The rule has a width

A sheet rewards the multiples of its sides found a clean rule for regular sheets of paper. On a regular sheet with $m$ sides, a regular polygon takes more of the sheet than the largest circle does exactly when its number of sides is a multiple of $m$: it can then face every side of the sheet with one of its own, share the sheet’s inscribed circle, and take $n \tan(\pi/n) / (m \tan(\pi/m))$ of it. Every other polygon to forty sides falls short. On a triangle that put the triangle first, the hexagon second at two thirds and the nonagon third.

The essay ended by asking about sheets that are only nearly regular, and the question is not a technicality. No sheet of paper is regular. A triangle cut from a square by eye is off by a degree or two, and a rule that holds only on the exact sheet is a rule about a shape nobody holds. So the measurement is how the ranking moves as an equilateral triangle is bent out of true — here along the simplest family, isosceles triangles with an apex angle on a dial, 60° being equilateral.

A corner and a hilltop #

Draw each polygon’s share of the sheet against the apex angle and the two kinds of competitor separate at once.

The multiples of three peak at 60° in a corner. Tilt the apex by one degree either way and the hexagon’s share falls by 0.45 of a percentage point, the nonagon’s by 0.27 and the dodecagon’s by 0.20, and it keeps falling at close to that rate. The circle peaks at 60° on a hilltop: its first degree costs it 0.005 of a point, a ninetieth of the hexagon’s loss, and four degrees cost it 0.14.

The reason is the reason the rule held in the first place. A hexagon beats the circle on an equilateral triangle because three of its sides lie along the triangle’s three sides, and that needs the triangle’s sides to point in three directions a third of a turn apart. Bend the apex and the directions stop matching. The hexagon can keep one side flat against the sheet and must touch the other two with corners; it has to shrink to do so, and the amount it shrinks is proportional to the mismatch. A coincidence of directions is lost at first order.

The circle has no directions to match. Every triangle has an inscribed circle, and among all triangles the equilateral one gives it the largest share of the sheet, $\pi/(3\sqrt3)$, about 60.5 per cent. A maximum over a smooth family is flat at the top, so a small bend costs the circle only the square of the bend. A property that is optimal is lost at second order.

Put the two together and the ranking cannot survive far from 60°. The multiples start with a lead over the circle — the hexagon’s 66.7 per cent against 60.5 — and spend it at a constant rate, while the circle stands still.

Where each multiple falls behind #

The lead and the rate settle how far each multiple of three can go.

The hexagon beats the circle from an apex of 44.4° to 79.1°, which is a window thirty-five degrees wide. The nonagon’s window runs from 49.2° to 72.5°, the dodecagon’s from 51.7° to 69.2°, the 18-gon’s from 54.4° to 66.1° and the 24-gon’s from 55.7° to 64.5°. The window narrows as the sides multiply, and for a reason that is visible in the formula: a multiple with many sides is already nearly a circle, so its lead over the circle is small — the dodecagon starts 1.4 points ahead and the 24-gon only a third of a point — and it has less to spend.

Every window is lopsided in the same way, with more room above 60° than below. It is not the circle that makes it so: the circle falls faster on the tall, narrow side, 2.4 points by an apex of 45° against 1.8 by 75°, which on its own would widen the windows below. The multiples fall faster there still — the hexagon loses 8.4 points by 45° and 7.1 by 75° — and their asymmetry wins. Both sides nevertheless close, and the rule of the regular sheet becomes a statement with a tolerance on it: the hexagon beats the circle on any isosceles triangle within about fifteen degrees of equilateral, and a many-sided multiple within about four.

The triangle loses fastest of all #

The first-order losses have an order of their own, and it is worth reading before the podium, because it says which polygons were relying most on the coincidence.

The regular triangle loses 2.0 percentage points for every degree the apex moves, four and a half times the hexagon’s rate. That is the extreme case of the same mechanism: on the equilateral sheet the triangle does not merely face every side with a side, it is the sheet, and every point of its perimeter lies on the sheet’s perimeter. Bend the sheet and none of its perimeter can stay there. The hexagon had half of its perimeter lying along the sheet’s sides, the nonagon a third, the dodecagon a quarter, and their rates fall in the same order: 0.45, 0.27 and 0.20 points a degree. Multiplied by the number of sides those rates come to 2.7, 2.4 and 2.3 — not constant, but close enough to say that a polygon’s loss is set by how much of its boundary the coincidence was holding against the sheet, and that a polygon with many sides, touching the sheet along three short stretches, has almost nothing to lose and almost nothing to gain.

