Which inflation factors exist

Inflation and the golden ratio grows a Penrose tiling by cutting every tile into smaller ones and rescaling, and reads the golden ratio off the dominant eigenvalue of the matrix that counts the pieces. The obvious next question is whether φ is special or merely first, and it has a sharp answer.

Almost no number can be the inflation factor of a tiling that diffracts to sharp spots. The admissible ones are a thin set with a name and an arithmetic characterisation, and the characterisation is an inequality between two integers.

Four steps from a substitution to a constraint #

The chain of reasoning is short and every link of it is computable.

A substitution has a matrix. Replace each tile by a word in the tiles and let M be the matrix whose (i, j) entry counts the copies of tile i inside the substitute of tile j. Applying M counts the tiles after one inflation; applying it n times counts them after n.

A primitive matrix has a leading eigenvalue. If some power of M has every entry positive — the substitution eventually mixes every tile into every other — then Perron and Frobenius guarantee a single largest eigenvalue λ, real and positive, with the tile counts growing like λⁿ and the relative frequencies given by its eigenvector. That λ is the linear inflation factor in one dimension and its square root in two.

λ is an algebraic integer. It is an eigenvalue of an integer matrix, so it satisfies a monic polynomial with integer coefficients — for a two-letter substitution, x² − (tr M)x + det M — and the other root of that polynomial is its algebraic conjugate.

Sharp Bragg peaks want the conjugates small. A quasicrystal’s positions are a projection of a lattice, and the deviation of the nth point from a perfect ladder is governed by the conjugate raised to the nth power. If the conjugate has modulus at least one the deviations never settle, the scattered waves never come into a fixed phase relation, and there is nothing for a Bragg peak to be built out of.

An algebraic integer greater than one whose conjugates all lie strictly inside the unit circle is a Pisot–Vijayaraghavan number. So the question “which inflation factors exist” is the question “which quadratic Pisot numbers are there”, and for a root of x² − tx + d the conjugate is d/λ, so the condition |λ′| < 1 is

$$|\det M| < \lambda.$$

Two integers and a square root, compared.

The smallest of them all #

The smallest Pisot number of any degree is the plastic number, about 1.3247, the real root of x³ = x + 1. The smallest quadratic one is the golden ratio, and that single fact is why φ is everywhere in this subject.

A quasicrystal’s inflation factor has to be Pisot, the arithmetic of a two-tile substitution makes it quadratic — and order is not periodicity is the reason the question is worth asking at all — and the smallest quadratic Pisot number is φ. An inflation factor near one is what a tiling wants — a large factor means each inflation multiplies the tile count enormously, so the structure is coarse and its hierarchy has few levels within any window. So the golden ratio is not a mystical constant that quasicrystals happen to like. It is the smallest number the constraint permits, and any structure that could use a smaller one would.

The next few are the silver ratio 1 + √2 = 2.414, which grows the octagonal tilings; 2, which is period doubling; and (3 + √5)/2 = φ², which grows the Penrose tiling when its inflation is applied twice. The dodecagonal tilings use 2 + √3. Every one of them is a quadratic Pisot number, and the list of the small ones is short.

What the eigenvector says, and which one to use #

The matrix carries two pieces of information and they are easy to confuse, so it is worth separating them once.

The right eigenvector of M gives the tile frequencies: how many long tiles there are per short tile in the limit. For the Fibonacci matrix that ratio is φ, which is the statement everyone knows — the golden ratio is the ratio of L’s to S’s in the infinite chain, and the convergents of its continued fraction are ratios of consecutive Fibonacci numbers.

The left eigenvector gives the tile lengths: how long the long tile has to be relative to the short one for the chain to be invariant under its own inflation. Use the wrong one and the chain that comes out is not self-similar at all — it has the right composition and the wrong geometry, and the inflated copy fails to lie on the original.

The two happen to coincide for the Fibonacci substitution, which is exactly why the mistake is easy to make and hard to catch: the matrix is symmetric, so its left and right eigenvectors are the same. For the silver-ratio substitution L → LLS, S → L they are not, and a chain built with the frequencies used as lengths comes out wrong.

The measurement that decides, with no threshold in it #

The Pisot condition is a theorem about the tail of a sequence, which is not something a finite computation can check directly. What a finite computation can do is build both kinds of chain and look at how their scattering behaves as they lengthen — and there is a measurement that separates them cleanly and involves no cutoff at all.

A genuine Bragg reflection is a sum of N terms that are all in phase, so its intensity grows like N², and the normalised intensity |F|²/N² settles to a constant as the chain grows. Anything else — diffuse scattering, a finite-size ripple, the broad maxima of a chain with no long-range order — has phases that do not conspire, grows like N, and its normalised intensity falls away like 1/N.

