Origami & Folding
what one sheet can do
Concept

Incircle — where it appears

A circle inside a polygon touching every one of its sides. A quadrilateral has one exactly when its opposite sides have equal sums, and a molecule's interior creases measure how far a polygon is from having one, which is why a circle that nearly fits marks a crease that is nearly too short to fold.

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

shrinking changes every side and not their alternating sumthe interior crease is 0.151, at least half of 0.301shrunk byfour sidess₁ − s₂ + s₃ − s₄nothing0.965 0.628 0.740 0.7770.3010.0820.766 0.471 0.597 0.5920.3010.1650.567 0.314 0.455 0.4070.3010.2470.368 0.157 0.312 0.2230.3010.3290.170 0.001 0.170 0.0390.301

Half the difference is a theorem

A quadrilateral's molecule has one crease that does not touch the outline, and on every quadrilateral measured it was at least half the difference between the sums of opposite sides. That was a measurement and has a proof. Shrinking the shape changes every side but not their alternating sum. The proof gives the crease exactly: the difference over twice the sine of half the two angles at the side that vanishes first. A projection argument extends the bound to every even polygon and explains why no odd polygon has one.

design · Molecules 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

Named alongside it

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

Inscribed polygonOptimisationRegular polygonStraight skeletonToleranceTree methodUniversal molecule

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