Origami & Folding
what one sheet can do
Concept

Constructive proof — where it appears

An argument that establishes something by building it. Much of this subject is proved that way — a pattern exhibited, a folded state found — which is why so many of its claims have figures attached.

Named by 3 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 3 equal partsestimated by eyethe left-hand divisions are exact — a consequence of the fold, not of carethe right-hand ones are a guess, and the error compoundsvalleymountain

The kindergarten was a geometry class

Froebel put paper folding into mass education in the 1830s, and did it as mathematics rather than as craft. His three categories — the folds of life, of beauty, and of knowledge — are the first systematic treatment of folding anybody wrote down, and the third one is a geometry syllabus.

history · Pedagogy the cut line10 straight edgesthe pattern10 skeleton arcs0 perpendiculars1 interior vertexassignments that fold420 of 102410 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 cut for a star

The fold-and-cut construction here could reach a triangle, a pentagon and a house, and refused everything that turned back on itself, because shrinking an outline with a reflex corner needs an event the shrink did not implement. With split events it reaches a five-pointed star — ten creases through one point, four hundred and twenty letterings that fold, and every edge of the outline landing on one line to a part in 10^16.

flat-folding · Flat-foldability

Named alongside it

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

The fold-and-cut theoremStraight skeletonAngle bisectorCrease patternEquidistanceExact divisionFlat-foldabilityFroebelFujimoto's methodInterior vertexPedagogyVertex degree

All concepts