The border is where the cranes come apart

Nearly every cutting fails at one crane exhausts the slit grid of connected cranes up to six by six and leaves two trends it could not finish reading. Almost every cutting that fails does so by leaving some crane attached to nothing, and the share of those failures rises with the size of the grid. Among the cuttings that pass that check — every crane keeping at least one of the joins at its corners — the share that actually hold the piece together falls: 65.6, 64.6, 54.3 per cent. Whether the second trend settles at some share or keeps falling to nothing was the question, and four sizes could not answer it.

Seven by seven has thirty-six joins and sixty-nine billion subsets, which is past what trying them one by one finishes. So the census below does not try them. It counts them, one join at a time, carrying forward only what the rest of the grid needs to know, and that runs to twelve by twelve — $2^{121}$ subsets — in a few seconds.

The answer is that the share does not settle. And the way it fails to settle says where on the sheet the failures are.

Counting what cannot be listed #

The grid is the one the 1797 book Hiden Senbazuru Orikata uses for its connected cranes. A square sheet is slit into an n-by-n grid of small squares, and each interior lattice point is either left joined, holding the four squares that meet there, or cut. A cutting is a choice of which lattice points to keep. It is sound when every square — every crane — keeps at least one joined point at its corners, and it holds when the kept joins leave the whole sheet in one piece — the question which cranes can stay joined first counted.

Listing every cutting is hopeless past six by six. Counting them is not, because whether a cutting holds is decided by information that passes through a narrow opening. Take the joins in order, down one column of lattice points and then down the next. At any moment some cranes have been touched by a join already decided and will be touched again by a join still to come; call them the frontier. Everything the undecided joins need to know about the decided ones is contained in two facts about the frontier: which of its cranes are already held, and which of them are already joined to one another through kept joins behind them.

So instead of carrying the cuttings, the count carries those two facts, with a tally of how many cuttings produce each. When a crane leaves the frontier for good it must already be held, or every cutting with that history is discarded. When the last crane of some piece leaves while other cranes remain, that piece has been cut off from the rest for ever, and those cuttings are discarded too. What survives to the end is exactly the sound cuttings that hold.

The frontier is about a column and a half of cranes, so the number of distinct histories the count carries is tiny beside the number of cuttings: 52 at six by six, where there are $2^{25}$ cuttings; 1,758 at ten by ten, against $2^{81}$; 11,024 at twelve by twelve, against $2^{121}$. The device is the transfer matrix, which Hendrik Kramers and Gregory Wannier introduced in 1941 to compute the Ising model of a magnet one row at a time, and it is the standard way to count configurations on a strip whose interactions are short.

It is also checked where it can be. At four, five and six by six the count and the exhaustion share nothing except the rules of the grid, and they agree exactly: 21 of 32, 785 of 1,215, 141,621 of 260,625.

The share keeps falling #

The first figure is the result, and the trend it answers is the one the six-by-six census left open.

From five by five the share of sound cuttings that hold falls at every size: 64.6, 54.3, 51.1, 46.7, 43.6, 40.5, 37.7 and 35.1 per cent at twelve by twelve. There is no sign of a floor. A maker who checks each crane for a kept corner, and finds every crane held, has an arrangement that holds together a little over a third of the time at twelve by twelve — and, if the trend continues, less often at every size beyond.

That settles the question in the direction the earlier census feared. The look at each crane does not remain a coin toss as the grid grows; it decides less and less of the answer. What it still does is catch almost every failure, since the failures it cannot see are a vanishing share of all failures. Both statements are true at once, because failures outnumber successes by more and more as well.

The more informative thing is the rate, which is not visible on a bar chart.

A straight line on a logarithm #

The second figure draws the logarithm of the share against the size of the grid, and from eight by eight on the points lie on a line.

Below eight the steps alternate with the parity of the grid — 0.062 from six to seven, 0.090 from seven to eight — an alternation the count measures and does not explain. From eight on they settle: 0.069, 0.073, 0.070, 0.072. Their mean is 0.071, so each time a row and a column of cranes are added, the share of sound cuttings that hold is multiplied by about 0.931.

