A tube refuses round itself

The cheap refusal on a map has a simple mechanism and, on a flat sheet, a simple census. A map folds to a single cell, so every panel lies over every other, and each crease’s letter says which of its two panels is on top. If those statements run round a circle — this panel over that one, that one over the next, and back to the first — the lettering cannot be folded. A map refuses in small pieces found that the circles live in windows: a lettering is refused almost exactly when some three-by-three window of it is one of four refused patterns, so the share refused is one minus $63/64$ to the power of the windows. The long circles go round the middle found what the windows miss — circles round larger rectangles of panels — and that two window sizes explain all but 0.07 per cent of a twenty-by-twenty map.

So on a flat map the test is local. The tube a map makes glued a map’s left edge to its right, made a real object that folds in the hand, and found a parity the flat map does not have: a tube whose circumference is odd comes back turned over and has no folded state at all. The question here is what the cheap refusal does on a tube whose circumference is even — whether it stays local, and what happens to the loop that the gluing has made.

Building the tube cell by cell #

A glued sheet has no flat drawing to fold, so the tube is built directly: cells in rows round the circumference, a crease between each pair of neighbours, and a column of seam creases between the last column and the first. Panels are face up and face down in a checkerboard, which is consistent round the tube only when the circumference is even. Every vertex is a right-angled crossing whose letters split three to one, and each vertex forces the crease above it, so the letterings that pass every vertex are exactly the free creases lettered at will — $2^{cr}$ of them on a tube $c$ round and $r$ long.

The construction is checked before it is trusted. Built the same way but not glued, the three-by-three map gives 4 of its 256 letterings refused and the four-by-four map 2,112 of 32,768, 64 of them by no three-by-three window — exactly the counts the flat-map essays reported from the crease pattern. Only then are the tubes measured.

Every refusal the windows miss goes round #

Small tubes can be counted completely, and the counts are not subtle.

The tube four panels round and four long has 65,536 letterings that pass every vertex, and the loop test refuses 41,728 of them, 64 per cent. The flat four-by-four map, built of the same panels, refuses 6.4 per cent. Of the tube’s refusals, 7,552 are refused by a three-by-three window — 11.5 per cent of all letterings, where the windows’ rule predicts 11.8, counting the four windows that straddle the seam. The other 34,176 are refused by a circle that goes round the tube, and on every tube counted exactly there is no third kind: every refusal no window explains has a shortest circle that winds once round.

The same holds on each small tube in the table. Four round and three long: 2,096 refused of 4,096, 240 by windows, 1,856 round. Six round and three long: 88,184 of 262,144, 23,616 by windows, 64,568 round. And the narrowest tube, two panels round, refuses exactly half of its letterings at both lengths counted, none of them by windows, since a window three panels wide does not fit on it.

That last number has a direct reading. A tube two panels round, flattened, is a band with two lengthwise folds, one at each edge, and both folds say which of the two layers is on top. They must say the same thing, and on this tube that means carrying the same letter. The tube is refused exactly when the two folds of its first row carry different letters — checked on every lettering at three, four and five panels long — because the vertex conditions carry that row’s agreement or disagreement up the tube unchanged. The whole tube stands or falls on one coin, and a coin comes down wrong half the time.

What a circle round the tube looks like #

The circles come in two shapes, and drawing one of each shows why no window can see them.

The simplest is a ring: one row of panels, each lying over the next all the way round, closed by the seam crease. It needs every lengthwise crease in that row to point the same way round the tube, which is why it is common on a narrow tube and rare on a wide one. The other kind changes rows as it goes, stepping up or down through a crossways crease and continuing round. On the tube four round, rings are about three fifths of the round refusals; on the tube six round, under a third, and the rest are circles of eight or ten creases that wander between rows.

Neither fits in a rectangle of the sheet, because a circle that goes round the tube crosses every column, and a rectangle of the flat map crosses only some. That is the whole of why the windows miss them. On a flat map every circle bounds a region, and the windows are the regions a circle is likely to bound; on a tube a circle round the tube bounds nothing, and the windows have no way to be large enough.

