The smallest quasicrystal

A quasicrystal in two dimensions is a Penrose tiling and takes some drawing. In one dimension it is a row of two kinds of segment, and everything that made quasicrystals an argument in 1982 is already present in it: an order that never repeats, a diffraction pattern of sharp peaks, and reflections that cannot be labelled by one integer each.

Being small makes it checkable. The same chain can be built three unrelated ways, and two of those are required on this page to produce the same sequence of tiles — not a similar one.

Construction one: a rewriting rule #

Start with a single long tile. Replace every long tile by a long followed by a short, and every short tile by a long. Repeat.

The lengths of successive generations are 1, 2, 3, 5, 8, 13 — each the sum of the two before it, because a generation’s tiles are the previous generation’s long tiles expanded into two and its short tiles into one, which is the previous generation plus the one before that. The counts of long and short tiles are consecutive Fibonacci numbers, so their ratio approaches the golden ratio, and the figure measures that ratio off the string rather than asserting it.

The sequence never becomes periodic. If it did, the ratio of long to short tiles would be rational — a repeating block has whole numbers of each — and the ratio is the golden ratio, which is not. That is the whole proof, and it is the one-dimensional form of the argument that a Penrose tiling cannot repeat.

Construction two: a strip through a square lattice #

Now forget substitutions. Take the ordinary square lattice, draw a strip across it at a slope of one over the golden ratio, keep the lattice points inside the strip, and project them onto the strip’s direction.

Two features of the construction are choices and both matter.

The slope is irrational. A rational slope gives a strip that repeats — after some number of steps the lattice presents the same configuration again — and the projected chain is periodic. Irrational slope, no repeat, and the chain inherits the failure to repeat from the line rather than from anything about tiles.

The strip’s width is one unit cell, measured across. That is what makes the projection produce exactly two gap lengths. A wider strip admits a third gap length; a narrower one leaves gaps where no lattice point qualifies. The width is the window, and choosing it is the step the general cut-and-project construction spends most of its care on — the last section of this page sweeps the width and counts the gap lengths at each setting, which is what makes that a measurement rather than a claim.

The agreement between the two constructions is asserted rather than admired. The projected sequence is read off as a word in long and short gaps, and that word is required to appear verbatim inside the substitution word — which it does, and which nothing but the two constructions describing one object would produce. One side is a rewriting rule on strings; the other is integer points in a strip.

Construction three: what it scatters #

The third route is the one an experimenter has, and it is a measurement rather than a decision — a chain with no repeat has no lattice, so the site’s exact machinery has nothing to work in.

Two things are worth separating in that picture.

The peaks are sharp. Sharp peaks are what “long-range order” means experimentally: a pattern with no long-range order scatters diffusely. The chain has no repeating unit and scatters sharply, which is precisely the combination that had to be admitted as possible in 1982 and which Shechtman’s measurement forced.

The peaks need two integers. A periodic chain’s peaks sit at integer multiples of one wavevector. These do not. Fitting the strongest peaks to integer combinations of the two strongest wavevectors succeeds with a residual well under one per cent — and the ratio of those two basis vectors comes out at the golden ratio, which nothing in the fitting assumed. That is the one-dimensional form of the indexing problem: a quasicrystal’s reflections are labelled by more integers than the space has dimensions, and the extra ones are what the higher-dimensional description accounts for.

Reading the chain as a number #

There is a third description, which is neither a rule nor a projection, and it is the one that makes the chain’s arithmetic obvious.

Number the tiles from the start. The nth tile is long when the fractional part of n divided by the golden ratio falls in one interval and short when it falls in the other — so the sequence is generated by repeatedly adding an irrational number and asking which side of a threshold the result lands on. That is the same construction as the strip, written as arithmetic instead of geometry: the position across the strip is the fractional part.

Two consequences follow immediately from that description.

The chain is determined by one irrational number, and different irrationals give different chains. The golden ratio’s chain is the one substitution produces because the golden ratio’s continued fraction is all ones, which is the same fact that makes it the hardest number to approximate by fractions.

The chain has no free parameters. A periodic chain of two tile lengths can have any ratio and any arrangement; this one has neither choice available once the slope is fixed. That rigidity is what “quasiperiodic” means as against “aperiodic”: the sequence is not merely non-repeating, it is generated by a rule with no room in it.

Why two integers rather than one #

The two constructions explain each other at exactly this point, which is the reason for putting all three on one page.

The chain was cut from a two-dimensional lattice. A two-dimensional lattice has a two-dimensional reciprocal lattice, and every reciprocal point projects to a wavevector on the line. So the chain’s diffraction is the projection of a two-dimensional set of reflections onto a line, and each surviving peak carries the two indices its parent reflection had.

