The crystal you get by rounding τ off

A Fibonacci chain is what a strip of the square lattice projects to when the line of projection has irrational slope. Everything aperiodic about it comes from that one fact. An irrational slope never returns a lattice point to the same place in the window, so the sequence of tiles never repeats, and the chain has long-range order and no period.

Tilt the line to a rational slope and the construction survives entirely — the same lattice, the same strip, the same window, the same rule for which points are kept. What changes is that the sequence now repeats, and its period is a denominator.

These are the approximants, and they are not a computational convenience. They are real phases: they sit in the phase diagrams of the alloys that form quasicrystals, with unit cells of tens of ångströms, and their diffraction patterns look ten-fold at low resolution and turn out on closer inspection to be cubic.

Two constructions of one object #

An approximant can be built in two ways that share no arithmetic, and requiring them to agree is what makes the object real rather than defined.

By substitution. The Fibonacci rule — a long tile becomes a long and a short, a short tile becomes a long — applied n times to a single letter gives a word of F(n+2) letters. Repeat that word for ever and the result is the n-th approximant. Its long and short tiles number consecutive Fibonacci numbers, so their ratio is a convergent of the golden ratio, and its period is a Fibonacci number.

By projection. Take the same strip of the square lattice and cut it along a line of slope F(n)/F(n+1) instead of 1/τ. The chain that falls out is periodic — because the slope is rational — and its period is what the construction is for.

The two are then compared. The projected word is periodic; one period of it must be a rotation of the substitution word — not the same string, since the two constructions have no reason to start in the same place, but the same cyclic word. That is checked by searching the rotations rather than by lining the two up by hand, and it holds.

Why the agreement is evidence. One construction is string rewriting on two letters; the other is arithmetic on lattice points in a window. Nothing about the first knows there is a lattice and nothing about the second knows there is a substitution. Two calculations of that kind agreeing is a fact; either alone would be a definition.

The object being approximated is the same construction with the slope left alone, and it is worth keeping its two properties in view while reading the rest. Its slope is irrational, so the strip never returns to the same position relative to the lattice and the sequence of tiles never repeats; and no finite piece of it is a period of the whole, so there is no cell to find however far one looks. Every figure below that draws an approximant draws that chain underneath it, tile for tile, which is where the comparisons come from.

Right for a while, then wrong #

The interesting property of an approximant is not that it is periodic but where it stops being right.

The infinite Fibonacci word begins with every finite Fibonacci word. So the n-th approximant’s first cell is correct, letter for letter — it is a genuine piece of the quasicrystal — and the disagreement begins only at the seam where the cell repeats.

How far past the seam it stays right is measurable, and it is more than one might expect: the approximant of period 34 agrees for 87 tiles, the one of period 55 for 142, the one of period 89 for 231. In each case the agreement runs for about two and a half periods before the sequences part.

The agreement length is exact, and it is a Fibonacci number less two #

About two and a half periods is a rounding, and the underlying statement is an identity worth having, because it says the agreement length is arithmetic rather than empirical.

Read the three numbers above beside the Fibonacci sequence. The approximant of period 34 agrees for 87, which is 89 − 2. The one of period 55 agrees for 142, which is 144 − 2. The one of period 89 agrees for 231, which is 233 − 2. The agreement length is the Fibonacci number two places past the period, minus two, and it holds at every order the ladder is run to — 3 against a period of 2, 985 against a period of 377.

So the ratio of agreement length to period is not two and a half and does not settle at any rational. It is F(n+2)/F(n) with a small correction, and F(n+2)/F(n) tends to τ², which is 2.618. The observed multiples climb through 2.375, 2.462, 2.524, 2.559, 2.582, 2.596 and are still climbing at the last order drawn. What looked like a constant is a convergent, and the constant it converges to is the square of the number the whole construction is about.

That is the same τ² that governs the composition error. The error in the ratio of long tiles to short falls by a factor of $\tau^2$ at every step up the sequence of approximants, so the correct region grows at exactly the rate at which the composition becomes right. Neither of those is a coincidence and neither is an extra fact: both are F(n+1)/F(n) → τ differentiated once, which is why one number does two jobs here.

That is the property that makes an approximant a good model. A diffraction experiment sees a region of finite size, and inside a region smaller than the agreement length an approximant is the quasicrystal — not similar to it, identical. Telling the two apart is an experiment about long distances, and the longer the cell the longer the distance has to be.

The slope that reproduces the word #

There is a small arithmetic point in the correspondence between the two constructions that is worth recording, because getting it wrong is easy and the error is silent.

The obvious guess is that the n-th substitution word corresponds to the cut at slope F(n)/F(n+1). It does not: it corresponds to the cut one step earlier. Cutting at F(n)/F(n+1) gives a chain whose period is the next Fibonacci number after the denominator, so the words come out one generation apart and every comparison fails.

