discreteness has to be assumed rather than deduced, and both are close to situations this collection treats.

Rank is not enough. The group generated by 1 and √2 has rank two over the integers, exactly as ℤ² does. It is isomorphic to ℤ² as an abstract group. What differs is where it sits, and no property of the group as a group can see the difference — which is why Bieberbach’s theorem needs the word crystallographic and would be false without it.

Closure is not enough either. A subgroup that is topologically closed as a subset of the plane is a lattice times a subspace, which is the fourth and fifth cases excluded. But “closed” is not a condition anybody checks on a set of translations found in a physical structure; what is checked is that the spots are separated, which is discreteness measured.

What decides it: a ratio being rational #

For a rank-two subgroup of a line, the whole question is whether the ratio of the two generators is rational. Rational gives a lattice of rank one, with the two generators being multiples of a common unit. Irrational gives density.

That reduces the classification to a question about a number, and it is a question with a procedure — the continued fraction — which terminates exactly when the number is rational.

And here is where the machinery reaches its own limit honestly. Run the continued fraction of √2 far enough on a computer and it terminates, because the double nearest √2 is a rational number with a denominator of about a thousand million million. The test decides about a machine number rather than about a real one.

So the classification of an actual set of measured spot positions cannot be made this way. What can be done is the measurement: watch the shortest combination as the search widens, and report the rate. The irrationality is brought in from outside; the density is measured. Those are different kinds of statement and the essays in this field keep them apart.

The same question one dimension up #

In space the classification has one more case and no more ideas: a subgroup is a lattice of rank nought to three, or is dense in a line, a plane or the whole of space, or is a lattice in some directions and dense along a subspace. The proof is the same argument applied to the subspace spanned by the small elements.

The extra case is the one crystallography actually meets. A structure periodic in two directions and quasiperiodic in the third — which is what a modulated crystal with an incommensurate wave along one axis is — has a translation group that is a rank-two lattice crossed with a dense rank-two group on the remaining line. It is neither discrete nor dense; it is both, in different directions.

That is why the satellites need a second integer along one axis and not the others, and why the extra dimension a superspace description adds is one rather than three. The classification says how many extra integers a structure needs, by saying which directions the density lives in.

What a physical structure is entitled to claim #

An experiment measures peak positions with an uncertainty, and no experiment distinguishes a rational ratio from an irrational one. What it can distinguish is the number of integers needed to index the peaks within the measurement’s precision, and that is the operational content of the classification.

A periodic crystal: every peak indexed by three integers, and no peaks appearing between as resolution improves.

A modulated or quasiperiodic structure: peaks needing four or more integers, with new ones appearing between the old as resolution improves, and their intensities falling systematically with the size of the extra indices.

The second is what Shechtman measured, and the argument that it was not a periodic crystal with a very large cell is precisely the argument above: a large cell gives a lattice with a small but constant spacing, and what was seen was spots at positions that a lattice could not hold at any spacing.

One consequence for the counts #

It is worth naming what the hypothesis buys, since every essay in this collection spends it.

A discrete group has a fundamental domain of positive area, so the quotient is compact and the point group is finite. Without discreteness the point group could be infinite — the rotations of a dense module are a dense set of angles — and then the crystallographic restriction says nothing, because there is no integer matrix argument to make.

A discrete group has a shortest vector, so the reduction algorithms terminate. Gauss’s reduction walks towards a shortest basis and stops because there is one to reach; on a dense group it would walk for ever.

And a discrete group has a well-defined index, so the subgroup counting works. Every count of domains, every coset enumeration and every statement about a superstructure needs the index of one lattice in another to be a whole number, which it is because both are discrete. On a dense group the index of a subgroup is not a whole number and usually not a number at all: ℤ inside ℤ + √2 ℤ has infinitely many cosets, and they are not indexed by anything a count could reach.

So the hypothesis is not one assumption among many; it is what makes the subject finite. Take it away and the failures are not at the edges — the machinery has nothing to grip on anywhere. That is worth remembering whenever a result in this collection is stated for “a crystal”: the word is carrying the discreteness, and every finite list that follows is finite because of it.

Where the theorem comes from #

The classification of closed subgroups of ℝⁿ is standard nineteenth- and twentieth-century material and has no single author; it is the first non-trivial thing one proves about topological groups, and the version here — a subgroup is a lattice times a subspace, or is not closed — is due in its general form to Pontryagin and the structure theory of locally compact abelian groups.

The measurement half is much older and belongs to number theory. Dirichlet’s approximation theorem of 1842 says that for any irrational α there are infinitely many fractions p/q with |α − p/q| < 1/q², which is exactly the statement that the shortest combination falls without limit, and the continued fraction is the algorithm that produces them.

