A sheet rewards the multiples of its sides

An odd polygon fits like its double closed the question of the odd polygons on rectangular sheets by showing it had already been answered. A box measures a shape by its widths, and a shape averaged with its half-turn has the same widths as the shape; an odd polygon averaged with its half-turn is the regular polygon with twice its sides, so the two fit the same boxes and the odd polygon’s share follows from its double’s. On a rectangle, no odd polygon ever uses more of a sheet than a circle does.

It ended on the sheets where that argument has nothing to say. A sheet with no half-turn — a triangle, a pentagon — does not measure widths alone, and the essay expected the odd polygons to rank well there at last: “a triangular sheet holds a triangle perfectly and a hexagon badly.”

The first half is true and the second is not. A triangle of paper holds a hexagon at exactly two thirds, second only to the triangle itself. What decides the ranking on a regular sheet is not parity at all, and not the half-turn. It is whether the polygon’s number of sides is a multiple of the sheet’s.

The ranking on a triangle #

A regular polygon inside a regular sheet can be slid and turned freely. For each turn, staying inside is one straight-line condition per side of the sheet — the polygon’s reach towards that side, times its size, plus how far it has been slid that way, may not pass the side — so the largest size at that turn is the corner of a small linear programme in three unknowns, and the turn is searched over one symmetry step of the polygon. Nothing in that needs the sheet or the polygon to have a centre of symmetry, which is exactly what the rectangle’s width argument needed.

The triangle takes all of it. The hexagon takes two thirds, the nonagon 63.0 per cent, the dodecagon 61.9, the fifteen-gon 61.4, the eighteen-gon 61.1, and the circle 60.5. Then the ranking crosses the circle and never comes back: the twenty-four-gon, the last multiple of three drawn, is at 60.8, and the best polygon that is not a multiple of three is below the circle.

The line between the two groups is divisibility by three, and nothing else. Every polygon from three to forty sides whose number of sides is a multiple of three beats the circle, and every one that is not falls short of it — the pentagon, the square, the octagon, the thirty-nine-gon alike. The odd polygons are split right down the middle by it: the triangle, the nonagon, the fifteen-gon and the twenty-one-gon rank above the circle and the pentagon, heptagon and eleven-gon below.

Sharing the sheet’s circle #

The reason fits in two sentences, and the second of them is a bound that holds for every polygon.

A polygon whose number of sides is a multiple of the sheet’s can be turned so that one of its sides lies along each side of the sheet. Every third side of a nonagon points the way a side of the triangle points, so a nonagon centred in the triangle, with those three sides pushed out until they lie on the triangle’s, is inside the triangle — it is the triangle with its corners cut off twice more — and its inscribed circle is the triangle’s inscribed circle.

And no polygon can do better than that. Whatever polygon sits inside the sheet, its own inscribed circle sits inside it too, so that circle is no larger than the largest circle the sheet holds. A regular $n$-gon with inradius $r$ has area $n\tan(\pi/n)\,r^2$, so its area is at most $n\tan(\pi/n)$ times the square of the sheet’s inradius. The multiples of $m$ reach that bound exactly, and on a regular $m$-gon sheet their share is

$$\frac{n\tan(\pi/n)}{m\tan(\pi/m)}.$$

A polygon drawn round a circle is always larger than the circle, $n\tan(\pi/n) > \pi$, so every multiple beats the circle, and by less as $n$ grows. That accounts for the shaded half of the ranking, exactly: the closed form and the search agree to fifteen decimal places on every multiple of every sheet tried.

The other half — that no polygon which is not a multiple ever reaches the circle — is measured, not proved. The same bound applies to it, but it cannot meet it: it cannot face every side of the sheet with a side of its own, so its inscribed circle is smaller than the sheet’s. How much smaller is what the search measures, and it is always enough.

Cutting the corners, three at a time #

The multiples have a picture that makes the formula unsurprising. The hexagon in the triangle is the triangle with its three corners cut off at a third of each side. Each corner cut away is a small equilateral triangle a third the size of the sheet, so a ninth of its area, and three of them leave $1 - 3/9 = 2/3$. The nonagon is the triangle with each corner cut twice instead of once, and the dodecagon is the hexagon with each of its six corners cut; every one of them is a polygon drawn round the same circle, with its sides in more directions. The sequence of shares — one, two thirds, 0.630, 0.619, 0.614 — is the area left as the corners are cut ever more finely round that one fixed circle. Its limit is the circle’s $\pi/(3\sqrt3)$, 0.6046.

