Origami & Folding
what one sheet can do
Concept

Evidence — where it appears

What a measurement or a document actually establishes, as against what it is taken to show. Separating the two is most of what the history field on this site does, and it is why its claims carry tests.

Named by 5 essays across 4 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 what the observer is givenambiguities separated, of 71the outline alonewhat a silhouette carries0the outline and the colourwhat a photograph of duo paper carries, from both sides0the complete layer orderwhat taking the model apart carries69the middle row is what anybody can actually see, and it is the top row

Which side is showing

Two earlier rungs asked what a folded object records about the pattern that made it, first from its outline and then from its complete layer order. Neither observation is one anybody can make. A photograph of duo paper carries the outline, the thickness and the colour showing at every point — and over the whole census the colour separates nothing at all.

flat-folding · Inverse problem the bar is how many patterns the population holdseach one sampled forty times and then searched, to see whether the two methods ever disagreethe printed patterns80 never lettered by 40 draws · all 8 settled by search · worst 60 nodestwist tessellations70 never lettered by 40 draws · all 7 settled by search · worst 19 nodesquadrilateral meshes60 never lettered by 40 draws · all 6 settled by search · worst 6 nodesfold-and-cut patterns70 never lettered by 40 draws · all 7 settled by search · worst 14 nodesthey never do here — the patterns that separate them are not in any of these four

Four populations with nothing to separate

This collection keeps four standing populations of crease patterns to test its machinery against. Twenty-eight patterns, sampled forty times each for a lettering that agrees with itself and then searched for one — and on every single member the two methods return the same verdict in the same breath. The patterns that separate them are in none of the four, and the reason they are not is what the populations are for.

complexity · Typical instances each cell is one patch, searched to a verdictgreen: a lettering exists · magenta: none exists, by exhaustion0.150.250.350.50.70.91.11.3turn angle, in radianssquare2626262626262626elongated1515323231313232hexagonal1515394545464545triangular1515393939373737the number in a cell is the nodes the search visited; 6 of 32 patches have no lettering at all

A population nobody chose

Five crease patterns were measured over and over because somebody had drawn five. Ninety-six drawn from a stated grid of tiling, turn and pleat width say something the five could not: nine of them have no consistent lettering at all, and the phenomenon the collection had spent so long measuring belongs to the one tiling the grid leaves out.

complexity · Typical instances Paper is made in ChinaPaper reaches JapanPaper is made in EuropeFolded paper is used ceremonially in Japan400 yrPaper is folded for amusement in Japan980 yrThe thousand cranes897 yrThe pajarita is folded in Spain293 yrPaper folding is taught as geometryOne fold solves a cubicThe diamond pattern in a crushed cylinderThe conditions at a flat-foldable vertexThe dashed-and-dotted diagram notationThe Miura foldA five-pointed star from one straight cutAny straight-line drawing, from one straight cutyear of the source500100015002000the date generally giventhe oldest source that says somedian overrun 201.5 years

The cure was named first

A heavy-tailed search runtime, the arithmetic for cutting it off and restarting, and the reason restarts work at all were established in the study of search between 1993 and 1998. This collection imported all three, and inherited with them the phenomenon they answer — which is that randomising a search's choices is what makes the tail.

history · Rediscovery

Named alongside it

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

MeasurementSamplingSearchSearch costTypical instancesAttributionThe decision problemDistributionDocumentary recordDuo paperEnumerationGenericity

All concepts