The correction is one index, and the way it was found is the point. The substitution period is known exactly — it is a Fibonacci number, asserted when the word is built — so the slope can be identified by requiring the projected period to equal it, rather than by reasoning about which slope ought to work. That is the same discipline the rest of this collection applies to counts: a number that can be checked against an independent calculation should be, and an index error that survives a check is a rarer thing than an index error that survives an argument.

What the diffraction does #

The two structures diffract differently, and the difference is exactly as small as the previous section suggests.

A quasicrystal’s reflections are indexed by two integers rather than one and sit at positions no single lattice contains. An approximant’s sit on one lattice of spacing one over its period. So the question is how far each of the quasicrystal’s strong reflections sits from the nearest reflection the approximant has to offer, and the answer falls away fast as the approximant grows.

These are measurements — a Fourier sum over a finite chain evaluated on a grid of wavevectors — and they carry the chain length and the grid with them. That distinguishes them from the rest of this collection’s claims about diffraction, most of which are exact statements about where an amplitude vanishes.

The physical reading is worth stating plainly. An approximant with a large enough cell is, to any real diffraction experiment, a quasicrystal — its peaks are in the right places to within the instrument’s resolution, and the extra reflections that would betray a lattice are too weak or too closely spaced to see. Deciding which one a specimen is turns on high-resolution work, and the history of the field contains several structures that changed sides.

What survives the rounding, and what does not #

It is worth being precise about which properties of the quasicrystal an approximant keeps, because the list is longer than the phrase periodic approximation suggests.

Kept: the local patches. Every patch of the approximant shorter than the agreement length is a patch of the quasicrystal, and — this is the stronger statement — every patch of the quasicrystal up to some size appears somewhere in the approximant. The two structures have the same local configurations; they differ only in how those configurations are arranged over long distances.

Kept: the composition. The ratio of long tiles to short is a convergent of τ, so the densities of the two tile types are right to within the convergent’s error.

Lost: the self-similarity. A Fibonacci chain is invariant under inflation — deflate it by τ and the same chain reappears. An approximant is not: inflating it gives a chain of a different period, one step further along the ladder. The infinite sequence of approximants is closed under inflation in a way no single one of them is.

Lost: the two-integer indexing. A quasicrystal’s reflections need two integers to be indexed; an approximant’s need one, because it has a lattice. The extra reflections a quasicrystal shows between the approximant’s are the visible sign of the second index.

That list is the reason approximants are studied rather than merely tolerated. A structure with the same local order and a period is a structure whose local order can be solved by ordinary crystallography, and the results transfer.

Where the numbers come from #

The sequence of approximants is the sequence of continued-fraction convergents of the golden ratio, and the golden ratio’s continued fraction is all ones — the slowest-converging there is.

That has two consequences worth separating.

The approximants are the worst possible. A convergent of a continued fraction is the best rational approximation for its denominator, and the golden ratio’s convergents improve most slowly of any irrational’s. So the Fibonacci quasicrystal is the hardest to approximate by a periodic structure, which is one way of saying it is the most irrational.

And they are the ones that appear. A structure whose composition must be close to τ will settle on a Fibonacci ratio, because those are the ratios available at small denominators — so the approximant phases observed in real alloys are the ones the continued fraction names, and their cell sizes are Fibonacci multiples of the quasilattice constant. The 1/1 and 2/1 approximants of the icosahedral aluminium-manganese-silicon system are exactly this, in three dimensions rather than one.

What is exact and what is measured #

Exact: the words, the periods, the tile counts, the ratios, the position of the first disagreement, and the identity between the two constructions. All of it is integer arithmetic on strings and on lattice points, with the tile ratios being ratios of Fibonacci numbers and the agreement lengths being positions in a string.

Measured: the diffraction. Peak positions come out of a Fourier sum on a grid, and every shift quoted carries the grid it was measured on.

Asserted and checked, in the negative direction: that the irrational cut gives a chain with no period inside the window measured. That is not a proof of aperiodicity — a finite window cannot prove one — and it is stated as what it is: the shortest period of a chain of several hundred tiles, which is none.

The window, and what it decides #

One parameter of the construction has been left unstated and it deserves a paragraph, because it is the only choice in the whole procedure.

The strip has a width, and the width chosen here is the projection of one unit cell onto the perpendicular direction. That is not arbitrary in the way a bonding cutoff is arbitrary: it is the width that makes the projection a bijection between the kept lattice points and the points of the line, so that no point is lost and none is doubled. A narrower window drops points and leaves gaps of three lengths rather than two; a wider one keeps too many and the tiles overlap.

