How much pattern is enough

The detector this site runs on is handed two things: the contents of one cell, and the fact that the cell repeats for ever. The second is doing a great deal of work and no essay here has asked how much.

A reader is handed neither. What a reader sees is a patch — a plate in a book, a floor, a photograph of a mineral surface, a field of view down a microscope. There is no infinite pattern in front of them and no guarantee that anything repeats outside the frame.

So: what can a window decide?

What “the window admits” has to mean #

The definition is the whole of the care here, and getting it wrong makes the measurement meaningless in one of two ways.

An operation is admitted when the window does not refute it. Take every point the operation carries from inside the window to inside the window, and require each image to be a point of the pattern. Points carried out of the window say nothing at all: there is nothing out there to compare against, and an image that left the frame is not evidence of anything.

The alternative — requiring every point’s image to be inside and correct — refuses the group’s own operations, since every symmetry of a pattern moves points across any boundary one cares to draw. A test that fails on the true answer is not a test.

Two guards keep it from being vacuous. The operation must carry the window’s central point to somewhere still inside, so that an operation shunting the whole patch off to one side is not admitted on an empty intersection; and at least half the points must be carried inside, so that a verdict rests on evidence.

And the window is round. A square patch of cells has its own symmetry, and a window more symmetric than the pattern is a window that flatters it — the four-fold symmetry of the frame would be lending four-fold symmetry to whatever is inside. A disc has every symmetry, so it adds none.

Two ways to be wrong, and they are opposite #

Under-reporting is the commoner failure. Thirteen of the seventeen show fewer operations than they have at the smallest window measured. That is not the window being fooled; it is the window being unable to look. A glide carries a point half a cell along its line, so a window that cannot hold both halves has nothing to compare and the operation goes untested. The groups that take longest to settle are exactly the ones distinguished by a glide — pg, pgg, pmg, p4g — together with p2, whose half-turns move points furthest.

Over-reporting is rarer and much more interesting. One group does it: cm.

This is the accidental-symmetry failure in a new place. That essay is about a motif too symmetric for its group — a dot whose orbit under p3 acquires mirrors nobody put in. This is about a window too small for its group, and the two produce exactly the same experience for a reader: a picture that is well formed, correctly computed, and describes a group other than the one it is under.

The order the groups settle in #

Reading the census by settling radius rather than by name gives an ordering that is not the one the classification uses, and it is worth setting out because it says what a window is actually good at.

Settled at the smallest window measured: cmm, p4, p4m. These have many operations, several of them rotations about points inside any window, and a small patch already refutes everything else. More symmetry makes a group easier to identify, which is the opposite of the intuition that a complicated pattern is harder to classify.

Settled next: p1, pm, cm, pmm, and all five of the three-fold and six-fold groups. A window of 0.8 cells is enough — nine groups at that size.

Slowest: p2, pg, pmg, pgg, p4g — needing a full cell’s radius. Four of the five are the groups with glides, and the fifth is the group whose only operation is a half-turn. The common feature is that the distinguishing operation moves a point a long way: half a cell along a glide line, or across a centre to the far side.

So the difficulty is a fact about displacement, not about complexity. p6m, with twelve operations, settles sooner than p2, with two. A window can test an operation only when it can see both a point and the point’s image, and what matters is how far apart those are.

That also explains the one over-reporting case. cm’s two operations are the identity and a mirror; a small window round a general point sees a fragment that happens to be consistent with a rectangular arrangement, and it takes a full cell to find the point that refutes it. Why cm is not pm argues the same distinction from the lattice’s side — and the window measurement says how much of the pattern a reader must see before the argument has any evidence in it.

The pair, one more time #

The two groups this site returns to most often are distinguishable from a small patch, which is worth knowing given how much trouble they cause. At a radius of 0.6 cells one admits five operations and the other two; at 0.8 both admit exactly their own six. A reader with a patch a cell or so across has enough to separate them, and their difficulty is a difficulty of notation and of seeing where the mirrors are, not of evidence.

