A window that is not an interval

Cut and project makes an aperiodic chain out of a periodic lattice. Take the square lattice, draw a line of irrational slope through it, keep the lattice points lying within a strip about that line, and project them onto it. The strip’s cross-section is the window, and in that construction it is an interval — a segment of the line across the strip, which is all a second dimension has room for.

Which inflation factors exist reaches the case where that stops being true and does not compute it. A substitution on three letters has an inflation factor that is a cubic algebraic integer, and its two other roots may be complex. Two complex roots span a plane, the window lives in that plane, and a window in a plane is free to be any shape at all.

Why three letters put the window in a plane #

The chain here is the tribonacci chain, from the substitution 1 → 12, 2 → 13, 3 → 1. Its matrix — how many of each letter each letter becomes — has characteristic polynomial $x^3 = x^2 + x + 1$, whose real root is $\beta = 1.839287$, the tribonacci constant. The other two roots are $-0.4196 \pm 0.6063\,i$, of modulus 0.737353.

That modulus is what decides everything. An inflation factor whose conjugates all lie inside the unit circle is a Pisot number, and a Pisot substitution gives a chain that diffracts in sharp peaks — which is the whole reason a quasicrystal looks like a crystal to a diffractometer, as what Shechtman measured records. For the Fibonacci chain the conjugate is the single number $-1/\varphi$, and the contracting direction is a line. Here there are two conjugates and they are a complex pair, so the contracting direction is a plane.

The construction that follows is Rauzy’s. Walk along the chain and count the letters: each prefix gives a point of ℤ³, and the walk goes off in the direction of the matrix’s leading eigenvector. Project each point onto the contracting plane, which for a complex pair is one complex number rather than two coordinates,

$$z(x) = x_1 + (\lambda - 1)\,x_2 + \lambda^{-1} x_3,$$

and the closure of those numbers is the window. The picture at the head of this essay is 223,317 of them.

The areas are the letter frequencies #

The window comes in three pieces, one for each letter that can follow, and they are not equal.

The three shares come out 0.542, 0.296 and 0.162, and the frequencies of the three letters in the chain are 0.544, 0.296 and 0.161, which are the normalised powers $1, 1/\beta, 1/\beta^2$. The agreement is to three decimals, which is what a grid of cells can resolve.

That is not a coincidence arranged for the essay; it is the statement that the window’s pieces measure how often the chain visits each letter. A cut-and-project set puts a point wherever the projected lattice point lands in the window, so the proportion of points of a given kind is the proportion of the window’s area of that kind. The eigenvector of a matrix, the frequency of a letter in an infinite word, and the area of a fractal are three readings of one vector.

Not an interval, and not by a little #

An interval is convex: a straight line meets it once. The three-letter window is not.

A straight cut across the window meets it in up to six separate pieces, and in 1.53 pieces on average over every cut taken. The window fills 71.5 per cent of its bounding box, the missing 28.5 per cent being the bays and notches around the boundary. The Fibonacci window, measured the same way, is one interval filling 100 per cent of itself in a single run, of length 2.618009 — which is $\varphi^2$.

The obvious worry is that the holes are an artefact of drawing an infinite set from a finite sample, and the measurement answers it: at 66,012 points the filling is 71.2 per cent and at 223,317 it is 71.5 per cent. Tripling the points moves the number by three tenths of a per cent, where an artefact of sampling would close the gaps as the sample grew. The boundary is genuinely rough — for the tribonacci window it is known to have dimension about 1.09, which is a statement this computation is too coarse to confirm and does not claim.

The three pieces are also almost disjoint: 236 cells of 28,598 carry more than one letter, which is 0.83 per cent and is the boundary where the pieces meet. They tile the window rather than overlapping it.

Reading the chain off the window #

The window is not only a picture of the chain; the chain can be read back out of it, and that is what a cut-and-project description is for.

Take the line through the origin in the direction of the leading eigenvector, and walk along the lattice points nearest it. Each step moves by one of the three unit vectors, and which one it moves by is decided by where the current point sits in the window: the three pieces are labelled by the three letters, and the point’s piece names the letter. Walk, look up the piece, take the step, look again. The chain is the itinerary of a point wandering in the window, and the substitution is the rule that says how the window maps to itself.

