The long circles go round the middle

A map refuses in small pieces found that the cheapest refusal of a folded state behaves on a map like a local test. The refusal reads each crease as a statement that one of the two panels it joins lies above the other, and a circle in those statements is a proof that no folded state exists. On a three-by-three map it fires on four letterings of 256, all of them pinwheels. On larger maps nearly every refusal contains one of those four pinwheels in some three-by-three window, and the share refused tracks $1 - (63/64)^w$ for a map with $w$ windows.

Nearly every. A four-by-four map has 64 refused letterings with no refused window, and the sampled maps had a few per cent more. The essay closed on what those were: larger pinwheels, recurring the way the small one does, or long circles running across the map? If the first, the test is local at a larger scale. If the second, it has a genuinely global part, and that part might come to dominate.

The answer is a third thing, and it is more useful than either. The long circles are not pinwheels, and they are not global either. Every refusal on every map enumerated exactly is a circle round a rectangle of panels, as short as a circle round that rectangle can be, and on the sampled maps all but a handful are — so the test is a family of window tests, one for each size of rectangle, and the sizes above four contribute almost nothing.

Sixty-four drawings, four shapes #

The 64 letterings can simply be listed, since a four-by-four map has only 32,768 letterings that pass every vertex. Each passes all four of its three-by-three windows and is refused only as a whole. Grouped by the map’s own symmetries — its four turns, its four mirrors, and swapping every mountain for a valley — they fall into four shapes of sixteen each.

Not one of the 64 is unchanged by a quarter turn, a half turn or any mirror. A pinwheel is defined by the symmetry: the four refused three-by-three letterings are each unchanged by a quarter turn about the middle panel, the letters round the centre repeating all the way round. A larger pinwheel would repeat the same way round a two-by-two block of panels. Nothing does. The sixteen versions of each shape are sixteen genuinely different letterings, which is the most a shape can have — eight placements times two letterings, with no two coinciding.

What they share is the circle. In every one of the 64 the shortest circle in the forced arcs crosses exactly twelve creases, visiting twelve of the map’s sixteen panels. It runs round the outside of the map on three sides and cuts inside it at one corner, stepping through a middle panel instead of the corner one, and it goes round eight of the map’s nine interior vertices. It is a ring of panels round the middle of the map. That is the answer to the question the earlier essay asked: the non-local refusals are long circles — as long as a circle on a four-by-four map can be while still going round the middle — and not repetitions of a small pattern.

A circle has to reach every side #

Twelve is not an accident of the four-by-four map. It is the shortest a circle can be if it touches all four sides of a four-by-four rectangle of panels.

A circle here is a closed walk from panel to neighbouring panel, one crease per step. A walk that visits a panel in the leftmost column and one in the rightmost has to cross the three columns between them and come back, and the same for the rows. So a circle whose panels span a rectangle $w$ panels wide and $h$ high crosses at least $2(w - 1) + 2(h - 1) = 2(w + h) - 4$ creases — the length of the ring of panels round the rectangle’s rim. Eight for a three-by-three rectangle, twelve for four by four, fourteen for five by four, sixteen for five by five.

Every refused lettering of every map enumerated exactly — three by three, four by three, five by three, six by three, four by four and five by four, 61,500 refusals between them — has a shortest circle of exactly that length. The circle behind a refusal is as short as a circle spanning its rectangle can be. It does not wander into the rectangle and out again, and it does not use the rectangle’s interior for anything but the cut-off corners. Whatever makes a lettering contradict itself, it does so along the tightest loop that encloses the vertices involved.

That is what turns a question about circles of every size into a question about windows. A refusal can be charged to the smallest rectangle its circle fits in, and a rectangle of a given size either closes such a circle on its own or does not. A map’s refusals are the union of its windows’ refusals, over every size of window — the three-by-three pinwheels and then the four-by-four rings and then the rest, each window read on its own letters.

Each size refuses on its own schedule #

The rectangles do not refuse equally often, and the rates are the quantity that decides how the test grows.

A three-by-three rectangle refuses one lettering in 64. Those are the four pinwheels. A four-by-three rectangle refuses nothing a pinwheel does not, and neither does any rectangle three cells across: every refusal on the maps three cells wide, up to six long, is a pinwheel in a three-by-three window. A circle round a four-by-three rectangle never closes on its own. A four-by-four rectangle refuses one lettering in 512 on its own — the 64 of the census, out of 32,768. A five-by-four refuses one in 5,461, and a five-by-five, counted over all 16,777,216 of its letterings, one in 2,849.

