Origami & Folding
what one sheet can do
Concept

Axiom — where it appears

One of the fold operations the subject takes as primitive, each aligning stated points and lines in one move. Seven of them exhaust what a single fold can determine, which is a completeness statement rather than a list.

Named by 6 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 distinct fold lines this axiom specifies and no other doesaxiomafter 4 pointsafter 9 pointsafter 565 pointsA1 — through two points08121054A2 — one point onto another08142649A3 — one line onto another4564994A4 — perpendicular through a point001661distinct lines in all1292274300the four operations name 38 folds at the first round and draw 12 lines with them

What each axiom is worth

The list of seven folds is complete, and the proof of that says nothing at all about whether its members are independent or equal. Measured on a bare square, one of the four elementary axioms supplies every fold the others cannot and the other three supply nothing. Two rounds later the ranking has inverted, and the one that carried the first round is the least productive of the four.

construction · The axioms the axiom is satisfied by both, and says nothing about whichthe two are square to one another, alwaysthe thin lines are the two the axiom is given; the two heavy ones are its answers

The axiom that names two folds

Bring one line onto another and the operation is satisfied by either of two folds, always exactly perpendicular to one another, creasing the paper in completely different places. Over four thousand random line pairs on a square, both folds land on the sheet two thousand nine hundred and sixty-five times, and the statement of the axiom does not say which one is meant. The fifth axiom is worse: its two answers are at any angle at all, from half a degree apart to square.

construction · The axioms 565 references · worst gap 0.089 of a sheet · under 0.092the disc is where a folder has the least to align againstthe faint lines are the folds themselves; a reference is where two of them cross on the paper

How far from the nearest reference

One fold puts nine reference points on a square sheet and two folds put five hundred and sixty-five. That is sixty-three times as many points, and it brings the worst-covered spot on the paper from a third of a sheet away to a twelfth — four times closer. A count of references is not a measure of what a fold buys, because a set of points can be arbitrarily crowded and still leave most of the sheet out of reach.

construction · Reference points axiom 1through two pointslinearaxiom 2point onto pointlinearaxiom 3line onto linelinearaxiom 4through a point, square to a linelinearaxiom 5point onto a line, through a pointquadraticaxiom 6two points onto two linescubicaxiom 7point onto a line, square to a linelinearthe degree each axiom can solve — one of them is why paper beats the compass

Which of the seven survive

The seven axioms are the complete list of ways one fold can be specified by aligning marked things. Every one of them names points and lines on a sheet, three of them quietly assume that a line has two sides, and on a closed sheet a line need not — so the list is complete for a disc and shorter for anything else.

construction · The axioms the bar is the share of each round's new references that lie on the sheet's own edgeafter one fold80%4 on the edge · 1 inside · 12 fold linesafter 2 folds9%48 on the edge · 508 inside · 92 fold linesan edge is a line a fold can cross twice, so it yields marks with the folds and not with their pairs

A reference on a sheet with no corner

Every construction in this subject begins from the sheet's own boundary: two edges meet at a corner, a corner is a point, and a point is what an axiom takes as input. A cylinder has two circles of edge and no corners at all, so a construction on one has nothing to start from and the seam is not a mark.

construction · Reference points

Named alongside it

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

Reference pointConstructionThe Huzita–Hatori axiomsSheet shapeBoundaryConstructibilityFold lineGluingThe axiomsAmbiguityAngle bisectorApproximation

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