Generator

More sides is not more paper

A generator in the axioms and construction library, called 44 times across 7 essays. Below: what it draws at its defaults and at the arguments the essays give it, what it checked while drawing, and everywhere it is used.

inscribed-polygons is one function. Everything below came out of it during this build, at arguments taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and when the generator changes, this page changes with it.

At its defaults

`view: "tools", show: "ranks"`

`view: "tools", show: "crossover"`

`view: "tools", show: "fit", ns: [8, 6], h: 1.1284`

What it checked while it drew

Collected by running this generator with a listener on the assertions, not written here. The count is how many separate times this build put that claim to the test.

each polygon is drawn at the rotation and size the search found, and every vertex lies inside the 1 by 1.128 sheet ×7
the 8-sided polygon uses more of the square than any of the 4 polygons with more sides ×3
where the hexagon passes each polygon, the proportion found by searching the two share curves agrees with the closed form √3⁄2 + ½√(8K⁄3√3 − 1) to 7e-16, on all 10 polygons ×2
a corner construction solving a quadratic agrees with the width maximisation to nine figures ×1
across every sheet tried, the best place each tool's polygons reach is held by its first polygon of 4k + 2 sides on that polygon's own sheet — 14 sides 3rd, 22 sides 5th, 46 sides 11th — against 4th, 9th, 12th on the seven sheets named ×1
and the best polygon only a fold can build never ranks higher than fourth — 21-gon 7th on square, 14-gon 4th on 11 : 10, 14-gon 5th on 6 : 5, 14-gon 5th on A series, 14-gon 5th on 3 : 2, 14-gon 5th on 2 : 1, 14-gon 5th on 3 : 1 ×1
and the odd primes in each crossing's degree are exactly the odd primes in the polygon's own totient — 17: 64, 18: 12, 19: 72, 20: 16, 21: 24, 22: 40, 23: 88, 24: 8 — so marking the proportion takes the same tool as building the polygon ×1
and the odd primes in each crossing's degree are exactly the odd primes in the polygon's own totient — 7: 24, 8: 8, 9: 12, 10: 16, 11: 40, 12: 4, 13: 48, 14: 24, 15: 16, 16: 16 — so marking the proportion takes the same tool as building the polygon ×1
each polygon is drawn at the rotation and size the search found, and every vertex lies inside the square ×1
each polygon passes the circle part of the way from the square to its own sheet — 61.2%, 62.6%, 62.9%, 63.2% of the way — and the share tends to 2√(2/3) − 1 as the sides grow ×1
each polygon's line K⁄h meets the hexagon's curve at the proportion the closed form gives — 8 sides at 1.1284, 7 sides at 1.0696, 11 sides at 1.0853 ×1
every share is a maximisation over the polygon's own rotation on each sheet, so no proportion is being compared at somebody else's best angle ×1
followed across every proportion from the square to 1.06, each polygon's best rank is the one it holds on its own sheet — 14 sides 3rd, 22 sides 5th, 18 sides 4th ×1
not one of the 3 rectangles has the same best polygon as the square — every one of them prefers the 6-gon ×1
on a sheet 3 times as long as it is wide every polygon's share times the length is its area over its least width squared, to 1e-15 — above π⁄4 for every even polygon and below it for every odd one, so every even polygon beats every odd one ×1
on a square the best polygon after the square itself is the octagon, at 82.8% of the sheet — computed here rather than carried over ×1
on every sheet the best polygon after the square is one a compass already builds, so the best a fold builds is the same polygon — the 8-gon on square, the 8-gon on 11 : 10, the 6-gon on 6 : 5, the 6-gon on A series, the 6-gon on 3 : 2, the 6-gon on 2 : 1, the 6-gon on 3 : 1 ×1
on its own sheet each polygon uses exactly (1 + L)/2 times the share it used on the square, L being the own sheet's length ×1
on its own sheet, 1⁄cos(π⁄n) long, every polygon of 4k + 2 sides from 6 to 46 uses (n⁄4)·sin(π⁄n) of it to 8e-16 and ranks exactly (n − 2)⁄4 among every polygon up to 60 sides ×1
on the 22-gon's own sheet every polygon of 4k + 2 sides with fewer sides is still turning, below its own peak, and every such peak is below the 22-gon's share there ×1
the closed-form share agrees with a direct search over rotations to a part in a billion, at 7 sheets for each of 4 polygons ×1
the even-sided polygons rank 8 > 10 > 6 on a square and 6 > 8 > 10 on a rectangle — the order is reversed, not merely disturbed ×1
the even-sided polygons rank 8 > 12 > 10 > 6 on a square and 6 > 8 > 10 > 12 on a rectangle — the order is reversed, not merely disturbed ×1
the hexagon's share searched at 21 proportions agrees with the closed form to 7e-16 — held by both widths up to 2⁄√3 and by the short side after ×1
the largest equilateral triangle comes out at 2√3 − 3 of the sheet, tilted by 15° ×1
the largest square inside a square is that square, which the search finds without being told ×1
the octagon and the hexagon use the same share of a sheet 1.1284 times as long as it is wide, 73.4%, and from there on the whole ranking of polygons stops moving ×1
the tilt the search settles on is the one the closed form gives, at four sheets for each polygon ×1