The square rectangles are more productive than the oblong ones between them, which is not what an argument about size alone would predict. A five-by-five window is less likely to close a circle of sixteen than a four-by-four is to close one of twelve, but more likely than a five-by-four is to close one of fourteen. The rings round a square block of vertices have a symmetry the oblong rings lack, and the letterings that go all the way round without contradiction at any single vertex are easier to arrange in the square.

The practical content is the fall from each rate to the next. Three by three to four by four is a factor of eight; four by four to five by four is a factor of about ten, and five by five sits between. Meanwhile the number of windows of each size a map contains grows only slowly as the size goes up: a ten-by-ten map has 64 three-by-three windows and 49 four-by-fours. So each larger size of window contributes roughly an order of magnitude less, and a test made of the first two sizes is nearly the whole test.

One size up #

The four-by-four ring turns up on large maps just as the pinwheel does — as a window.

A lettering like this one would have been counted as non-local by the earlier reading, since no pinwheel is anywhere in it. Cut out its four-by-four window, and the window is one of the 64 drawings above, turned or mirrored. The contradiction is local; it is local at the next size. That is the sense in which the lead question’s second alternative was wrong: the long circles do not run across the map. They run round a four-by-four block and would refuse the same letters anywhere on any map large enough to hold them.

The same is true one size further out. Of the exact five-by-five enumeration’s 148,224 refusals with no pinwheel, 130,048 are four-by-four rings in one of its four four-by-four windows, 12,288 are circles round one of its five-by-four or four-by-five windows, and 5,888 need the whole map. No refusal on any map up to five by five needs anything but a window — once windows of every size are allowed.

Fifteen and twenty a side #

The earlier essay’s second question was what happens to the share refused as maps grow past ten a side: whether the part that small windows do not explain stays at a few points, or grows until it dominates. Maps of fifteen and twenty a side answer it, drawn uniformly as before.

The part beyond four by four peaks and then falls. It is 0.30 per cent of letterings at six a side, 1.38 at eight, 1.57 at ten, 0.77 at fifteen and 0.07 at twenty. The four-by-four part does the same: 1.6, 3.0, 3.5, 1.9 and 0.3 per cent. Neither is shrinking because large circles become rarer; each is shrinking because a lettering refused by a pinwheel is not counted again, and pinwheels crowd out everything. At twenty a side, 99.4 per cent of letterings contain a refused three-by-three window, 0.3 per cent more a refused four-by-four one, 0.07 per cent something larger, and 0.22 per cent pass.

The independent-window estimate, extended to both sizes as $1 - (63/64)^{w_3}(511/512)^{w_4}$, predicts 23.6, 46.0, 66.8, 94.7 and 99.7 per cent against the measured 24.8, 47.7, 70.9, 96.2 and 99.8. It runs one to four points low in the middle, where overlapping windows are refused together a little more often than apart and the larger rectangles are still adding a point. At the ends it closes. A map of twenty a side is refused all but a fifth of a per cent of the time, and nearly all of that by a pinwheel — which is the first of the two outcomes the earlier essay set out, and the curve’s shape it predicted.

A ring map inside every map #

The map that is not a rectangle ended on a question it did not pursue: the map with a hole. Squares arranged in a ring have a folded state whose layers must come back to themselves, which is a condition the rectangular counting rule never has to express. It asked whether the rule survives, and called finding out a modest amount of work.

The refusals are that question, asked inside every rectangular map. A circle round a rectangle of panels is exactly a ring of squares, and the condition that refuses the lettering is exactly the ring’s failure to come back to itself — the panels round the ring each lying above the next, all the way round. Every contradiction a map’s letters can contain, of this kind, is a ring map’s contradiction, and the census here is a census of which letterings of a ring of $2(w + h) - 4$ squares fail to close, weighted by how often a rectangular map’s own vertices force them. The rectangular map has no hole, and the circles behave as if it did: the interior vertices a circle goes round are the hole, and their letters are the part of the map the circle cannot see except through the creases they force.

That also connects the refusals to a map with no edges and the tube a map makes, where a gluing gives the sheet a loop that cannot be shrunk. There the loop is a property of the surface and some sizes have no folded state at all; here the loops are drawn round interior vertices and can always be shrunk, and it is the letters that decide whether one closes. The two kinds of refusal are the same statement about layers that must come back to themselves, arrived at from the topology in one case and from the lettering in the other.