Read that way, the ranking on a regular sheet is a statement about how much of the sheet lies outside its own circle, and where. A triangle carries nearly two fifths of its area in its three corners, which is why its circle takes only 60.5 per cent and why there is so much room between the circle and the triangle for the multiples to occupy. An octagon carries only about a twentieth in its eight shallow corners, and the multiples of eight have almost nothing to rank in.

The non-multiples are shut out of that sequence for a geometric reason that is easy to see in the drawings. A pentagon inside a triangle can put at most one side flat against a side of the sheet at a time, since its sides point in five directions and the sheet’s in three, and only one pair can agree. So it touches the sheet at a side and at corners, and a polygon held by its corners cannot open out to the circle the sheet holds: some of its corners reach the sheet’s sides before its own circle has grown to the sheet’s.

Two families, one limit #

As the number of sides grows both families close on the circle, one from each side. The multiples of three come down as $n\tan(\pi/n)/\pi$, which is $1 + \pi^2/3n^2$ and a little more. The others climb: at forty sides the forty-gon reaches 0.9984 of the circle and has still not touched it. Nothing about the non-multiples’ approach is smooth — neighbouring polygons climb by different amounts, because each meets the sheet’s sides in its own way — but none of them is ever above the line.

That is where the triangle differs most sharply from the rectangle. On a rectangle the ranking was frozen by parity, and even polygons came down to the circle from above while odd ones climbed from below. On a triangle the same picture appears with three in the place of two. The rectangle’s rule was a special case: a rectangle’s rotations are its half-turn, the multiples of two are the even polygons, and “every even polygon beats every odd one” was the statement that the polygons which can face both pairs of a box’s sides with sides of their own are the ones that win.

The same rule on five sheets #

On every sheet the polygons that beat the circle are exactly the multiples of the sheet’s sides. Second place is always the polygon with twice the sheet’s sides — the hexagon on the triangle at two thirds, the octagon on the square at 82.8 per cent, the decagon on the pentagon at 89.4, the dodecagon on the hexagon at 92.8, the sixteen-gon on the octagon at 96.0 — since the formula falls as $n$ grows and the next multiple after the sheet itself is its double. The circle’s share climbs with the sheet’s sides, from 60.5 per cent of a triangle to 94.8 of an octagon, so a sheet with more sides leaves less between the circle and the polygons that beat it.

The square row is a check against the essays on rectangles. On a square the winners are four, eight, twelve and sixteen — the multiples of four — and the octagon’s 82.8 per cent is the value the square is in the answer found and the sheet a polygon fits exactly explained. The hexagon, which every even polygon beats every odd one found overtaking the octagon on a sheet 1.1284 long, is not a multiple of four and does not beat the circle on the square: a sheet has to be stretched out of its own symmetry before a polygon outside its multiples can win.

A hexagon of paper #

The hexagonal sheet deserves a word of its own, since it is the regular sheet folders reach for most after the square — cut from a square, or bought ready cut, for tessellations whose twists and pleats run in three directions. Its multiples are six, twelve, eighteen and twenty-four, taking all of it, 92.8, 91.6 and 91.2 per cent, with the circle at 90.7. The margin second place has over the circle is barely two points, against six on the triangle and over four on the square, because a hexagon is already so nearly round.

The hexagon also shows the rule’s reach beyond parity and beyond three. Its winners are the multiples of six, so the triangle — a divisor of six, not a multiple — is not among them, and neither is the nonagon: a nonagon has sides pointing in nine directions and the hexagon’s in six, and only three of the nonagon’s can face the hexagon’s sides at once. A polygon wins on a sheet when it can repeat all of the sheet’s directions, not when it shares some of them, and a triangle repeats half a hexagon’s.

Where folding first takes a place #

The question the earlier essay carried was whether folding ever buys a place on such a sheet that a compass cannot. The heptagon a compass cannot reach sets out which polygons each tool builds: a compass needs the polygon’s totient to be a power of two, a single fold needs only that it have no prime factor above three, and a nonagon — totient six — is the smallest polygon a fold builds and a compass does not after the heptagon.

On a triangle of paper the nonagon is third, at 63.0 per cent, behind the triangle and the hexagon and ahead of every other polygon and the circle. On a hexagon of paper the eighteen-gon, also a one-fold polygon, is third. Those are the first sheets in this collection on which a polygon only a fold can construct stands on the podium; on every rectangle from the square to three to one, the best such polygon placed fourth at best.

