Origami & Folding
what one sheet can do
Concept

Angle bisector — where it appears

A line through the point where two lines cross that makes equal angles with both. Two crossing lines have two of them, always perpendicular to each other, and folding along either one carries the first line onto the second.

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

the cut line3 straight edgesthe pattern3 skeleton arcs3 perpendiculars1 interior vertexassignments that fold30 of 646 creases in allarcs one way, perpendicularsthe other: fails Maekawaequidistance off by 1.9e-16mountainvalleyevery node sits the same distance from each edge that formed it,which is why one fold can carry several edges onto the line at once

One straight cut

Any drawing made of straight lines can be folded so that the whole drawing lands on a single line, and one cut releases it. The construction is a shrinking process, and it explains itself the moment the shrinking is drawn.

flat-folding · Flat-foldability 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

Named alongside it

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

AmbiguityAxiomConstructionConstructive proofEquidistanceThe fold-and-cut theoremFold lineThe Huzita–Hatori axiomsReference pointSheet shapeStraight skeleton

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