A constant factor per size is a strong clue, and it helps to say what the alternative would look like. If a sound cutting could come apart anywhere on the sheet with some small chance per join, the log of the share would fall in proportion to the number of joins, and the number of joins is $(n-1)^2$. Each size adds $2n - 3$ of them — thirteen at eight by eight, twenty-one at twelve — so the steps would grow by three fifths across the range the figure draws. They do not grow. A loss that is the same at every size is a loss proportional to something that grows by a fixed amount per size, and on a square grid that is the border: four more lattice points on the outermost ring each time a row and a column are added.

So the straight line predicts where to look. The failures should be at the edge.

What a stray piece looks like #

The six-by-six grid is small enough to try every cutting, and it can be asked what the sound cuttings that come apart actually look like.

A sound cutting that does not hold has kept joins falling into more than one cluster, where two kept joins belong to one cluster if they share a crane. The third figure sorts the 119,004 such cuttings at six by six by the size of their smallest cluster. The commonest by far is a cluster of one join: 76,331 of them, 64.1 per cent, are cuttings in which some kept join shares a crane with no other kept join. Two-join clusters account for 19.8 per cent, three-join clusters 6.3, and the tail thins quickly after that.

A lone join is a local object in exactly the sense local is not global uses the word. It is one lattice point with every lattice point around it cut — the eight around it in the middle of the grid, fewer at the edge — and the four cranes it holds are held by it alone. The check at each crane cannot see it, because those four cranes are held; they are merely held by something that holds nothing else.

Five cuts at the edge, eight in the middle #

The fourth figure draws one of these cuttings, and it is the kind the counts say is typical.

Twenty of the twenty-five joins are kept. Every crane keeps at least one join, so the look at each crane passes it. And the piece is in two, because one kept join on the border of the grid has all five of its neighbouring joins cut: its four cranes hang together and touch nothing else.

The count of neighbours is the whole mechanism. A lattice point in the middle of the grid shares a crane with eight others, one on the outermost ring with five, and a corner point with three. To leave a point standing alone, every one of those neighbours has to be cut — and with joins kept or cut at even odds, eight cuts in a row are an eighth as likely as five. The border has fewer neighbours to cut through, so a piece comes loose there far more easily.

The fifth figure checks that the census agrees, and it agrees overwhelmingly. At five by five every one of the 430 sound cuttings that come apart does so through stray pieces touching the outermost ring of joins. At six by six, 118,379 of the 119,004 do — 99.5 per cent. Of the 76,331 six-by-six failures with a lone join, 75,706 have their lone joins only on the border; 625 have one in the middle.

The 625 are worth seeing, because they show why they are rare. The sixth figure draws one: seventeen joins kept, and one of them in the middle of the grid with all eight neighbours cut. That many cuts in one place spend a large part of the sheet’s joins, and the cranes around the hole still have to be held by something else — which the other kept joins manage, just, in this cutting and in very few others.

Why a border makes a straight line #

The two findings — a constant factor per size, and failures at the border — are one finding, and the argument connecting them is short enough to state.

Suppose each lattice point on the outermost ring has some small chance of being the root of a stray piece, and suppose those chances are roughly independent, as they are for points far enough apart along the border. Then the chance that a sound cutting has no stray piece anywhere on its border is roughly a product of one factor per border point, and a product of equal factors is a power. The border of an n-by-n grid of joins has $4(n-2)$ points, so the share that holds behaves like $c^{\,4n}$ for some $c$ just below one, and its logarithm falls by the same amount — four times $\ln c$ — every time n grows by one.

Matching that to the measured step of 0.071 puts each border point’s contribution at a little under two per cent: $e^{-0.071/4} \approx 0.982$. That is an estimate from a heuristic rather than a measurement, and it should be read that way. What the heuristic does firmly is explain the shape. A loss that lives on the edge is a loss proportional to the perimeter, and a loss proportional to the perimeter is a straight line on a logarithm.

The same shape turns up wherever an object is summarised by a count taken over a finite piece of something regular. A count is not a length cuts rectangles out of a tessellation and finds them reporting fifty per cent more creases than the pattern has, then twenty-five, then seventeen: the excess is the divided creases along the cut, which scale with the perimeter while the true count scales with the area, so the error shrinks as one over the size. The rim adds up prices the rim of a drawn cell by the edge. And in statistical physics the logarithm of the number of states of a large region splits into a term proportional to its area and a correction proportional to its boundary — the surface energy of a crystal is exactly that correction. The slit grid sits at sizes where the boundary term is the part that moves.

