Origami & Folding
what one sheet can do
Concept

Simple fold — where it appears

A single fold along a straight line across the whole of what is being folded, made in one motion. Restricting a machine to these is what turns folding into a model of computation with something to prove about it.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

folded 8 times, then unfolded33 interior vertices, all of degree 433 of 33 satisfy Kawasakithe folding is the reason, not the drawing45 creases drawn at random485 interior vertices, all of degree 40 of 485 satisfy Kawasakisame count, same sheet, nothing folded

The creases a sheet gives itself

A crease pattern drawn at random satisfies the flat-folding condition at essentially none of its vertices. A sheet crumpled at random satisfies it at every single one, on every seed, at every size — and the reason is a tautology that is very easy to miss.

material · Crumpling 24681012foldsinterior verticesfacetscrease lengththe median facet falls from 2.6e-1 to 4.0e-3 of the sheet

Every facet is a layer

Fold a sheet at random as many times as patience allows, then count three things: the creases it carries, the facets they cut it into, and the layers in the stack. The last two are the same number, always, and it is one more than the first — so how deep a crumpled sheet folds can be read off the flattened pattern without folding anything.

material · Crumpling the sectors are 40°, 140°, 40°, 140° — each row is one way the vertex can start to foldcrease 1crease 2crease 3crease 4mode 10.7100.7102 of the four creases movemode 200.7100.712 of the four creases movethe numbers are the four fold angles' ratios to one another as the vertex leaves the flat state

Two mechanisms at one point

Two creases drawn across each other cannot fold flat — Maekawa's count refuses them at every angle. They move perfectly well as rigid panels, and they move in two ways: bend along one line while the other stays flat, or the reverse. Every other developable vertex of degree four has two ways too, and in both of them all four creases move together at a fixed ratio. The crossing is the case where the two motions have nothing to do with each other.

rigid · Rigid folding

Named alongside it

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

IdealisationAssignmentConservationCountingCrossingDegree-fourFacetFlat-foldabilityFold angleGenericityKawasaki's theoremLayer ordering

All concepts