The same question, where it is asked for real #

Three places outside this site ask exactly this question, and each answers it differently because each has a different kind of window.

An electron microscope images a patch a few nanometres across, which for most structures is a handful of cells. That is on the edge of what this measurement says is enough, and it is why lattice imaging is usually reported together with a diffraction pattern rather than alone — the two have different windows and different failure modes.

A diffraction experiment has no window at all in this sense, and that is its great advantage: the beam illuminates an enormous number of cells and what it measures is a property of the average. The symmetry a diffraction pattern reports is not the crystal’s, because Friedel’s law adds a centre, but it is a property of the whole illuminated volume rather than of a patch. The trade is exact: an image gives a small window on the real structure, and diffraction gives an unlimited window on a partly obscured one.

And a real crystal is finite, which turns the question round. A fivefold axis in an ordinary crystal is the case where a local arrangement has a symmetry the crystal does not have — a cluster with an exact fivefold axis inside a group that permits none. Read through this essay’s measurement, that is a window small enough to admit an operation the pattern refutes further out, occurring not as an artefact but as a fact about the material. The window failure and the local-symmetry phenomenon are the same arithmetic seen from opposite ends: one is a limitation of looking, the other is a property of the thing.

Which is why the site’s own habit is what it is. Every plate here is generated from a group and handed back to a detector that knows the lattice, so the pictures cannot be wrong about their captions. What this essay adds is the other half: the plate has to be large enough that a reader, without the detector, is looking at evidence rather than at a coincidence.

What a settled window has and has not established #

A settled window has failed to refute anything. That is not the same as having established the group, and the difference is not pedantry.

Everything outside the window is unconstrained. A pattern could agree with p4 inside a window of radius five and do something entirely different at radius six — it would not be a wallpaper pattern at all, but nothing inside the window says so. The measurement’s honest statement is: of the operations this lattice permits, these are the ones no visible point refutes.

That asymmetry is the same one the aperiodicity count runs into from the other side. There, a finite measurement can prove periodicity and cannot prove aperiodicity; here, a finite window can refute a symmetry and cannot establish one. In both cases the negative direction is decidable and the positive one is not, and in both cases the temptation is to report the positive as though it had been shown.

Which is what makes the numbers useful rather than merely interesting. A window of a cell’s radius is enough to rule out every group but one, given a lattice — and ruling out is what a classification does. The seventeen are seventeen because sixteen of them can be refuted for any given pattern, not because one of them can be proved.

Where the exactness stops #

The point comparison is exact. Coordinates are integers in hundred-and-twentieths of a cell, and an image is a point of the pattern or it is not; there is no tolerance anywhere in the test. Only the window’s boundary uses Cartesian distance, and where the boundary falls is a choice rather than a claim.

The linear parts come from the lattice. The candidates tested are the operations of the lattice’s holohedry, which is the same restriction the site’s detector works under. A window being asked which operations of this lattice it admits is a different and easier question than a window being asked which isometries of the plane it admits — the second would require finding the lattice from the patch first, which is a measurement this site does not make.

The window is centred on a general point, not on the origin. The origin is a lattice point and a special position in most of these groups, and a window centred on a symmetry element is a window chosen to show that element.

The radii are a sample. Eight sizes from 0.6 to 3 cells, and the settling radius is reported as the smallest measured size from which every larger one also settles — one lucky size is not a decision. A finer sweep would move the reported numbers a little; the ordering between groups is what the measurement is for.

Nothing here finds the lattice. The measurement assumes it, which is the largest of the assumptions and the one a reader with a real patch does not get for free. Finding the lattice from a patch is the first step of any actual pattern analysis and it has its own failure modes — a patch can be consistent with a lattice twice as fine as the true one, and with any sublattice of the true one. What is measured here is the second step, and it is the easier of the two.