What a census of the cheap test does not see #

Every refusal counted is a refusal by the one-pass test, which is a necessary condition for a folded state and not a sufficient one. A lettering that passes every window of every size may still have no folded state; the test that never fires on a map found the test silent on almost every small map for exactly that reason. The census here classifies what the cheap test catches. It says nothing about the letterings it lets through.

The circle drawn is the shortest one. A refused lettering usually has several circles, and a refusal is charged to the rectangle of its shortest. On every exactly enumerated map that circle is as short as its rectangle allows; on the sampled maps, 3 of the 156 refusals that needed a rectangle larger than four by four had a shortest circle longer than the ring round its rectangle — circles that bend inward round a notch. They are counted with their rectangle all the same. The claim that every circle hugs its rim is a finding about maps up to five by four, not a theorem.

And the larger maps are samples. Four thousand letterings at most sizes, three thousand at fifteen a side; shares under a per cent carry proportionally large sampling error, and the peak-and-fall of the beyond-four-by-four share rests on differences of a fraction of a point at the largest sizes. The five-by-five rate of one in 2,849 was counted over every lettering; the figure that tabulates it draws sixty thousand and reads one in 2,857.

The idealisation underneath #

The map is the idealised one of the whole series: a rectangle of identical square panels of zero thickness, every crease a straight line through the whole sheet, every interior vertex a right-angled crossing where three creases of one letter and one of the other is the only flat-foldable choice. On that map one crease at each interior vertex is forced by the other three, so a map with $E$ creases and $V$ interior vertices has exactly $2^{E - V}$ letterings that pass every vertex, and choosing the unforced creases by coin samples them uniformly. Every share here is a share of those letterings, not of foldings — the count counts labels is the reminder that the numbers depend on what is being counted — and a uniformly drawn lettering is not what a folder produces.

Which rectangles, and how they were checked #

The four shapes were found by listing every lettering of the four-by-four map, keeping the refused ones with no refused three-by-three window, and grouping them under the map’s eight symmetries and the exchange of mountains and valleys; the grouping is required to give four shapes of sixteen with no lettering fixed by any symmetry, and every circle is required to be twelve creases long and to go round eight of the nine interior vertices, counted by testing which vertices lie inside the loop through the crossed creases’ midpoints. The rectangle rates are exact for every rectangle up to five by four, charged to the smallest rectangle containing the shortest circle, and each exact map is required to have no refusal whose circle is longer than its rectangle’s rim. The rates of one in 64 and one in 512 are required exactly. On the sampled maps a refusal is charged to a three-by-three window if any refuses on its own, then to a four-by-four window, and only then to its circle’s rectangle.

Still open: whether the rim rule is a theorem #

Why the shortest circle hugs its rectangle’s rim is the question this leaves. It holds on every refusal of every map enumerated exactly and on 153 of the 156 sampled refusals that needed a larger rectangle, and the exceptions are circles that bend round a notch. An argument would have to say why a lettering that passes every vertex cannot close a circle that doubles back — presumably because the vertex conditions at a doubling-back force the two arcs there to disagree — and why a notch is sometimes allowed. A proof would make the window family exact rather than measured.

The rates of the larger rectangles are the other open quantity. One in 64, one in 512, one in 5,461, one in 2,849: there is no obvious formula, and the square-beats-oblong pattern invites one. Counting the six-by-six rate needs either an enumeration of $2^{35}$ letterings or a count of ring letterings weighted by what the interior forces, and the second is the more interesting, since it treats each rectangle as the ring map it is.

Sideways from here, where the exponent comes from is about how fast the number of foldings grows, and the window family says the cheap test takes a fixed fraction of letterings per window and therefore nothing from that growth rate that the area does not already set. The map counted from the layers counts foldings by ordering panels; whether its search spends its time on letterings the windows would refuse at once is a measurement of how much a window pre-check would save it.

The habit worth carrying is about what “local” means. A test is local if every failure it finds fits in a bounded piece — not if every failure fits in the smallest piece. The four-by-four refusals looked global because they did not fit in a three-by-three window. They fit in the next size, as do the five-by-fives in theirs, and the sizes fall off fast enough that two of them are almost the whole test.