Origami & Folding
what one sheet can do
Concept

Verification — where it appears

Checking that a computed answer is the answer, by machinery that had no part in producing it. Every check in this collection compares a computation against itself — a count against a closed form, a folded state against a theorem — which is why a claim about the world needs a different kind of evidence.

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

computed herenot established herethe curvature a growth field forcesthe growth solverthat any leaf grows that waythat a corrugation with tapered columns folds flatthe pattern librarythat a leaf's creases are where these arethe packed fraction of four folding geometrieswing-packingthe packed fraction of any wingthat one freedom needs one driverdof-censushow an insect actually deploys a wingthe surface a fold count fits in a boxsurface-in-volumethe surface area of any organwhich lattice shapes admit a single routethe routing modulethat any published design used this routeevery row's left side is a computation in this repository; every right side is a claim it does not make

The organism is not the model

Every figure in this field draws a fold this repository computed. Not one of them measures a leaf, a wing or a gut. That is the rule the field was built to, and it is worth stating as a table rather than as a preamble — because a field about living things is where a computed geometry is most likely to be read as an observation.

biology · Idealisation the same 2×2 glued cell, searched under two rulesa cycle is a contradictiona cycle whose steps add to zero isand what the loops dothe square gridnothing, in 359 nodesevery loop travels (2 directions)the triangular gridnothing, in 12,143455 nodesevery loop travels (2 directions)the honeycombnothing, in 9,6191,043 nodesevery loop travels (3 directions)the elongated triangular tilingnothing, in 9,123162 nodesevery loop travels (5 directions)the rhombille tilingunfinished at 200,000unfinished at 200,000—“nothing, in n” is an exhausted search: a proof that the pattern has no consistent lettering, which is false

The cost of asking the wrong sheet

A test written for a sheet with an edge, run on a sheet without one, does not fail. It exhausts — proving, at three, thirty-five and three thousand four hundred and fifty-five nodes, that no lettering exists — and the letterings it proved impossible fold, on the collection's own machinery, at every size they were tried at.

complexity · Hardness of folding

Named alongside it

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

BoundaryDecision procedureEvidenceExhaustive searchGluingIdealisationLocalityModel limitsPatchScopeSearch cost

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