That is the same construction the Fibonacci chain makes with an interval. There the window is a segment, the two pieces are two sub-segments, and the wandering point moves by an irrational rotation — the classic picture of a Sturmian sequence. Here the window is a fractal, the three pieces are fractal, and the motion is an exchange of three pieces rather than two. Everything about the description survives the change of shape; what does not survive is the picture of a strip through a two-dimensional lattice, which has no room for a boundary like this one.

Why the walk is a staircase #

There is a way of seeing why the window is bounded at all, and it makes the Pisot condition feel less like a technical hypothesis.

The walk visits the lattice points closest to the line. Its position after n letters is n times the direction vector, plus an error — the part of the position that is not along the line. That error is a vector in the contracting plane, and it is exactly the $z$ plotted above. The window is the set of errors, and asking whether the window is bounded is asking whether the walk stays within a fixed distance of the line.

Each application of the substitution multiplies the along-the-line part by β and the error by λ. Since $|\lambda| < 1$, errors shrink under the rule faster than new ones accumulate, and the total stays bounded — which is what a geometric series with ratio 0.737 does. With $|\lambda| > 1$ the same series diverges and the walk wanders off, which is the measurement at the end of this essay. So the Pisot condition is the convergence condition of a series, and the window is its sum.

It is really the window #

Calling a region “the window” means something specific, and it can be checked. A cut-and-project set is the set of lattice points whose projection onto the internal space falls inside the window — so for every lattice point, landing inside must be the same thing as being on the chain.

Of 17,576 lattice points tried, 47 land inside the window and all 47 are on the chain; none of the remaining 17,529 lands inside. No exception in either direction. That is the definition satisfied, and it is the step that turns a picture of a fractal into a statement about the chain: the tribonacci chain is a cut-and-project set, with a window that no strip through a two-dimensional lattice could have produced.

Forty-seven sounds few against seventeen thousand, and it is the number to expect: the walk is a one-dimensional staircase through a three-dimensional block, so the points on it are about the cube root of the points in the block, and the block was chosen large enough that its corners are far from the line. What the test checks is not how many points are on the chain but that the two descriptions — walk the substitution, or project and test membership — agree on every one of the seventeen thousand, including the seventeen thousand five hundred and twenty-nine that belong to neither.

It is worth saying what this does not settle. The chain was built by substitution and the window was computed from it, so the check confirms that the two descriptions agree on this chain. That every Pisot substitution has such a window is the Pisot substitution conjecture, open since the 1980s for three letters and more; what is verified here is one case, computed.

The substitution, seen in the window #

The last measurement is the one that explains the shape.

Multiplying every point of the window by λ gives a shrunken, rotated copy, and that copy is the piece of the first letter — it covers 100.0 per cent of that piece, and 99.7 per cent of the image lies within it. The substitution rule 1 → 12 says that every letter’s image begins with a 1; in the window that reads as: the whole window, contracted by λ, is the part followed by a 1.

So the window is the fixed set of a contraction, which is what a fractal is. The substitution builds the chain by expanding along β; the same rule, seen on the contracting plane, builds the window by shrinking by λ — and a set that is the union of shrunken copies of itself has a boundary with no tangent. The roughness is not decoration. It is the substitution rule, drawn.

Why Penrose’s window is a polygon and this one is not #

The Penrose tiling is a cut-and-project set too, and its window is a regular decagon — a shape with straight edges and ten corners, as far from a fractal as a window can be. The difference is not that the Penrose case is two-dimensional and this one is one-dimensional; it is the arithmetic.

Penrose’s lattice is four-dimensional and its internal space is a plane, so its window is a plane region, like this one. What makes it a polygon is that the lattice is the one where five-fold becomes legal — the root lattice whose symmetry group contains a ten-fold rotation — and the window is the shadow of a unit cell of that lattice, which is a convex polytope. Project a convex body and a convex body comes out.

