Origami & Folding
what one sheet can do
Concept

Tractable restriction — where it appears

A special case of a hard problem that can be decided cheaply, carved out by restricting the input rather than by improving the method. The restriction is where the interest lies: it says which feature of the general problem the hardness actually comes from, and a case that stays hard under it has been narrowed to that feature.

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

MVMV123455 segments, 4 creases12345the stack, solvedassignmentsMVMVvalid stacks1decided byexhaustive searchover the orderingsthe folded positions come from the crease spacing; the assignment only decides which way each turn wraps

A strip is decidable

Take the same problem down one dimension and it stops being hard. The reason is not that strips are small — it is that overlaps on a line form a chain, and chains cannot contain the cycles that make the two-dimensional question intractable.

flat-folding · Flat-foldability how deep into the pile the machine has to be allowed to reach before it reaches every stateone layer is the patient machine and the full depth is the machine that may choose3 equal stamps22 of a possible 3 · 12 states4 equal stamps33 of a possible 4 · 32 states5 equal stamps44 of a possible 5 · 100 states6 equal stamps55 of a possible 6 · 288 statescreases at .20 .55 .7022 of a possible 4 · 8 statescreases at .15 .40 .50 .8522 of a possible 5 · 16 statescreases at .13 .31 .62 .7844 of a possible 5 · 24 statescreases at .40 .50 .62 .7233 of a possible 5 · 12 statescreases at .08 .24 .28 .35 .7255 of a possible 6 · 48 statesthe even strips are the ones that need the most, and they are the ones the machine that takes everything does best on

The easiest strip needs the deepest reach

The patient machine and the machine that may choose are the two ends of one number: how many layers of the pile a machine is allowed to hold. At one it reaches four states whatever the strip; at the pile's full depth it reaches everything. In between it is a machine nobody has defined, and measuring where completeness arrives inverts these essays' own ordering — the evenly creased strip, which the machine that takes everything folds perfectly, needs the deepest reach of all, and one uneven strip is complete at two.

complexity · Simple foldability

Named alongside it

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

DecidabilityLayer orderingThe machine modelMap foldingOne-dimensional foldingReachabilitySimple foldabilityStackingStamp folding

All concepts