A second look, at the joins #

If the failures are stray joins, a maker has a second check available that is as local as the first. Look at each kept join and see whether any neighbouring join is also kept. A join with none is a piece on its own, and the cutting does not hold.

The seventh figure prices that second look. It never refuses a cutting that holds — in a cutting that holds, every kept join shares a crane with some other kept join, or the piece would be in two — so it is a necessary condition like the first. And it removes most of what the first let through. At four by four the share of passing cuttings that hold goes from 65.6 to 91.3 per cent; at five by five from 64.6 to 81.0; at six by six from 54.3 to 76.8.

The second look is still not the answer, and it cannot be. The two-join and three-join clusters survive it, and a check on clusters of every size is no longer a look at one place: it is the global question again. What the second look shows is how much of the gap between local and global is made of the very smallest global failures — about half of it at six by six — and that those smallest failures are overwhelmingly at the edge, where a maker working inward from the border of a sheet would meet them first.

The size of the thing counted #

It is worth seeing what twelve by twelve is as an object, because the count reaches a size nobody would cut by hand. The last figure draws the twelve-by-twelve sheet slit for cranes: 144 squares, 121 lattice points left as joins, 264 sides of slit — a great deal of cutting for a subject that prices a cut one at a time.

The census does not claim that the 1797 book’s arrangements are sound, minimal or chosen at random, and it does not read the plates at all; that is a documentary question of the kind a record is not a proof keeps separate from anything computed. What it says is what a slit grid permits, and how much of that a maker’s two easy checks could find.

What the counts leave out #

Joins are counted as kept or cut with even odds. The shares are shares of all cuttings, which is a way of describing the space rather than a model of anybody’s choices, and a maker who kept joins with some other habit would face different shares. The exact counts of cuttings do not depend on this; the percentages do.

Holding means being one object. A piece connected through a single join counts the same as a piece joined everywhere, and paper at a single uncut lattice point tears, which is the fragility one cut short of falling apart measures in a sheet slit for folding. A census weighted by how much of each crane’s edge stays attached would be a better guide to what survives folding and handling, and it would need a model of how a join fails, which nothing here supplies.

The heuristic behind the straight line is a heuristic. The independence it assumes between stray pieces at different points of the border holds only roughly, and the estimate of two per cent a border point comes from fitting the line rather than from counting border failures one at a time. The counts are exact; the account of why they fall at that rate is an explanation that fits them.

And the second look is priced only where the grid can be exhausted, at four to six by six. The count that reaches twelve by twelve carries whether cranes are held and joined, not whether a kept join has a kept neighbour, so the share after the second look is not known past six.

Still open: where the middle takes over #

The straight line cannot be straight for ever. Stray pieces in the middle of the grid are rare — 625 of 119,004 at six by six — but they have a chance per lattice point, and the number of interior lattice points grows with the area. Eventually that term must bend the logarithm downward, and the steps must start to grow.

There is a hint that it has begun. The same count carried two sizes further, which takes about forty-five seconds, gives steps of 0.0725 from twelve to thirteen and 0.0741 from thirteen to fourteen, both above the mean of the four before them. Two steps are not a trend, and the parity alternation has not fully died away. At what size a slit grid starts to come apart more often in its middle than at its border is the question these counts now point at, and answering it needs either a count carrying the second look as well, so that border and interior failures can be tallied separately at every size, or a size well beyond what a frontier of a column and a half can reach.

The count has one more use waiting. The earlier census found that the fewest joins holding a four-by-four and a six-by-six grid meet a counting floor exactly and in one way, and predicted that eight by eight would need exactly twenty-one. A count that carried the number of kept joins alongside the frontier would test that prediction at eight, ten and twelve by twelve with no change to the method.

The habit worth carrying is about reading a rate. When a share falls by the same factor at every size, look at the border before looking at the interior. A constant factor per size is a quantity growing with the perimeter, and the perimeter is where a large regular object is least like its own middle.