Here the window is not the shadow of anything. It is the closure of a set of errors, assembled by a contraction, and nothing forces it to be convex or even to have a rectifiable boundary. A window is a polygon when it comes from a projection and a fractal when it comes from a limit, and the tribonacci chain’s window comes from a limit because its substitution has no lattice symmetry to inherit convexity from.

That distinction also says which construction is the more general. Every polygon window can be described as a limit of this kind; not every limit is a polygon. Order without periodicity is a property both kinds have, and the shape of the window is what separates the cases a picture makes obvious from the cases only a computation reaches.

How far the staircase strays #

The window’s size is a statement about the chain, and it is the one a reader can check against the picture.

The walk’s error — its distance from the line, measured in the contracting plane — never exceeds 1.6376, and averages 0.800. Those numbers settle as the chain lengthens: at 19,513 points the maximum is 1.6328, at 66,012 it is 1.6369, at 223,317 it is 1.6376. The walk is not merely near the line on average; it is never further than a fixed distance from it, and that distance is the radius of the window drawn above.

A set of points within bounded distance of a line, spaced so that no two are closer than some minimum, is what makes a chain a quasicrystal rather than a random arrangement: it is order without periodicity stated as a pair of inequalities. The upper bound comes from the window being bounded, which comes from the Pisot condition; the lower bound comes from the lattice being discrete. Neither has anything to say about repetition, and the chain does not repeat.

The same numbers bound the error of an approximation. Truncating the chain and repeating it periodically — the rational approximant inflation makes visible — displaces atoms by at most the window’s diameter in internal space, which is why an approximant’s diffraction pattern sits close to the quasicrystal’s and why the closeness improves as the approximant grows.

What Pisot buys #

If the conjugate root lies outside the unit circle, none of this exists. The projected points do not fill a bounded region — they run away, because each step multiplies their spread instead of shrinking it.

Measured on a substitution with inflation factor $(1+\sqrt{13})/2 \approx 2.303$, whose conjugate is $-1.303$: the spread of the projected points goes 23.1, 42.2, 74.6, 129.7 as the chain lengthens — a factor of 5.6 across the four lengths. The tribonacci chain over the same range goes 2.613, 2.646, 2.658, 2.669, a factor of 1.022, creeping up as a longer chain reaches further into the corners of a window that is already there.

A bounded window is what the Pisot condition buys, and a bounded window is what sharp diffraction needs. That is the same fact the inflation-factor essay reaches from the diffraction side, arriving here from the geometric one: no window, no cut-and-project description, no reason for the peaks to be sharp.

The checks, and what they refuse #

The refusal that matters most is the second: the filling is not an artefact of sampling. It is the one a reader should be suspicious of, because a picture of a fractal drawn from a finite set of points always has holes, and the answer is not an argument but a second measurement at three times the sample.

The picture also stops short in a way worth naming. A window is an infinite set and every drawing of one is a finite sample, so the boundary in these figures is drawn to the resolution of the grid and no further. Its true dimension — about 1.09 for this window, by results this computation does not reproduce — is a statement about scales far below the ones drawn here. What the figures do settle is the part that survives coarsening: the areas, the number of pieces a cut meets, and the membership test.

Who found it #

Gérard Rauzy introduced this construction in 1982, for exactly this substitution, in a paper about the tribonacci sequence’s ergodic properties; the fractal has carried his name since. Pierre Arnoux and Shunji Ito generalised it in 2001 to a wide class of substitutions, and the connection to cut-and-project sets — that these fractals are windows and the chains are model sets — is the content of the Pisot substitution conjecture, which holds for two letters and is open beyond. The tribonacci constant itself is older than any of it, and the chain is a standard example precisely because everything about it can be computed.

Where this goes: four letters, and what the conjecture would need #

With four letters the window sits in a three-dimensional internal space, and a Pisot quartic with two complex pairs would give something no picture can show whole. The measurement that would carry over is the one this essay leans on hardest — the lattice test, which is a statement about membership rather than about shape, and which is what a proof of the Pisot conjecture would have to establish in general rather than case by case. The shape is the part that is beautiful; the membership is the part that is a theorem.