Origami & Folding
what one sheet can do
Concept

Support width — where it appears

The distance between the two parallel lines that touch a convex shape from opposite sides, across a stated direction. A shape fits inside a rectangle exactly when its widths across the two axes are small enough, which turns fitting a polygon into a maximisation over one angle.

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

tilted 15.00°side 1.0352846.41% of the sheetand the same answer twicea corner construction givesside 1.03528from a quadratic, sharing no codethe width maximisation and the closed form agree to nine figures

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.

construction · Optimal constructions the same census, on four sheetsshare of the sheet used, at each polygon's own best rotationsidessquareA series3 : 26 : 5346.4%40.8%38.5%48.1%4100.0%70.7%66.7%83.3%567.4%51.4%48.4%60.5%669.6%61.2%57.7%72.2%772.9%53.5%50.5%63.1%882.8%58.6%55.2%69.0%975.1%54.4%51.3%64.1%1075.3%57.4%54.2%67.7%1176.3%54.8%51.6%64.5%1280.4%56.8%53.6%67.0%best after the square: square 8 · A series 6 · 3 : 2 6 · 6 : 5 6each polygon at its own best rotation on each sheet · the square uses all of a square and 70.7% of the next sheet along

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.

construction · Optimal constructions

Named alongside it

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

Inscribed polygonOptimalityAxiomConstructibilityPaper proportionQuadraticSheet shapeSymmetry

All concepts