The polygons that are not multiples of three behave differently in kind. They were never aligned with the sheet, so the equilateral triangle is not a peak for them and they change at first order with no corner at all: the octagon gains 0.12 of a point for the first degree the apex opens, the pentagon 0.21, and each loses or gains by a different amount going the other way. On the equilateral sheet they sat below the circle; bending the sheet moves them up and the multiples down, and the ranking inverts within twenty degrees.

A degree on a real sheet #

The rates translate directly into what a folder should expect from a triangle cut by hand. A triangle whose apex is off by a degree — its tip about two and a half millimetres out of place on sides of fifteen centimetres — gives its largest hexagon 66.2 per cent of the sheet instead of 66.7, and its circle 60.45 instead of 60.46. The hexagon is still well ahead at a degree, and at five degrees; the rule of the regular sheet is safe for any triangle a careful folder produces. It is the 24-gon’s lead that a sloppy cut destroys, at four or five degrees, and the 24-gon is not a polygon anybody folds from a triangle.

So the practical content of the windows is not a warning about hexagons. It is that the regular-sheet ranking is exact at one shape and robust near it for the polygons that matter, fragile for the ones that nearly tie with the circle anyway, and meaningless twenty degrees away, where a different set of polygons leads and some of them need a tool single folds do not have.

Who takes the podium instead #

When the multiples fall, something has to take their places, and the ranking at each apex angle shows what.

At an apex of 80° the regular triangle is still first, with 68.8 per cent: it is no longer the sheet’s own shape, but it is still the polygon whose three corners can reach into the sheet’s three corners, and it keeps first place until the apex passes about 92°. Second is the hendecagon, at 58.6 per cent, then the octagon at 58.2 and the 14-gon at 57.9. The hexagon, which held second place on the equilateral sheet by more than five points, has fallen below the circle.

That hendecagon is worth pausing on. The eleven-sided one nobody can fold is the essay about why no sequence of single folds constructs a regular hendecagon — its totient has a factor of five, and a fold solves cubics, not quintics — and two creases at once is where it becomes reachable, by two folds made at the same instant. So on a triangle twenty degrees from equilateral, the second-largest regular polygon the paper holds is one that single-fold origami cannot construct at all. On the equilateral sheet, the regular-sheet essay recorded the first polygon only a fold can build, the nonagon, reaching the podium in third place. Off it, the podium can require a tool beyond single folds.

The dial shows the whole run. At 70° the hexagon and nonagon still hold second and third. By 75° the nonagon has gone and the 14-gon — a polygon one fold can build and no compass can — is third. At 80° the hendecagon leads the field behind the triangle; at 85° and 90° the octagon does; and at 95° the pentagon takes first place from the triangle itself, with 55.0 per cent against 52.9. Below 60° the story is the same in the other direction: at 45° the places behind the triangle go to the 16-gon, the 13-gon and the octagon, all within three tenths of a point of one another.

What the polygons look like there #

The numbers are close enough that a picture helps to see why a polygon with no relation to three can win.

The hexagon at 80° lies flat along the base and touches the two slanted sides with corners, because its sides point every sixty degrees and the triangle’s slanted sides no longer do. It is a hexagon wedged into a triangle it was not cut for. The hendecagon and the octagon have no special relation to the triangle at all; they sit much as the inscribed circle does, touching three sides at a corner or a side each, and since each is nearly as round as a circle while being allowed to push a corner into the triangle’s corners, each takes a little more. Away from the equilateral, being round is worth more than being aligned.

That is also why the places behind the triangle are crowded and noisy there. Every polygon with many sides is within a point or two of the circle, and which of them happens to fit the sheet’s three angles best changes from degree to degree. The regular sheet had a structure — the multiples first, in order, and everything else below the circle — and the nearly regular sheet has a crowd.

