Origami & Folding
what one sheet can do
Concept

Distribution — where it appears

The whole spread of values a repeated measurement produces, rather than any summary of it. A search run from many seeds has one; a search with no seed in it does not, which is why a distribution has to be shown to belong to the problem before a strategy is built on it.

Named by 5 essays across 3 fields — each of them below, with the objects they name alongside it.

the curve is stop-and-restart; the rule is a constant letter order1001e+31e+41001e+31e+4563 at a cutoff of 10080 nodes, deterministic, nothing to restartexpected nodes in totalcutoff, in nodes

Restarting what cannot be restarted

Stopping a search early and starting it again with a fresh seed costs five hundred and twelve steps in expectation against sixteen thousand for patience. Every number in that is right. The distribution it is right about was made by the search's own coin, and taking the coin out costs eighty — with nothing left to reseed.

complexity · Hardness of folding 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 the angles that admit a thirdφ₁ − φ₂ + φ₃ a multiple of a straight angle30°30°60°60°90°90°120°120°150°150°60°, 120°the first crease's angle, against the secondevery other pair of angles folds nothing,at any length and any positions

How rare a band that folds is

Almost every crease pattern fails to fold flat, and the usual way of saying so is a count over discrete choices. A glued band fails for a reason that no count can reach: its crease angles have to satisfy an equation, and a set defined by an equation has no volume in the space it sits in.

flat-folding · Genericity the grid, 2×2 cellsone drawing, four sheetscutoutgluedacrossgluedalonggluedboth waysvertices4444free letters1210108panels9664V − E + F1000the vertex row is the control: identifying edges can neither make nor destroy a vertexand Euler's number is the cheapest check that the gluing did what it says

One population, four sheets

A population of patterns is a way of asking what is typical, and it has always been a population of drawings. Put the same drawings on four different sheets and the verdicts move — not because the drawings changed but because the sheet did, which means a population has two halves and only one of them was ever chosen.

complexity · Typical instances the share of each folded footprint at every depth, shallowest firsteach block is one depth, its length the share of the footprint that deep; the shading cycles so neighbours differThe preliminary base8 deep on 99.8%The Miura fold16 deep on 11.7%The square twist9 deep on 17.4%The hexagon twist7 deep on 24.4%The Yoshimura pattern60 deep on 100.0%Fold and cut — the triangle7 deep on 2.91%The tapered corrugation16 deep on 0.78%The waterbomb tessellation32 deep on 98.1%the tick marks where the deepest depth begins

A proud hundredth buys almost nothing

A folded design's sheet is set by its mean layer count and its paper by its deepest point, and the ratio of the two moved the size at which a design uses both equally. That ratio is two numbers of a distribution. Read the whole distribution of every printed pattern's folded footprint and the shelf splits in two: six patterns reach their deepest depth over at least a tenth of the footprint, and two reach it only on a sliver — the fold-and-cut triangle on 2.9 per cent, the tapered corrugation on 0.78. A bound that lets part of the footprint stand proud of the stack's ceiling does nothing at all until the proud share passes the deepest depth's share, so a hundredth moves one pattern of eight, and a twentieth moves two.

history · Paper as substrate

Named alongside it

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

Search costGluingRandomised algorithmSearchTypical instanceAttributionBoundaryClosureCountingDeterminismDocumentary recordEvidence

All concepts