The windows keep their rate #

The windows themselves behave exactly as they do on the flat sheet, which is worth checking separately because a tube has more of them.

A tube $c$ round and $r$ long has $c(r - 2)$ three-by-three windows, against $(c - 2)(r - 2)$ for the same sheet flat, because the seam adds two columns of windows. On seven tubes from four round to twelve round, drawn fifteen hundred letterings at a time, the windows refuse at one minus $63/64$ to the power of that count, within two points — 16.2 per cent against 17.2 on the tube four round and five long, 69.5 against 67.8 on twelve round and eight long. A window that straddles the seam is an ordinary window; it cannot tell it has been glued.

So the gluing does two separate things. It adds windows, which is a small change in the local test’s rate and exactly predictable. And it adds a loop, which the local test’s census does not contain at all.

Narrow tubes refuse most #

How often the round circles refuse depends almost entirely on the circumference.

On tubes five long the round refusals take 49 per cent of letterings at two panels round, 55 at four, 32 at six, 17 at eight, 10 at ten and 5.6 at twelve. Every two panels of circumference cut the share by a factor between about 0.46 and 0.63, on every length measured, which on the logarithmic scale is three nearly parallel straight lines after the first step. A circle round a wider tube needs more creases to agree, and each additional pair of columns roughly halves the chance.

The four-round tube refuses slightly more round itself than the two-round one does. The two-round tube has exactly one way to fail round — its two folds disagreeing — and a four-round tube has rings in each row and longer circles besides, and at four round the extra ways outnumber the extra creases they need.

Length adds windows, not circles #

The tube’s length is the other dimension, and it behaves the opposite way.

From six panels long to eight, the share refused round the tube moves by two points or less at every circumference measured, while the windows’ share climbs by ten to fifteen. A circle round the tube needs only one band of the tube, a row or two, and adding length adds more bands in which it might occur without making any one of them more likely. The windows grow with the area, so a long tube is eventually refused mostly by windows, as a large flat map is; a short, narrow tube is refused mostly round itself.

That is a clean separation, and it says where the non-local part of the test lives. On the flat map the circles that escape the windows go round rectangles of the sheet, and their share falls away as the rectangles they need get larger. On a tube the escape is the tube’s own loop, of a fixed size set by the circumference, and it does not fall away with the size of the sheet at all.

Why the flat map could not show this #

The flat map’s census had every ingredient of this result except the loop, and it is worth seeing how close it came. The test that never fires on a map began the measurement by finding the cheap refusal silent on the smallest maps, because a map smaller than three by three has no circle of panels for letters to close. The map that is not a rectangle then asked about a map with a hole — squares in a ring — and noted that a ring’s layers have to come back to themselves, which the rectangular counting rule never has to express. It did not measure the ring.

The tube is that ring, made with no hole cut: every row of a tube is a ring of squares, and the ring’s failure to come back to itself is exactly the refusal counted here as a ring. On a flat map the same ring exists only round a rectangle of interior vertices, where the vertices inside it constrain its letters and make its closing rare. On a tube there is nothing inside the ring, because what it goes round is the hole of the cylinder, and its letters are constrained only by the rows above and below.

That is also why half a rim matters here. A tube keeps the two open ends of the flat map and loses the two glued edges, and the rim is where cheap refusals on a flat sheet are rarest, since a circle cannot pass beyond it. Gluing removes two sides of rim and replaces them with the seam, across which a circle can pass freely; the refusal rate a flat map gets from its interior, the tube gets from its whole circumference.

A pre-check worth one row #

The practical reading is for anybody searching a tube’s letterings for one that folds. The loop test is cheap — it builds the arcs and looks for a circle — but on a tube there is a cheaper test inside it. Check each row as a ring: its lengthwise creases, read round the tube, must not all point the same way. That costs one pass over the tube’s creases, and on a tube four panels round it catches three fifths of every refusal the windows miss before any window is examined.

On the tube two panels round it is the whole story: one comparison between two letters decides every lettering. On wider tubes the rings are a shrinking share of the round refusals, since the wandering circles take over, but they remain the cheapest part of the test to run. The map counted from the layers counts flat maps’ foldings by building layer orders; on a tube, a search built the same way would spend much of its effort on letterings a ring check would have refused in one line, and how much is exactly the round share measured above.