Weyl’s equidistribution theorem of 1916 says more: the points of ℤ + αℤ are not merely dense but uniformly distributed, so the fraction of them in any interval is proportional to its length. That is the fact underneath the Fibonacci chain’s tile ratio being exactly the golden ratio, and it is the point at which this classification stops being an obstruction and starts being the machinery of the aperiodic field.

The three-distance theorem, as a picture of the fourth case #

The points of ℤ + αℤ inside an interval are not merely dense; they are arranged in a way with a name, and it is the cleanest picture available of what the fourth case looks like.

Take the first N multiples of α, reduced modulo one. They cut the unit interval into N + 1 pieces, and those pieces take at most three distinct lengths — never four, at any N, for any α. The theorem is Steinhaus’s, proved in the 1950s, and the three lengths are read off the continued fraction of α.

Two consequences matter here.

The gaps do not become uniform. The three lengths persist at every N, so a finite window of a dense group never looks like a lattice with a small spacing; it looks like a set with two or three spacings, which is exactly what the Fibonacci chain is and exactly what a quasiperiodic structure’s diffraction shows.

And the three lengths are not independent. The largest is always the sum of the other two, at every N and for every irrational — which is the theorem’s sharp form and is what makes the picture of the fourth case so rigid. It is measured directly on the points drawn here: sort them, take the gaps, and the distinct values come to three, with the largest equal to the sum of the smaller two to the last digit the arithmetic holds. A lattice would return one gap length. A random set of the same size would return as many distinct gaps as it has points. Three, with an addition relating them, is the signature of a group generated by two incommensurable steps and of nothing else.

And the number three is a symmetry statement. It is three rather than two because the interval is a circle and a circle’s rotations by α have that structure; the same theorem in higher dimensions has no such clean answer, which is why one-dimensional quasiperiodic order is understood and the general case is not.

The same alternative for the rotations #

The classification above is about translations, and the group of rotations has the same shape of answer — which is where several of this collection’s other counts come from.

A subgroup of the rotations of the plane is a subgroup of a circle, and the same argument applies unchanged: it is either finite or dense, with nothing in between. A subgroup containing a rotation by an irrational multiple of a full turn contains rotations by arbitrarily small angles, because the multiples of an irrational number are dense modulo one — which is the same continued-fraction fact the measurement above uses on translations.

So a pattern’s rotations are finite in number or come arbitrarily close to every angle, and the second is not something a discrete point set can have while remaining discrete. That is why a crystallographic point group is finite, and it is an assumption exactly as the lattice’s discreteness is: the classification of the finite rotation groups begins by supposing finiteness, and the alternative is not a longer list but a different kind of object.

It is also why the pinwheel tiling’s orientations are the strange case they are. Its tiles point in a dense set of directions, so the set of orientations is the second case, and the pattern’s “symmetry” is not a group of the kind any classification here handles.

What replaces periodicity when it goes #

The hypothesis this essay is about was, until recently, part of the definition of a crystal. It is not any longer, and what replaced it is worth stating because it is the definition the aperiodic field works under.

The condition that survives is about the points rather than about the translations. A Delone set is a set of points that is uniformly discrete — no two closer than some fixed distance — and relatively dense — no gap larger than some fixed size. Every crystal is one, every quasicrystal is one, and the definition says nothing about periodicity.

That is enough to keep most of what discreteness bought. A minimum separation is what makes a diffraction pattern’s peaks resolvable at all, and a maximum gap is what makes the structure’s density finite. What it does not buy is a lattice, so the counting arguments in this collection do not transfer — which is exactly why the aperiodic field has modules and windows in place of lattices and cells.

The definition crystallography actually adopted goes one step further and is a statement about the measurement: a crystal is a solid with an essentially discrete diffraction pattern. That is a deliberate move from a property of the structure to a property of what an experiment shows, and it was made because the structures found in 1982 satisfied the second and not the first.

Where the ladder goes next #

Into the aperiodic field, where the fourth case is the subject rather than the exclusion. Cut and project builds a quasiperiodic structure by taking a lattice in a higher-dimensional space and keeping the shadow of a strip — and the shadow is a dense projection of a discrete object, which is how a structure gets to be ordered without repeating.

And back into this field, where the hypothesis is used. Every essay here that begins “let L be the lattice of translations” is invoking the third case, and the seventeen, the fourteen and the two hundred and thirty are all counts of what can happen inside it. The classification above is the fence those counts sit inside, and it is worth having looked over it once.