No threshold appears anywhere in that comparison. It is not is this peak sharp enough — it is does this number stay put, and the answer is visible at chain lengths of a few hundred points. That is the reason this is the measurement the essay uses rather than a peak count or an integrated intensity, both of which need a cutoff chosen by hand.

The two chains, side by side #

The non-Pisot chain is grown by the substitution A → B, B → AAAB, whose matrix has trace one and determinant −3. Its eigenvalue is (1 + √13)/2 ≈ 2.3028 and its conjugate is (1 − √13)/2 ≈ −1.3028 — outside the unit circle, and not by a small margin.

The two chains are built by identical procedures and their spectra are computed by identical code. Everything that differs between the pictures comes from one integer comparison made at the start.

Where the condition is necessary and not sufficient #

This is the part that needs stating carefully, because the temptation to run the implication backwards is strong and the counterexample is famous.

The Pisot condition is necessary for a one-dimensional substitution sequence to have a pure point diffraction spectrum: Bombieri and Taylor proved that a non-Pisot factor rules it out. The converse — that a Pisot factor guarantees it — is the Pisot substitution conjecture, and it is open in general and proved only in special cases.

The standing counterexample is the Thue–Morse substitution, A → AB and B → BA. Its matrix is [[1, 1], [1, 1]], its eigenvalue is 2, and 2 is trivially a Pisot number, having no conjugates at all. Yet the Thue–Morse sequence has a singular continuous spectrum: no Bragg peaks, and no smooth diffuse background either, but a measure concentrated on a set of measure zero. Its diffraction is neither of the two things this essay has been separating.

What fails is a further condition — the substitution has to satisfy a coincidence condition that Thue–Morse does not — and the details are a research subject rather than an essay. What matters here is the shape of the claim: an integer comparison rules a factor out, and nothing this simple rules one in. An essay that said otherwise would be claiming a conjecture as a theorem.

Reading the plot of factors #

The scatter of factors against conjugates repays reading, because three features of it are statements rather than decorations.

The admissible region is a horizontal band, not a disc. For a quadratic factor the conjugate is real, so the unit circle appears as the pair of lines at ±1 and the condition is simply that the conjugate lies between them. Every point in the band is a possible inflation factor and every point outside is not, and roughly half the factors a small matrix can produce fall outside — so the constraint deletes about half the candidates rather than a rare few.

Nothing sits near λ = 1. The smallest factor in the band is φ, and there is a gap below it: no substitution matrix with small entries produces an admissible factor between 1 and 1.618. That gap is the arithmetic reason quasicrystals have the hierarchy they do — an inflation cannot be gentle, and each level of the hierarchy is at least φ times the last.

The rational points are all admissible. A factor that comes out a whole number has no conjugates to be small, so every integer above one is trivially Pisot — which puts 2, 3 and 4 in the band and is why period-doubling substitutions diffract sharply. Those chains are periodic on average in a sense the irrational ones are not, and the sharpness of their peaks is the least surprising thing about them.

The one trap in the plot is the second of those: a matrix whose characteristic polynomial factors over the integers has two eigenvalues rather than an eigenvalue and a conjugate, and calling the second one a conjugate would report 2 as inadmissible. The distinction is made in the code by asking whether the discriminant is a perfect square, and getting it wrong deletes the period-doubling substitution from the list for a reason that is a confusion of two words.

Where the arithmetic runs out #

Three limits, each of them a real restriction on what has been shown.

Two letters only. Everything above is quadratic, because a two-tile substitution has a two-by-two matrix. A three-tile substitution has a cubic characteristic polynomial, two conjugates, and both must be inside the unit circle — which admits the plastic number and a great many others, and which this site has not enumerated.

One dimension only. The chains here are one-dimensional. A two-dimensional substitution tiling has the same matrix and the same eigenvalue, and its inflation factor is the square root of λ rather than λ; the Pisot condition applies to the linear factor. The Penrose tiling’s matrix has eigenvalue φ² and its linear inflation is φ, which is the arithmetic behind inflation and the golden ratio — but the general theory in two dimensions needs more than eigenvalues, and no figure here computes a two-dimensional spectrum.

Finite chains. Every spectrum here is of a chain of a few hundred to a few thousand points, so every peak has a width of order 1/L and there are finite-size ripples around the strong ones. The scaling measurement is designed to be insensitive to that — it asks how a number moves with L rather than what it is — and the peak widths in the figures are the window’s rather than the structure’s, which the captions say.