That also explains why the peaks are dense on the line and yet the pattern looks discrete: projecting a two-dimensional lattice onto a line of irrational slope gives a set with points everywhere, but the intensities fall away fast, so only finitely many are visible at any threshold. A quasicrystal’s diffraction pattern is dense and appears discrete, and the appearance is a fact about intensities rather than about positions.

Where the chain shows up #

The chain is not only a teaching example, and two of its appearances are worth naming because they arrived independently.

In real quasicrystals. The one-dimensional layer sequences of several quasicrystalline alloys are Fibonacci chains, measured directly by electron microscopy: the spacings between atomic planes take two values in the golden ratio, in exactly this sequence. The chain is a description of matter and not only of a construction.

In systems with two competing periods. Whenever a physical system is driven at two frequencies whose ratio is irrational, the sequence of events it produces has this structure — the same arithmetic of adding an irrational number and reducing modulo one. That is why the chain turns up in the study of quasiperiodic forcing, in the Frenkel–Kontorova model of an adsorbed layer, and in the spectra of one-dimensional Schrödinger operators, where the Fibonacci potential is a standard object with a Cantor spectrum.

None of that is derived here. It is worth stating because it is the reason a one-dimensional toy has a literature: the chain is the simplest object in which an irrational ratio becomes a discrete structure, and discrete structures with irrational ratios turn out to be common.

What recurs, and how often #

Aperiodic does not mean disordered, and the chain makes the difference concrete in a way that a two-dimensional tiling does not.

Take any finite block of the chain — say the seven tiles beginning at some position. That block recurs infinitely often along the chain, and the gaps between its occurrences are bounded: there is a distance within which a copy is guaranteed. A pattern with that property is called repetitive, and every quasicrystal in the mathematical sense has it.

So the chain is not “a pattern that fails to repeat”. It is a pattern in which every finite piece repeats, infinitely often, at bounded gaps — and in which no single translation maps the whole thing onto itself. Those two statements are compatible, which is the fact the whole subject turns on, and this is the smallest object that exhibits it.

The matrix behind the substitution #

Everything about the chain’s arithmetic follows from a two-by-two matrix, and writing it down connects the substitution rule to the golden ratio without any inspection of the sequence.

Count the long and short tiles in one generation. The rule turns each long into one long and one short, and each short into one long — so the next generation’s counts are L' = L + S and S' = L. That is a matrix acting on the pair of counts, with rows (1, 1) and (1, 0), and repeated substitution is repeated multiplication by it.

Its eigenvalues are the golden ratio and minus its reciprocal. The larger one dominates, so the counts grow by a factor of the golden ratio per generation and their ratio converges to it — which is why the tile ratio is what it is, and why it converges rather than oscillating. The same matrix is the one whose powers produce the Fibonacci numbers, which is the other way of seeing the same fact.

The eigenvector for the smaller eigenvalue is what the strip’s width comes from. In the cut-and-project picture, the perpendicular direction is the one the small eigenvalue contracts, so the window shrinks under inflation and the construction is self-consistent — inflating the chain and re-cutting the strip give the same points. That correspondence between an eigenvalue and a window is the general mechanism behind every substitution tiling, and the chain is the smallest place it can be watched.

What breaks with the wrong window. Widening the strip past one unit cell admits a third gap length and the sequence is no longer a two-letter word; narrowing it leaves stretches with no points at all. Neither result is a Fibonacci chain. Both are measurements rather than assertions here, and making the measurement corrects the usual statement of it.

The width that works is not unique, and the reason is the same self-similarity as everything else on this page. Scaling the window by φ scales the projected chain by 1/φ, so a strip φ times as wide gives the same Fibonacci chain at a finer scale: its long gap is the unit window’s short gap, exactly, which the figure checks. So the good widths are the powers of φ and the bad ones are everything between them, and “one unit cell” is the representative of that family rather than a magic number. What the eigenvector fixes is the family; which member of it is drawn is a choice of scale and nothing more.

Why the peaks are dense and the picture is not #

One feature of the diffraction figure deserves separating out, because it is the thing that most often confuses a first reading.

The chain’s diffraction peaks are dense: between any two of them there is another, everywhere along the axis. That follows immediately from the two-integer indexing, since the numbers formed by adding a whole number to a whole multiple of the golden ratio come arbitrarily close to any real number.

A picture of a dense set ought to be a solid block of ink, and the figure is not. What saves it is that the intensities fall away extremely fast as the indices grow: a peak indexed by large integers is present and unmeasurably weak. So the pattern appears discrete at any threshold, with the number of visible peaks growing as the threshold falls.

That is more than an artefact of drawing. It is what a real diffraction measurement sees, and it is the reason a quasicrystal’s pattern can be indexed at all — an experimenter records the peaks above their noise floor, gets a finite list, and finds two integers per peak. Lowering the noise floor adds peaks between the ones already found rather than extending the pattern outwards, which is a signature that distinguishes a quasicrystal from a crystal with a large cell.