The idealisations underneath #

One family of triangles. Every figure here uses isosceles triangles, which bend the equilateral one along a single line of shapes. A scalene triangle bends it in two directions, and the first-order losses of the multiples should be first order in every direction, since a coincidence of three directions is lost whichever way it is broken; but the windows measured here are windows along one line, not regions.

Exact regular polygons. A polygon’s share is its largest exact regular copy, found at every rotation by a linear programme on its size and position and then refined in the rotation. A folder building a heptagon or a hendecagon does not build an exact one, and what buys the reach costs the accuracy is the reminder that the constructions reaching past the compass are the ill-conditioned ones.

And share of the sheet is the measure. The largest polygon is the one that wastes the least paper, which is what the regular-sheet essay ranked by. A designer who wants a hexagonal model from a triangular offcut wants the hexagon whatever its share, and cares about the ranking only as a measure of how much paper the offcut wastes.

What this cannot show #

Whether the ranking at a given apex is exact past the first few places. Behind the triangle and the leading multiples, the polygons with many sides sit within hundredths of a point of one another, and the rotation search that finds each one’s best position is refined to far better than that but is still a search. The first and second places quoted are separated by more than the search’s error; the fourth and fifth places at some angles are not, and nothing here depends on them.

Why the windows are lopsided by the amounts they are. The circle’s decline differs on the two sides of 60°, and so do the polygons’; the net asymmetry is measured and not derived.

And whether other sheets behave alike. A nearly square rectangle is the obvious next case — the square is a choice is where the question was first asked whether the square was the right sheet at all — and it differs in one respect that matters: a rectangle’s largest circle is set by its shorter side and falls at first order too, so the multiples of four and the circle both lose at first order there. Whether the octagon’s window on a nearly square sheet is wider or narrower than the hexagon’s on a nearly regular triangle is a different comparison and has not been made.

How the claims were checked #

The corner and the hilltop are checked at the equilateral triangle: the hexagon, nonagon and dodecagon are each required to lose share on both sides of 60°, measured half a degree away, and the circle’s loss there is required to be less than a twentieth of the smallest of theirs.

The windows are found by stepping out from 60° in half-degree steps until each multiple first falls below the circle, then halving; each is required to exist on both sides, and each is required to be narrower on both sides than the window of the multiple before it.

The regular sheet is recovered at 60°: the podium there lists the triangle, hexagon, nonagon and dodecagon in the order and at the shares the regular-sheet essay found, and all eight multiples of three to twenty-four beat the circle, which is the old rule reproduced by a different solver on a sheet written as three arbitrary sides rather than as a regular one.

Still open: the sheets people actually cut #

The measurement has a practical successor and a theoretical one.

The practical one is a triangle folded from a square. The usual way to get an equilateral triangle from square paper is a short fold sequence, and each fold carries its own error. Exact is not accurate found how such errors compound in a division, and the same accounting run on the triangle-from-a-square sequence would say how far from 60° a folder’s triangle typically lands — which would turn the windows here into a probability that the hexagon a folder is planning is the best polygon their sheet actually holds.

The theoretical one is the first-order rate itself. The hexagon loses 0.45 of a point a degree, the nonagon 0.27, the dodecagon 0.20 — close to a rate inversely proportional to the number of sides, but not exactly. Those rates come from a linear programme whose tight constraints change at the equilateral triangle, and differentiating the programme there would give them in closed form and say whether the product of rate and sides tends to a limit.

Sideways from here, an odd polygon fits like its double found that on a rectangle a polygon and its double fit alike because a box sees only widths. A triangle has no half-turn, and the ranking here is ordered by alignment with three directions; the square has four and the rectangle two, so how quickly a nearly square sheet’s ranking dissolves toward the rectangle’s is the question that would connect the two results. And every even polygon beats every odd one on the square is the finding most likely to be fragile in the same way, since it too is a statement about alignment.

The habit worth carrying is about exact results. Before relying on a rule derived at a symmetric point, ask whether the rule is an optimum or a coincidence there. When symmetry costs asked what imposing a symmetry costs a design; this is the converse, what a symmetry that was quietly doing the work costs when it is lost. An optimum survives a small change, because the first-order loss is zero; a coincidence does not, because it is exactly the first-order term that breaks it. The circle’s share on a triangle is an optimum and the hexagon’s advantage is a coincidence, and that difference is the whole of the window.