Where it is called

Changing this generator changes every figure on this list, which is what makes the list worth publishing rather than keeping in a check script.

Every even polygon beats every odd one

Crossed with what each tool can build, the census of the largest regular polygon a sheet holds gives the same verdict on every proportion from a square to three to one: the best polygon is one a compass already builds, and the best polygon only a fold can build places fourth at best. The ranking itself stops moving at a proportion of 1.1284, where the hexagon overtakes the octagon. On every longer sheet only the short side holds a polygon, each polygon's share is a fixed constant divided by the length, and the constant — its area over the square of its least width — comes down to the circle's π⁄4 for even polygons and climbs up to it for odd ones. So every even polygon beats every odd one.

The biggest one that can also be folded

Which regular polygon uses a square sheet best, and which of them a fold can actually construct, are two questions with completely different pedigrees. Answered side by side over sixteen polygons, they turn out to agree — and the reason is that both are questions about the arithmetic of the same number.

The crossing is as hard as the polygon

Lengthen a square sheet and the largest hexagon it holds turns, pressed against all four edges, until it overtakes the polygons held by the short side alone. Every one of those overtakings happens at a proportion with a closed form, √3⁄2 + ½√(8K⁄3√3 − 1), and the number that comes out is exactly as hard to mark as the polygon being overtaken is to build. The octagon's 1.1284 is a compass number of degree eight. The heptagon's 1.0696 has degree twenty-four and needs a fold. The hendecagon's 1.0853 has degree forty and needs two folds at once.

The largest triangle in a square

The biggest equilateral triangle a square sheet holds is tilted by exactly fifteen degrees and uses 46.4% of the paper. Both numbers come out of a quadratic — which means a compass reaches this optimum too, and folding's advantage is not needed here at all.

The sheet a polygon fits exactly

A regular polygon with 4k + 2 sides has flat edges along one axis and corners along the other, so there is one sheet, 1⁄cos(π⁄n) long, that it touches on all four edges at once. On that sheet it is beaten only by the multiples of four with fewer sides, and so it ranks exactly (n − 2)⁄4. That puts the fourteen-gon, which only a fold builds, third rather than fourth; the twenty-two-gon, which needs two folds at once, fifth rather than ninth; and the forty-six-gon, beyond two folds, eleventh. Seven sheets from a square to three to one had missed all three.

The square is in the answer

The largest regular polygon a square sheet holds is not increasing in the number of sides, and the octagon's win is the striking part: it uses 82.8% of the paper against the twelve-gon's 80.4% and the hexagon's 69.6%. Run the same census over rectangles and the octagon's advantage is gone — on every proportion tried the hexagon leads, and the order among the even-sided polygons reverses outright.

Turning is uphill all the way

A regular polygon of 4k + 2 sides on a sheet a little longer than a square cannot lie flat: it turns, pressed against all four edges, until the sheet is exactly its own. Its share on the way has a closed form, and the closed form's slope is proportional to h² − 1 for every such polygon — flat on the square, rising all the way to the own sheet, and falling after it. So the own sheet is exactly the peak, the gain from the square to it is the average of one and the sheet's length, and the rank the census measured for polygons of this kind, (n − 2)⁄4, is now a theorem.

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