And the motif is the site’s own. Three points in general position, the same asymmetric unit every plate here uses. A different motif would change the numbers: a motif with more points gives the window more to check and settles sooner. What would not change is which groups are slowest, since that is a fact about how far their operations move a point.

Who found it, and when #

The question belongs to pattern analysis rather than to crystallography, and it is asked most sharply by people looking at real objects: archaeologists classifying pottery decoration, textile historians, and the long literature on which of the seventeen appear in ornament. Their patches are small, damaged, and drawn by hand, and the practical rule of thumb — a repeat and a bit — is what this measurement puts a number on.

Dorothy Washburn and Donald Crowe’s Symmetries of Culture (1988) is the standard treatment, and its flow charts for identifying a plane group from a fragment are exactly the refutation procedure above, run by eye: does it have a four-fold centre, is there a mirror, is there a glide not along a mirror. Each question is a refutation, and the chart terminates because sixteen refutations leave one group.

The crystallographic version is a measurement with a tolerance, and it is near-symmetry: a real crystal’s atoms are displaced, so the question becomes how nearly an operation holds rather than whether it holds. The window question and the tolerance question are independent — one is about how much is visible, the other about how exactly it agrees — and a real determination faces both at once.

The window a reader without a lattice needs #

The measurement assumes the lattice is known, and the essay names that as its largest assumption. It is worth asking what the answer becomes without it, because that is the situation a reader with a photograph is actually in.

Finding the lattice from a patch means finding the translations, and a translation can only be seen where the window contains two copies of something. So the window must be at least two cells across in each independent direction before any translation is visible at all, and it must be enough larger than that for a candidate translation to be distinguishable from a coincidence.

That doubles the requirement at a stroke. A radius of one cell settles the operations given the lattice; a radius of two or more is needed before the lattice itself is available, and the settling radii above then apply on top of it.

There is also a failure with no counterpart in the essay’s measurement. A patch can suggest a smaller lattice than the pattern has — if the motif’s decoration at the finer scale is not visible in the window — and a wrong lattice makes every subsequent operation test wrong. That is the same error a diffraction experiment makes when it misses a superstructure, arriving in real space and without the compensating advantage that the missing reflections would be weak rather than absent.

So the honest number for a reader is not one cell’s radius but three or four, and the essay’s figure is a lower bound obtained by giving away the harder half of the problem.

The patterns a window can never settle #

There is a class of patterns for which no window whatever decides the question, and putting them beside the seventeen says what periodicity is buying.

A Penrose tiling has the property that every patch occurring in it occurs infinitely often, in every other Penrose tiling of the same kind. So two such tilings — genuinely different, carried onto each other by no motion — are locally indistinguishable: any finite window shows a patch that both contain, and no amount of enlargement separates them.

That is exactly the opposite of the situation measured here. A periodic pattern is settled by a window of a couple of cells because its whole content recurs at that scale; an aperiodic one is never settled, because its whole content is not in any window and the parts that are recur everywhere.

The two facts are the same fact seen from either side. A finite window decides a pattern exactly when the pattern has a finite description that fits in it, and a lattice is such a description while an aperiodic order is not — which is why the aperiodic essays count patches instead of identifying groups, and why their conclusions are always about what a bounded measurement cannot conclude.

Where this leaves the plates #

Every wallpaper figure on this site draws two or three cells, which is a choice that had never been justified.

It turns out to be about right, and for a reason the measurement makes precise: a radius of one cell — a window two cells across — is where every one of the seventeen settles. A plate of two cells is showing a reader enough to refute the alternatives, and a plate of one cell is not, for the four groups that need the second.

There is a second reading of the same number, and it is the one a reader can use. A plate two cells across is not merely conventional: it is the smallest plate on which every one of the seventeen is decided, so the site’s figures are, without anybody having chosen it, drawn at the size where the evidence is complete.

The exception is worth carrying: cm at less than a cell admits five operations where the group has two. If a plate of this site ever draws less than a cell of cm, it will be drawing a picture of a group the caption does not name, and nothing about the picture will say so.