How much of the hat is a crystal

The hat and the turtle are one tiling takes a patch of hats and lays it out again at other edge lengths without changing which tile touches which. Everything combinatorial survives — the neighbours, the reflected tiles, the aperiodicity. What changes is where the corners are. At the hat’s own lengths and at the turtle’s the corners lie on a lattice; at equal lengths no lattice holds them. The essay ended by asking what that does to the pattern a tiling scatters, because positions are the one thing on which the members of the family differ.

A diffraction experiment reads positions and nothing else. Put an identical scatterer at every corner of every tile — every point where the kite grid’s lines meet a tile’s boundary — and the intensity scattered in each direction is the squared modulus of a sum of phases, one per corner. The tiles, their shapes and their contacts enter only through where the corners fall.

The answer for the hat has two parts, and the second is the surprise. Its pattern repeats exactly, like a crystal’s, because its corners are on a lattice. And inside each repeat the pattern is mostly a crystal’s too: sixty-three per cent of what the corners scatter beyond the lattice’s own reflections lands on the reflections of an ordinary periodic structure with partly filled sites, and the aperiodicity is confined to the rest.

Every corner on one lattice #

The hat is drawn on the kite grid, the tiling of the plane by kites cut from regular hexagons, and every corner of every hat is a corner of some kite. There are three kinds of kite corner: the centres of the hexagons, the midpoints of their sides, and their own corners.

All 1,217 corners of the patch, found by exact cover on the grid, lie on a single lattice. Written in the integer coordinates of the grid’s points, it is the sublattice of index three: the sites whose two coordinates differ by a multiple of three. Half of that lattice’s sites are not kite corners, so no tiling on the grid can ever use them; inside a disc well within the patch, 704 such sites and not one of them occupied, which is the arithmetic of the grid rather than anything about hats.

Of the kite corners that could be used, the hexagon centres are used without exception — 121 of 121 inside the disc — and that one is not a coincidence of the patch. A hat is eight kites, four from one hexagon and two each from two neighbours, so no hat ever holds all six kites round a hexagon’s centre. Every centre is therefore on the boundary between two tiles and is a corner of both. The other kinds are used some of the time: side midpoints between seventy-two and seventy-eight per cent of the time, depending on the direction of the side, and hexagon corners sixty-eight and fifty-six per cent, depending on which way the corner points.

A pattern that repeats #

A set of points on a lattice scatters a pattern that repeats on the reciprocal lattice, whatever the points are. Adding a reciprocal lattice vector to the scattering vector changes the phase of every lattice point by a whole number of turns, which changes nothing, so the intensity at the new vector is the intensity at the old one. That holds for a perfect crystal, for a random scattering of occupied sites and for an aperiodic tiling alike. The picture below is the hat patch’s pattern over four cells of the kite grid’s reciprocal lattice, and each cell holds the same arrangement of reflections.

The same tiling laid out at other lengths is the control, and it shows that repetition is a statement about positions and not about the tiling.

Adding a reciprocal vector changes the hat’s intensity by at most 1.1 × 10⁻¹⁵ of the central peak, which is rounding. The turtle’s corners lie on the grid shrunk by $\sqrt{3}$, so its pattern repeats on reciprocal vectors $\sqrt{3}$ times as long, and there the change is 1.1 × 10⁻¹⁶. The equilateral member has the same corners joined in the same way at positions no lattice holds, and its intensity changes by up to 7.4 × 10⁻² — seven per cent of the central peak. Sampled on a grid of ninety steps a cell, its strongest reflection after the centre is a peak of 0.261 at (0.711, 0.333) in the hat’s cell coordinates, where the hat has a reflection of full height at (2/3, 1/3). One set of contacts, three sets of positions, and only the two on lattices repeat. The difference is invisible to every count the tiling essay made, and it is the first thing a diffractometer would report.

The reflections at sixths belong to an average crystal #

Repetition says nothing about what is inside one cell, and the hat’s cell is not a random-looking cloud. Its strongest reflections sit at simple fractions of the cell: at the thirds, with a third of the central peak’s height, and at the halves and sixths, with an eighth or a little less.