At that width the chain has exactly two tile lengths, and the reason is the three-distance theorem: the gaps between the first n points of an irrational rotation take at most three lengths, and the window’s width spends the third. So the two-letter alphabet the substitution works in is not a modelling choice — it is forced by the geometry, and the same forcing applies to the rational cut.

That matters here because it is what makes the two constructions comparable at all. If the projection produced a three-letter word and the substitution a two-letter one, no rotation would ever match them.

Two things an approximant is not #

It is not a quasicrystal with defects. An approximant is a perfect periodic crystal; nothing in it is disordered, and its diffraction is as sharp as any crystal’s. The difference from a quasicrystal is in where the reflections are, not in how sharp they are — which is the opposite of the difference a faulted structure shows, where the positions are right and the sharpness goes.

And it is not an approximation in the numerical sense. The word suggests something computed to a tolerance, and there is no tolerance anywhere: an approximant is an exactly defined periodic structure, and the sense in which it approximates is that it agrees with another exactly defined structure on a finite region. Everything about the agreement is a statement about integers and positions in a string.

Nor is it unique for its period. Several distinct periodic structures share a given cell size and a given composition, and only the ones the continued fraction names have the local patches right.

The same move, elsewhere in this collection #

Replacing an irrational by a rational and watching a structure become periodic is a move this collection has made before under other names, and the family resemblance is worth noticing.

An incommensurately modulated structure has a modulation whose period is an irrational multiple of the lattice period; make the ratio rational and the structure becomes a superstructure with a large cell, which is an ordinary crystal. The same alloy can show either, depending on temperature, and the sequence of rational values it locks into as it cools is called a devil’s staircase.

Two chains with no common period are the same phenomenon with two lattices instead of one: rationalise the ratio and the composite becomes commensurate, with a cell that is a common multiple.

In each case the irrational case is the generic one and the rational cases are a countable set inside it, and in each case the rational cases are the ones with a cell, a space group and a solvable structure. That is the recurring bargain of this part of the subject: periodicity is what makes a structure tractable, and it is the property the interesting structures do not have.

The approximants that exist #

The construction here rounds a slope off and produces a periodic chain. Real materials do the same thing, they are named by the fraction used, and their existence is the strongest evidence that the construction describes something.

In the aluminium–manganese and aluminium–copper–iron systems, alongside the icosahedral phases, there are cubic phases whose local arrangements are the same clusters in the same relative orientations, arranged periodically. They are called 1/1, 2/1, 3/2 approximants after the Fibonacci ratio their structure corresponds to, and the cell edge grows with the denominator exactly as the period does here.

They are ordinary crystals. They have a space group, a unit cell, a finite list of atomic positions, and a diffraction pattern indexed by three integers. Nothing about them requires the aperiodic apparatus at all, and a crystallographer meeting one without knowing the context would solve it as a large intermetallic structure.

What makes them interesting is the comparison. Their diffraction patterns have strong reflections at very nearly the positions of the quasicrystal’s strong ones — the shift falling as the denominator grows, exactly as the measurement in this essay shows — and their local structures are the quasicrystal’s local structures. So the family is a sequence of ordinary crystals converging on something that is not one, and every member of it is solvable by methods the last member defeats.

Why the periodic cousin is how the aperiodic one is solved #

That last observation is a method rather than a curiosity, and it is how most of what is known about quasicrystal structures was found.

A quasicrystal cannot be solved by ordinary crystallography. It has no cell, so there is no asymmetric unit to list, and its structure has to be described in six dimensions with atomic surfaces whose shapes are the unknowns. That is a much harder determination than a periodic one, with far fewer constraints available.

An approximant is solvable. It has a cell, its reflections index on three integers, and every technique in the rest of this collection applies to it unchanged. So the route taken in practice is: find the approximant, solve it as an ordinary crystal, extract the clusters, and assume the quasicrystal is built from the same clusters arranged aperiodically.

The assumption is the weak link and it is worth naming as one. The two structures share their local arrangements as far as anybody can check, and as far as anybody can check is doing real work in that sentence — the check is a comparison of diffraction intensities and of a handful of measured environments, not a proof.

What makes the assumption reasonable is the construction on this page. The approximant and the quasicrystal come from the same lattice, the same strip and the same rule, differing in one number; so their local patches agree out to a distance that grows with the denominator, exactly and computably. That is a statement about the model rather than about the material, and it is the strongest thing available.

Where this goes #

The approximants are the bridge between the two halves of this collection’s treatment of order without repetition: the periodic side, where everything is decided by integer arithmetic in a lattice basis, and the aperiodic side, where nothing is. An approximant is on the periodic side by every test and on the aperiodic side by every measurement short of the finest.

That is also the honest summary of what Shechtman’s measurement had to rule out. A ten-fold pattern from a large-celled approximant would look very like a ten-fold pattern from a quasicrystal, and separating the two took better data than anybody had in 1982.