Origami & Folding
what one sheet can do
Concept

Fold line — where it appears

The line a single fold creases the paper along, before anything is said about which way it goes. An axiom specifies one by naming alignments rather than by giving its position, and an alignment can be satisfied by more than one line.

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

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

Named alongside it

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

AxiomConstructionReference pointSheet shapeAmbiguityAngle bisectorApproximationCovering radiusExact divisionThe Huzita–Hatori axioms

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