On the other sheets folding arrives much later. The square’s multiples of four are all compass polygons until twenty-eight, seventh; the pentagon’s multiples of five are compass polygons until twenty-five, fifth, and that one needs two folds made at once; the octagon’s first fold-only place is its sixth, the thirty-nine-gon, the best of its non-multiples. Which sheet rewards folding is decided by its first multiple whose totient has a factor of three. On a triangle that is its third multiple, nine, whose totient is six; on a hexagon it is eighteen, again the third; on a square the multiples of four keep totients that are powers of two until twenty-eight.

Turning a square in a triangle #

The polygons that are not multiples still have a best position, and it is not one a folder would guess.

The best square stands on the base with two corners on the sloping sides and takes 49.7 per cent of the triangle — the classical square-in-a-triangle, whose side is $2\sqrt3 - 3$ of the triangle’s. Every turn away from that costs it, since the square then touches the sheet at corners only and one of them reaches a side first. The pentagon does better, at 53.9 per cent, and still falls below the circle. These are the positions the linear programme finds; none of them shares the sheet’s circle, and that is the whole difference between them and the multiples.

Where the half-turn stops helping #

It is worth seeing what the rectangle’s argument would have predicted here, because it predicts wrongly and the error is instructive.

Applied to a triangle, the width argument says a triangle fits wherever its half-turn average, a hexagon, fits — and so takes the hexagon’s share times the ratio of their areas, 44 per cent. It takes all of it. The nonagon takes 63.0 per cent against a predicted 59.2, the pentagon 53.9 against 52.7. On a sheet with no half-turn a shape is not interchangeable with its half-turn, and the odd polygons that are multiples of the sheet’s sides do better than their doubles would suggest, because they are not being measured by width at all.

What the census cannot show #

Every sheet here is a regular polygon, and every polygon placed in it is regular. Real paper comes as rectangles almost always, and hexagons and triangles are cut from them; the census ranks what a regular sheet can hold, and says nothing about how the sheet was made. A triangle cut from a square by folding is itself a construction, with its own errors, and exact is not accurate measured what hands do to such constructions.

The polygons are points and lines. A nonagon drawn in a triangle with its incircle shared has three of its sides lying exactly on the sheet’s edges; a folded nonagon marked on a real triangle has creases, and a crease one fibre from the edge is not the same object as the edge. The ranking is the geometry’s, and a folder’s tolerance would blur the gaps near the circle, which are fractions of a per cent.

And the non-multiples’ side of the rule is a census to forty sides, not a proof: the inscribed-circle bound says no polygon beats the multiples of $m$, and the search says none of the others beats the circle, but no argument here says why their shortfall from the sheet’s circle is always large enough.

How the fits were checked #

For each polygon, sheet and turn the largest fit is the best corner of the three-unknown linear programme, every triple of the sheet’s sides tried; the turn is searched on a grid of 1,440 angles over one symmetry step and refined by golden section. The multiples of each sheet are required to match the shared-incircle share to a part in a billion, the polygons above the circle to be exactly the multiples on every sheet to forty sides, and the square’s best share in a triangle, turned through a whole quarter turn, never to exceed what the search reports.

Still open: sheets that are only nearly regular #

A sheet that is nearly regular rewards its near-multiples. A rectangle slightly off square keeps the octagon ahead for a while and then hands the lead to the hexagon at 1.1284; a triangle slightly off equilateral should hand its lead from the nonagon to something else in the same way. Where, and to which polygon, is the triangle’s version of that essay, and it needs the linear programme on a general triangle, which the one here already handles.

The tolerance at the peak is the other direction. A multiple’s fit is tight on every side of the sheet at once, so a sheet cut a little wrong costs it share to first order in every side’s error, while a non-multiple touches the sheet at fewer places. A sheet cut badly may therefore rank the polygons differently from a perfect one, and the nonagon’s podium place may be the first to go.

Sideways from here, the same rule appears where a folded twist is cut from paper of its own shape: a sheet that turns with the twist finds a twist giving one answer to a measurement only when its sheet shares enough of its rotations. And the biggest one that can also be folded asked which of the largest polygons a fold can mark; on a triangle the answer is that the first one a compass cannot is also the third largest there is.

The habit worth carrying is about which symmetry a measurement respects. Before asking what a shape does on a sheet, ask what the sheet can see of it. A rectangle sees widths and so cannot tell a shape from its half-turn; a regular sheet sees its own directions and rewards the shapes that repeat them, and parity was only the rectangle’s name for that.