Origami & Folding
what one sheet can do
Concept

Documentary record — where it appears

What survives in writing about a subject, as against what happened in it. It selects for what somebody had a reason to write down, which is why invented things are dated and traditional ones are not.

Named by 16 essays across one field — each of them below, with the objects they name alongside it.

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

Nothing here is as old as it sounds

Paper folding is described everywhere as an ancient art. The oldest surviving book of it was printed in 1797, the oldest reference to folding for amusement is from 1680, and the median claim in this subject is dated two centuries before anything that attests it.

history · Attribution 15 claims · 5 resting on one sourceartefacta surviving folded object, or a picture of one made at the timePaper is made in Europe1056The pajarita is folded in Spain1793manuscripta hand-written document that survivesFolded paper is used ceremonially in Japan1600Paper reaches Japan720one sourceprinteda printed book or paper with a publication datePaper folding is taught as geometry1838The conditions at a flat-foldable vertex1979The Miura fold1970The diamond pattern in a crushed cylinder1951The dashed-and-dotted diagram notation1954Any straight-line drawing, from one straight cut1998Paper is folded for amusement in Japan1680one sourceThe thousand cranes1797one sourceOne fold solves a cubic1936one sourcesecondarysomebody later reporting it, with no surviving primary sourcePaper is made in China105A five-pointed star from one straight cut1873one sourcea source is dated; it is not thereby rightthis ranks what a source can bear, not what it says

A record is not a proof

Every other claim here can be re-derived from the figure that makes it, and a wrong one shows. A date cannot: it is checked once, by hand, against a record that is itself a survivor. This field is the one most likely to be wrong and least likely to be caught, and saying so is the only defence it has.

history · Attribution Folded 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 geometryyear of the source500100015002000the date generally giventhe oldest source that says somedian overrun 400 years

Two traditions and a merge

Ceremonial wrapping, recreational folding and the kindergarten syllabus are three separate lineages with three separate purposes, and they were independent until the late nineteenth century. Told as one continuous tradition, the oldest date in any of them becomes the age of all three.

history · Two traditions patternclassdatedfoldingThe preliminary basetraditional—724 mmThe Miura foldpublished as mathematics19701049 mmThe square twistgenerated here—704 mmThe hexagon twistgenerated here—916 mmThe Yoshimura patternpublished as mathematics19552380 mmFold and cut — the trianglegenerated here—258 mmThe tapered corrugationgenerated here—1057 mmThe waterbomb tessellationtraditional—2290 mm2 dated, all of them published; 6 undated, none of them ownedthe fourth class — a designer's model — is what this shelf holds none of

The patterns nobody owns

This site prints crease patterns at true scale and prints no designer's work, and that has always been stated as a rule applied at the end. Read the printed shelf as a documentary record instead and the rule turns out to be a property of the record: every pattern that carries a date was published as mathematics, every undated one belongs to nobody, and the two silences are one silence.

history · Attribution each mark is one claim: right of the line means dated earlier than its evidencethe vertical line is agreement between the claim and the record0paper itselfwhen and where the material was made, which is archaeologymedian 0 · spread 204a thing people dida fold, a ceremony, a toy, a lesson — something with no first daymedian 347 · spread 979a thing somebody proved or designeda statement with a paper, a date and an authormedian -14 · spread 131

Two kinds of claim

This site has published a median overrun of two centuries and a mean of twenty-two years in the opposite direction, and both are right. Split the record by what each claim is about and the reason appears: every claim about a practice is dated earlier than its evidence, most claims about a result are dated later, and the two scatter by 979 years and 131.

history · Attribution the bar is the letterings with a folded statethe row is how many times the letter changes going round the central polygonthe ring reads as one letter032 pass every vertex · 28 have no order2 changes round the ring8192 pass every vertex · 184 have no order4 changes round the ring032 pass every vertex · 32 have no ordera twist looks like a twist when the ring reads as one letter, which is why this was never checked

Taught with a wrong reason

Four mountains and four valleys is what the preliminary base's symmetry suggests and Maekawa forbids it; a twist looks like a twist when its central ring reads as one letter, and no such lettering folds; a tessellation is verified because its unit is, and a forty-nine-panel patch of one had no folded state at all. In each case the conclusion taught is right and the reason offered for it is not, and the site that repeats them is this one.