The distinction between dense and discrete is therefore not visible in any single measurement. A crystal with a very large unit cell also has closely spaced peaks, and telling the two apart means examining how the pattern fills in as sensitivity improves. That was one of the serious objections raised against the 1982 result, and answering it took better data rather than better argument.

Where the exactness stops #

The chain is a measurement, not a decision. The site’s integer machinery decides symmetry for periodic patterns in the plane by working in a lattice basis. There is no lattice here. The tile ratio is measured, the diffraction peaks are located numerically, and the indexing residual is reported rather than asserted to zero.

The agreement between the constructions is checked on a finite piece. The projected word is required to appear inside a substitution word of finite depth. Both sequences are infinite and the check is on a window of twenty tiles, which is strong evidence and not a proof — the proof is a standard result about Sturmian sequences and it is not derived here.

One dimension. Nothing here establishes anything about Penrose tilings or icosahedral quasicrystals; the chain is the simplest member of a family and its simplicity is exactly what makes it unrepresentative of the difficulties in the others.

Where the ladder goes next #

The general construction the strip is an instance of, in the plane rather than on a line, is cut and project.

The measurement that made all of this a physical question is what Shechtman measured, and the point group it revealed — the one no crystal may have — is icosahedral symmetry.

The two-dimensional substitution, and the ratio it converges to, is inflation and the golden ratio.

What the pictures here cannot show. Every figure on this page is a finite piece of an infinite object, and every claim that matters — never repeats, every block recurs, the peaks are dense — is a claim about the infinite one. A drawing of forty tiles is compatible with a chain that becomes periodic at tile forty-one, and the reason to believe otherwise is the construction rather than the picture.

Running the rule backwards #

The substitution builds the chain outwards from a single tile, and the operation that matters most is the one that goes the other way. It is available, it is unique, and its uniqueness is what “hierarchical order” means concretely.

The chain can be read as super-tiles. Every long tile in a Fibonacci chain is followed either by a short tile or by another long one. Group each long-then-short pair into one block, and let every long tile not followed by a short one stand as a block on its own. The blocks are of two kinds, they occur in the ratio the tiles occurred in, and the sequence of blocks is itself a Fibonacci chain.

The grouping is forced. There is exactly one way to cut the chain into blocks of that kind — no choice is made at any point, and no other cutting produces a Fibonacci chain of blocks. That property is called recognisability, and it is not automatic: substitutions exist whose output can be decomposed in several ways, and they behave quite differently.

So the deflation can be iterated. Blocks group into super-blocks, super-blocks into super-super-blocks, and the process never terminates because the chain is infinite and the rule is invertible at every stage. A point of the chain therefore sits inside a nested sequence of blocks of every size, and its address in that hierarchy is unique.

That is the structure the diffraction is measuring. A chain with no repeat has no translations to give it long-range order, and the hierarchy is what replaces them: two regions of the chain agree because they occupy corresponding positions in super-tiles of a large enough order, not because a translation carries one to the other.

And it is why the chain is called self-similar rather than merely aperiodic. Inflate by the golden ratio and the chain maps onto itself, tile for tile. Almost no aperiodic sequence does anything of the kind.

The chain is one of a family #

The golden ratio is not the only number that produces a chain of this sort, and knowing the family makes clear which of the chain’s properties are about golden and which are about the construction.

Change the substitution’s arithmetic. Send the long tile to two longs and a short, and the short to a long. The counts satisfy a different recursion, the ratio of tile numbers converges to $1 + \sqrt{2}$, and the chain that results is aperiodic, self-similar and hierarchical in exactly the way the Fibonacci chain is.

The projection construction changes only in its slope. Cut the same square lattice with a strip of slope $\sqrt{2} - 1$ rather than $1/\varphi$ and the projected chain is the one that substitution produces. The strip’s width still has to be one cell across to give two gap lengths, and the slope still has to be irrational to prevent a repeat.

The diffraction changes only in its scale factor. Peaks still need two integers, and the second integer is still multiplied by the chain’s own irrationality — $1 + \sqrt{2}$ in place of the golden ratio.

So everything structural survives and one number moves. What does not survive is the extremal property: the golden ratio is the irrational hardest to approximate by rationals, so the Fibonacci chain is the one whose patches return soonest and whose deviation from periodicity is most evenly spread. Its relatives are perfectly good quasiperiodic chains that are slightly less well behaved, in a sense that the repetitivity measurement makes precise.

And not every irrational will do for a substitution. The projection construction accepts any irrational slope, while a substitution rule needs its expansion factor to satisfy an algebraic condition — which inflation factors exist is that question, and the answer is a good deal narrower than “any number bigger than one”.