Reflections at rational positions are what a periodic structure produces, and there is an obvious candidate. Take the lattice the corners lie on and give each site the probability with which the patch uses sites of its kind: one for a hexagon centre, zero for a site that is not a kite corner, and the measured fractions for the rest. That is a crystal — a periodic arrangement of partly occupied sites, repeating with the hexagons the kites were cut from — and it is exactly the kind of structure a crystallographer refines when a site is found to be occupied only some of the time.

The two structures — the patch itself, with each site used or not, and the average crystal, with each site used a fraction of the time — were diffracted side by side, as plain sums over the same disc of sites.

At every reflection whose position is a whole number of sixths the two agree exactly: 1.0000 at (1/3, 2/3), 0.3438 at (1/3, 1/3) and at (0, 1/3), 0.1250 at (1/2, 0) and (1/6, 1/3), 0.1094 at (0, 1/2). The agreement is not an approximation. At a scattering vector that is a whole number of sixths of the cell, every site of one kind has the same phase, so the sum over the sites of a kind depends only on how many of them are used, and the average structure uses exactly as many. Every reflection at the sixths is a reflection of the average crystal, and so is every strong reflection in the picture at the top.

Between the sixths the average crystal is silent, apart from a trace of the disc’s edge three millionths of the central peak high. The patch is not. At (0.227, 0.540) it scatters 0.0149, and at (0.313, 0.773) 0.0114. Those are what the hat has that a crystal of partly filled sites does not.

How much, by Parseval’s identity #

How much of the pattern the average crystal accounts for can be measured exactly rather than estimated from peak heights, and the measurement needs no diffraction at all.

Write the occupation of the lattice as a function that is one on a used site and zero on an unused one. Its average over all the sites is the fraction used, 0.375, and its variance is the fraction used times the fraction unused. Parseval’s identity says the total intensity scattered outside the lattice’s own reflections is that variance. Averaging the occupation over the sites of each coset of a coarser sublattice gives a periodic structure, and the variance of that average is exactly the intensity that lands on its reflections. The share of the variance the average keeps is the share of the pattern the periodic structure accounts for.

The share explained by period two is 0.203 and by period three 0.408; period five, which shares no factor with the grid’s six, explains 0.008. At period six the share is 0.6335, and averaging over the kinds of site gives the same number to every digit, which is the statement that the period-six average crystal and the kinds-of-site average crystal are one structure. Longer periods explain almost nothing more: 0.638 at twelve, 0.644 at eighteen, 0.653 at twenty-four. The random occupations are the control on how much of that rise is real, because averaging a finite sample over more and more cosets explains more of its variance by chance alone — and a random occupation of these 1,412 sites reaches 0.129 at period twenty-four by sampling. The patch’s rise after period six is smaller than the random rise at every period tried.

The shares can be taken apart further, because Parseval’s identity sorts the intensity by where it lands. An average over period two keeps the reflections at halves of the cell; an average over period three keeps the ones at thirds; an average over period six keeps both, and the reflections at sixths that are neither. So the period-six share is a sum of three parts, and each can be read off by subtraction: 0.203 at the halves, 0.408 at the thirds, and 0.023 at the sixths that are neither. Period four, which keeps the halves and the quarters, explains 0.205 — the quarters add two thousandths, less than the random control’s sampling at that period — and period five, which keeps only fifths, explains 0.008, below what random occupations of the same sites reach there.

That pattern of shares is a description of the hat’s corners in the plainest terms. Nearly two thirds of the order is carried by the reflections at thirds and halves, which are the reflections that tell the three kinds of kite corner apart from one another and from the sites that are never corners. The corners’ preferences between centre, midpoint and hexagon corner are most of what they have to say; which particular midpoints and corners are used is the rest.

So about sixty-three per cent of what the hat’s corners scatter, beyond the lattice’s own reflections, belongs to a crystal with partly filled sites and the repeat of the hexagons underneath, and no longer repeat within reach explains any more. The other thirty-seven per cent is spread over reflections like the two found between the sixths, each a hundredth of the central peak or less.

Where the aperiodicity is #

The thirty-seven per cent is the hat’s aperiodicity as a diffraction experiment would meet it. It is not noise: the random occupations show what noise looks like, and the patch’s residue is larger and concentrated. But a patch of 1,217 corners resolves it only roughly. The weak reflections between the sixths move as the patch grows — patches of 110, 183, 273 and 381 tiles put their strongest in-between reflections at different places — so their positions are not yet measurements.