The connection to cut and project #

There is a second route to the same constraint, and the fact that it arrives at the same place is worth having.

Cut and project builds an aperiodic chain by slicing a two-dimensional lattice with a strip of irrational slope and keeping what falls inside. That construction produces sharp diffraction automatically, because the points are a projection of a lattice and the scattering inherits the lattice’s reciprocal structure — which is the argument the extra dimension that makes it periodic makes at length.

Now ask when a substitution chain can also be produced that way. The slope of the strip has to be an eigendirection of the substitution matrix, so it is a quadratic irrational; the window has to be bounded, so the perpendicular components of the lattice points must stay in a finite interval; and those components are governed by the powers of the conjugate eigenvalue. A conjugate outside the unit circle sends them to infinity, the window is unbounded, and the construction fails.

So the Pisot condition read from the projection side says the window is bounded, and read from the diffraction side says the peaks are sharp. Those are two descriptions of one fact, and the second is the measurable one.

What this says about quasicrystals that exist #

The constraint is not decorative. Every quasicrystal ever measured has an inflation factor from this short list, and the reason is now visible from two sides.

From the diffraction side: what Shechtman measured was a pattern with sharp spots, and sharp spots require the Pisot condition — the sharpness being exactly what the Fibonacci chain demonstrates in the smallest case there is. From the structural side: the tilings whose factors are Pisot are exactly the ones expressible by cut and project with a bounded window, which is what allows a real material to grow one locally without global bookkeeping.

The two facts are the same fact, and that is why the classification of quasicrystals is so short. There are icosahedral quasicrystals with φ, decagonal ones with φ, octagonal ones with 1 + √2, dodecagonal ones with 2 + √3. That is nearly the whole of the experimental list, and its brevity is arithmetic rather than chemistry.

Where the ladder goes next #

The aperiodic anchor is now six rungs deep: the Penrose tiles, inflation, order against periodicity, matching rules, cut and project, and the constraint on the factor. The rung above is the cubic case — three-letter substitutions, where the Pisot condition needs both conjugates inside the circle and where the plastic number finally appears — which would need a computation this site has not made.

The rung beside it is the one the counterexample points at: what the spectrum of a non-Pisot substitution actually is. It is not pure point and it is not necessarily continuous either, and the third possibility, singular continuous, is a kind of order that neither periodicity nor quasiperiodicity describes. That would be a genuinely new thing to draw, and drawing it honestly needs a measure rather than a set of peaks.

Why the band begins at the golden ratio #

The gap below φ in the plot is reported as an observation, and for the two-letter case it is a short computation — worth doing, because it explains why one number turns up so relentlessly.

A two-letter substitution’s matrix has integer trace t and determinant d, and its factor is the larger root of x² − t x + d. The conjugate is the smaller root, and the Pisot condition asks for the larger to exceed one and the smaller to lie strictly between −1 and 1.

Multiply the two roots and the product is d; add them and the sum is t. With the larger root above one and the smaller inside the unit interval, the product’s modulus is less than the larger root — which is the essay’s inequality |d| < λ — and the smallest admissible case is t = 1, d = −1.

That polynomial is x² − x − 1, whose larger root is the golden ratio. So φ is the smallest quadratic Pisot number, there is nothing between one and it, and every two-tile substitution with a smaller factor fails the condition.

The smaller Pisot numbers exist and are not quadratic. The plastic number, about 1.3247, is a root of a cubic, so it needs a substitution on three letters — and a tiling built on it has three tile lengths rather than two. The gap in the plot is therefore a statement about the number of tiles rather than about the numbers, and it closes as soon as a third letter is allowed.

The borderline case #

The condition asks for the conjugates to lie strictly inside the unit circle, and the boundary is worth naming, because it is where the spectrum stops being one thing and does not become the other.

An algebraic integer above one whose conjugates all lie on or inside the unit circle, with at least one exactly on it, is a Salem number. The smallest known is about 1.1762, and whether there is a smallest at all is an open question of the same family as the one about Pisot numbers’ accumulation.

A substitution with a Salem factor sits exactly on the edge of this essay’s condition. Its deviations neither shrink nor grow: the conjugate raised to the nth power stays on the unit circle rather than tending to zero, so the points do not converge onto a ladder and do not run away from one either.

What the diffraction does there is not settled by the argument on this page, and it is not settled in general. Such chains are known to have spectra that are neither pure point nor absolutely continuous, with a singular component — a measure concentrated on a set of zero length and carrying no atoms. That is a third possibility the essay’s two chains do not exhibit, and it is the reason the Pisot condition is stated with a strict inequality rather than a weak one.