Origami & Folding
what one sheet can do
Concept

Regular polygon — where it appears

A polygon with all sides and all angles equal, which is constructible exactly when the cosine of its central angle is reachable. The degree of that cosine is Euler's totient halved, so which polygons an instrument reaches is a condition on a factorisation rather than on the polygon.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

the equation on an edge, at both of its endsas drawnevery vertex moved 12 per centratio − 1round a loopratio − 1round a loopthe square grid000.200.90the triangular grid000.170.75the honeycomb000.351.73the rhombille tiling2.0002.992.14the elongated triangular tiling000.171.27the rhombille's tiles are not regular, its ratios are three and a third, and they still multiply to one

The propagation that never had to work

The twist construction carries one equation per edge of its tiling and propagates the twist sizes outward from a seed. On every tiling anybody has drawn a twist on, every one of those equations is satisfied trivially — both ends of an edge read the same two numbers, because a regular polygon has one interior angle. The construction has been running and doing nothing, and the one tiling where it did something is the one whose tiles are not regular.

tessellation · Resch how many extension steps the shortest tower to each polygon takesa polygon of n sides needs the degree of two cosine of a turn over n, which is Euler's totient halved3 sides0degree 1 · square roots only, so a compass reaches it4 sides0degree 1 · square roots only, so a compass reaches it5 sides1degree 2 · square roots only, so a compass reaches it6 sides0degree 1 · square roots only, so a compass reaches it7 sides1degree 3 · 1 cube root8 sides1degree 2 · square roots only, so a compass reaches it9 sides1degree 3 · 1 cube root10 sides1degree 2 · square roots only, so a compass reaches it12 sides1degree 2 · square roots only, so a compass reaches it13 sides2degree 6 · 1 square root and 1 cube root14 sides1degree 3 · 1 cube root15 sides2degree 4 · square roots only, so a compass reaches it16 sides2degree 4 · square roots only, so a compass reaches it17 sides3degree 8 · square roots only, so a compass reaches it18 sides1degree 3 · 1 cube root19 sides2degree 9 · 2 cube roots20 sides2degree 4 · square roots only, so a compass reaches it21 sides2degree 6 · 1 square root and 1 cube root24 sides2degree 4 · square roots only, so a compass reaches it26 sides2degree 6 · 1 square root and 1 cube root27 sides2degree 9 · 2 cube roots28 sides2degree 6 · 1 square root and 1 cube root30 sides2degree 4 · square roots only, so a compass reaches it32 sides3degree 8 · square roots only, so a compass reaches it34 sides3degree 8 · square roots only, so a compass reaches it35 sides3degree 12 · 2 square roots and 1 cube root36 sides2degree 6 · 1 square root and 1 cube root37 sides3degree 18 · 1 square root and 2 cube roots38 sides2degree 9 · 2 cube roots39 sides3degree 12 · 2 square roots and 1 cube root40 sides3degree 8 · square roots only, so a compass reaches itthe pale bars are the polygons a compass reaches, and they are not the cheap ones — 4 of the one-step polygons need a cube root

Gauss's polygon is the expensive one

Which regular polygons a fold reaches is a condition on the factorisation of Euler's totient, and every polygon that passes it also has a height — the number of extension steps the shortest tower to it takes. Read that column instead of the verdict and the field inverts: the heptagon, which no compass reaches, costs one step; the seventeen-sided polygon that made Gauss famous costs three, the most on the list; and the polygons a compass finds easy are the ones a folder pays most for.

construction · Origami numbers the largest regular polygon a triangle of paper holds, for each number of sides, best firstshaded: sides a multiple of 3; plain: the rest; the upright line is the largest circle, 60.5%bars start at 50%3 sides100.0% · compass6 sides66.7% · compass9 sides63.0% · one fold12 sides61.9% · compass15 sides61.4% · compass18 sides61.1% · one fold21 sides60.9% · one fold24 sides60.8% · compass23 sides60.2% · neither22 sides60.1% · two folds at once20 sides60.1% · compass19 sides60.0% · one fold17 sides59.9% · compass16 sides59.9% · compasscircle

A sheet rewards the multiples of its sides

On a rectangle every odd polygon fits exactly as its double does, because a box measures widths and a width cannot see a half-turn. A triangle of paper has no half-turn, and the ranking it gives is a different kind of thing. On a regular sheet of m sides, a regular polygon beats the largest circle exactly when its sides are a multiple of m — it can then face every side of the sheet with one of its own and share the sheet's circle, taking n·tan(π/n) over m·tan(π/m) of it — and every other polygon to forty sides falls short. On a triangle that puts the triangle first, the hexagon second at two thirds, and the nonagon third: the first sheet on which a polygon only a fold can build takes a place on the podium.

construction · Optimal constructions the largest regular polygon's share of an isosceles triangle, against its apex anglemultiples of three in one colour, the rest in another; the dashed curve is the circle0.5000.600406080100apex angle, degrees (60 is equilateral)share of the sheethexagonnonagondodecagonoctagon11-goncirclethe multiples of three touch every side only on the equilateral triangle, so their peak is a corner

The rule has a width

On an equilateral triangle of paper the regular polygons that beat the largest circle are exactly the multiples of three. Tilt the apex a degree and every one of them loses share in proportion, while the circle loses almost nothing — a multiple's advantage is a corner and the circle's is a hilltop. So the rule holds only in a window, 44° to 79° of apex for the hexagon and 56° to 65° for the 24-gon, and past it the second place goes to polygons like the hendecagon that no single fold can build.

construction · Optimal constructions the hexagon: 66.67 per cent at 60°, losing 0.448 points a degreedashed: the slope 2n tan²(π/n) ⁄ 9√3 a radian, either way; below, the inscribed circle59616365675256606468apex angle, degrees (60 is equilateral)share of the sheet, per centcircle

The worst angle sets the price

On a triangle of paper that is nearly equilateral, the largest regular hexagon loses 0.448 percentage points of the sheet for every degree the triangle is out of true, the nonagon 0.267 and the dodecagon 0.193. Those rates were measured. They have a closed form, 2n tan²(π/n) / 9√3 a radian, and it depends only on the worst of the triangle's three angles, whatever its shape. Rate times sides falls to 2π² / 9√3, and the window in which each polygon beats the circle shrinks as ninety degrees over its number of sides.

construction · Optimal constructions the formula, the fitted polygons, and the two edges of the windowsolid: the one-line share, where the polygon keeps its base; dots: fitted; lower curve: the inscribed circle545658606264664550556065707580apex angle, degrees (60 is equilateral)share of the sheet, per cent

The window has an exact edge

How far an isosceles triangle may stray from equilateral before its largest regular hexagon falls behind its inscribed circle was measured — 15.6 degrees closing the apex, 19.1 opening it — and estimated to first order as the polygon's lead over its rate. Neither number needs measuring. The fitting problem has three sides and three unknowns, so its bound is an equality, the polygon keeps a side on the base, and its share is one line. That line carries the same factor as the circle's, so the circle's own loss cancels out of the comparison entirely, and each edge of the window is an angle in closed form. The opening half is wider because opening the apex lengthens the perimeter. And the window does not shrink as ninety degrees over the sides, as the first-order estimate says, but as (2 − √2)·180° over them — about 105 — because the corner's reach bends back as fast as it rises.

construction · Optimal constructions

Named alongside it

The objects these essays reach for when they reach for this one.

Inscribed polygonToleranceConstructibilityIncircleOptimalityOptimisationSheet shapeConstructible polygonField extensionKawasaki's theoremOrigami numberPleat

All concepts