What they converge to is known from the other direction. The hat tilings have been shown to be cut-and-project sets — the construction a quasicrystal is built by, a slice through a lattice of higher dimension — and a structure of that kind diffracts into sharp reflections at positions forming a dense set. That result is quoted here and not computed. What the computation adds is the proportion: the reflections a crystal of partly filled sites would give are most of the hat’s pattern, and the dense set of reflections the aperiodicity requires carries about a third of it.

That is a sharper statement of what order without repetition looks like when the order sits on a lattice. The Penrose tiling’s corners lie on no lattice, and its pattern has none of this structure: nothing repeats, and no average crystal exists to take a share. The hat’s corners are on a lattice, its pattern repeats, and its aperiodicity is a correction to a crystal rather than a replacement for one.

A refinement that would be two-thirds right #

The measurement has a practical reading, and it is the surprising connection in the subject.

A crystallographer given a diffraction pattern from a crystal of hats — say, molecules shaped to tile that way, sitting on a lattice — would index the strong reflections on the cell of the hexagons underneath, find that several sites in that cell are only partly occupied, refine the occupancies, and obtain a structure that fits every strong reflection exactly. The refined structure would be the average crystal above, and it would be wrong about the thirty-seven per cent — which would appear as weak intensity between the reflections, easy to take for diffuse scattering from disorder.

That is the symmetry of an average meeting an aperiodic structure instead of a disordered one, and the same warning applies in a stronger form. The occupancy does not name the disorder found that one occupancy fits many different disorders; here one set of occupancies fits a structure that is not disordered at all, only aperiodic, and nothing in the refined occupancies says so. The information is in the weak reflections and nowhere else, which is also where what diffraction cannot tell apart puts it.

What one patch cannot say #

The patch is not the plane. It is the first tiling the exact cover finds on a disc of hexagons, and its fractions are its own. A smaller patch of 110 tiles, found the same way, gives a share of 0.638 against this patch’s 0.634, with 0.203 at the halves and 0.409 at the thirds, so the proportions are not an accident of one disc — but two patches from one search are not a sample of the plane either. The clearest sign is the two kinds of hexagon corner, used sixty-eight and fifty-six per cent of the time: a tiling of the whole plane has every orientation of tile equally often, and a turn by sixty degrees exchanges the two kinds of corner, so in the plane they would be used equally. The share of sixty-three per cent is this patch’s share, and the plane’s value — which the frequencies of the hat’s substitution would give exactly — is not computed.

The scatterer is a choice. A unit scatterer at every tile corner is one convention; a scatterer at every tile’s centre, or one weighted by how many tiles meet at a corner, gives a different occupation and different numbers. The lattice and the repetition survive any choice made on the grid; the share does not.

The weak reflections are unresolved. Their existence is measured; their positions are not, and nothing here shows that they are sharp rather than broad.

The checks, and what they turn away #

The statements above rest on tests that could fail, two of which exist to fail on purpose.

The two controls are what make the share meaningful. A measure that reported a large share for a random occupation would be measuring the number of cosets; one that could not reach a share of one would be unable to recognise a crystal when given one. Neither happens: the random occupation explains half a per cent at period six, and the periodic one explains all of its variance at period twelve.

The hat, and what was said about its diffraction #

The hat was found by David Smith and announced in March 2023 with Joseph Myers, Craig Kaplan and Chaim Goodman-Strauss, and within the year Michael Baake, Franz Gähler, Jan Mazáč and Lorenzo Sadun described the hat tilings as cut-and-project sets with pure point diffraction, which is the quoted half of this essay. The average-structure reading of partly occupied sites is the working method of every structure refinement, and Parseval’s identity is older than crystallography; putting the two against a tiling that is known to be aperiodic is what turns them into a measurement of how aperiodic it is.

Where this goes: the plane’s share, and the reflections between #

Two numbers are left open, and they are the next measurements rather than the next arguments. The first is the share for the plane rather than for one patch: the substitution that builds hat tilings gives the frequency of every kind of corner exactly, and with it the average crystal and its share, which may or may not be a simple number. The second is the set of weak reflections, which a patch grown by the substitution to tens of thousands of corners would resolve, and which the cut-and-project description predicts in advance — so that the thirty-seven per cent could be checked reflection by reflection against positions computed from the slice.