The two refusals round one loop #

There is a connection worth making explicit, because the tube now has two refusals that both live on its loop.

The first is the parity the tube a map makes found: a path round the tube crosses every lengthwise crease once, each crossing turns the paper over, and an odd circumference brings it back face down. That refusal is about orientation coming back to itself round the loop, and it is decided by the geometry alone. The second is the one here: going round the loop, each crease says which panel is on top of the next, and the statements must not come back to their start. That refusal is about order coming back to itself round the loop, and it is decided by the letters.

The long circles go round the middle noted that every refusal on a flat map is a ring map’s refusal — a ring of panels round some interior vertices whose layers fail to close — and that the tube’s loop and the flat map’s circles were the same statement arrived at from topology and from lettering. On the tube the two meet on one loop: the even circumference passes the orientation check, and then the letters face the order check round the same path, and on a narrow tube they fail it about half the time.

What the census cannot show #

Whether a lettering that passes folds. The loop test is necessary and not sufficient; a lettering with no circle in its forced arcs may still have no folded state, as the map counted from the layers and the search essays measure. The shares here are what the cheap test refuses, not what fails to fold.

Anything about the torus. Gluing a tube’s two open ends as well adds a second loop along the tube, and the forcing that makes a tube’s valid letterings easy to draw — each vertex settling the crease above it — no longer runs from one end to the other. The torus’s letterings need a different sampler, and its second loop presumably carries circles of its own.

Anything about odd circumferences, which have no folded state and no letterings to count.

The idealisations underneath #

Letterings are drawn uniformly from those that pass every vertex, by choosing the free creases at random, which is the same population the flat-map essays sampled; it has no reason to resemble the letterings anybody would choose. Each refused lettering is filed under the first test that refuses it, in the order three-by-three window, four-by-four window, circle round the tube, anything else, so a lettering refused both by a window and round the tube counts once, as a window. The round share is therefore the share the windows cannot explain, which is what the question was. Samples are fifteen hundred letterings per tube, so a share near a half carries about a point and a quarter of sampling error and the small ones proportionally less.

How the numbers were checked #

The cell-by-cell construction reproduces the flat maps’ exact counts before any tube is measured: 4 of 256 on three by three, 2,112 of 32,768 on four by four with 64 of them outside every window.

On every tube counted exactly, every refusal outside the windows must wind once round the tube, and the share a three-by-three window refuses must agree with the windows’ rule to within a point; both hold on all five.

On sampled tubes the windows’ share must agree with the rule to within four points, the round share must fall by a factor under 0.85 for every two panels of circumference, and the leftover — refusals that neither a window nor the tube’s loop explains — must stay under three per cent on every tube drawn. It does, and it grows with the tube’s width and length, which is what the long circles round rectangles that the long circles go round the middle found on flat maps would do.

Still open: the second loop, and a rule for the rate #

The torus is the obvious next object, and it needs one thing first: a way to draw letterings that pass every vertex when no crease is free to be forced last. A map with no edges found what gluing both pairs of edges does to the folded states; the loop test on it would have two families of circles round two loops, and the question is whether their rates simply multiply or interfere.

A formula for the round rate is the other lead. The share falls by about half per two panels of circumference, and the ring alone accounts for a known fraction at each width — every lengthwise crease in one row pointing the same way round, weighted by what the vertices force. Counting the wandering circles the same way would give the rate as a sum over paths round a cylinder, which is a transfer-matrix calculation over one row of the tube at a time, and it would say whether “about half” is a constant or a slowly moving number.

Sideways from here, where the exponent comes from is about how fast the number of foldings grows with a map’s area. On a tube the cheap test takes a share set by the circumference out of every length, so if the same is true of foldings, a tube’s growth rate would carry a correction that depends on its circumference alone — a measurable prediction about a quantity nobody has counted.

The habit worth carrying is about locality and topology. A test that is local on a disc need not be local on a surface with a hole in it. The windows were enough on the flat map because every circle bounded something small; the tube has a circle that bounds nothing, and the test found it about half the time.