history · Pedagogy 34560%20%40%60%80%100%creases in the stripreachable by simple folds72%30%17%13%the basic symbolsa dashed line — valleya dotted line — mountainan arrow — fold it nowand what they missreverse, squash, sink,petal — every one of thema move no dashed linecan ask forevery assignment of 68 seeded spacings

A file has no paper

The field's interchange format is three arrays — where the vertices are, which pairs of them an edge joins, and a letter for each edge — and that is exactly the object every computation on a crease pattern starts from. A list of edges cannot say that two of them must not cross, because crossing is a property of the drawing and the list has no drawing in it. So a pattern that no paper could carry is a perfectly well-formed file, and four of this collection's own were.

history · Notation the vertex nobody listedthe two lines meet at 22.9°sectors 157.1° 22.9° 157.1° 22.9°alternating sums 314.2° and 45.8°Kawasaki fails — it holds only at a right angle2 mountain and 2 valleyMaekawa fails — a crossing can only be 4–0, 2–2 or 0–4mountainvalleyraw edge

The reader decides the junction

Five of the eight patterns printed here have places where one crease ends on another — four on the preliminary base, eight on the square twist, twelve on the hexagon twist, nineteen on the Yoshimura, three on the fold-and-cut triangle. At each of them a reader has to decide whether two lines meet or pass through one another, and no notation, caption or teaching text in the subject mentions that the decision is being made.

history · Pedagogy 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 3 creases on a Möbius bandthe panels take two coloursseamthe same seam123the right edge onto the left, turned over3 creases, 3 panelsinterior vertices: 0two-coloursthe reflections closeand turn the paper the right waymountainvalleyraw edge

Found by people not folding paper

The shortest strip that makes a Möbius band has a literature, and it is in differential geometry rather than in origami. The two subjects have the same number, they reached it by completely different routes, and neither of them cites the other — which is the fourth time this collection has found that shape.

history · Rediscovery 34560%20%40%60%80%100%creases in the stripreachable by simple folds72%30%17%13%the basic symbolsa dashed line — valleya dotted line — mountainan arrow — fold it nowand what they missreverse, squash, sink,petal — every one of thema move no dashed linecan ask forevery assignment of 68 seeded spacings

No format has a gluing

A crease pattern file records vertices, edges, assignments, faces and layer orders. Every one of those is a feature of the paper's interior, and the boundary appears only as a kind of edge — so there is nowhere in the scheme to say that two boundary edges are the same edge, and the sheet a pattern is on cannot be written down.

history · Notation the year the record can put two lineages in the same worldthe later of two surviving sources, which is what a joint claim rests onFolded paper is used ceremoniawith paper is folded for amusement 1680Folded paper is used ceremoniawith the pajarita is folded in spai1793Paper is folded for amusement with the pajarita is folded in spai1793Folded paper is used ceremoniawith the thousand cranes1797Paper is folded for amusement with the thousand cranes1797The thousand craneswith the pajarita is folded in spai1797Folded paper is used ceremoniawith paper folding is taught as geo1838Paper is folded for amusement with paper folding is taught as geo1838The thousand craneswith paper folding is taught as geo1838The pajarita is folded in Spaiwith paper folding is taught as geo1838all 5 are jointly attested only from 1838 — every earlier joint claim is a claim about at most 4 of themeach row is the later of two surviving sources · Paper folding is taught as geometry at 1838 is what the whole set waits for

When two of them are first attested together

Every single date in this field is argued about by centuries, and a pairwise one is not. The earliest year at which two claims are both attested is simply the later of their two surviving sources, so it inherits the better-attested half of each pair rather than the worse — and the year at which all five lineages of practice are jointly on the record is 1838, one year from where the same claims are popularly dated together.

history · Two traditions how many witnesses each claim has5 of 15 have one, and a loss removes them from the record rather than weakening themclaimsurviving sourcesafter one lossPaper is made in China21 leftPaper reaches Japan1nothing attests itPaper is made in Europe32 leftFolded paper is used ceremonially in Japan43 leftPaper is folded for amusement in Japan1nothing attests itThe thousand cranes1nothing attests itThe pajarita is folded in Spain21 leftPaper folding is taught as geometry43 leftOne fold solves a cubic1nothing attests itThe diamond pattern in a crushed cylinder21 leftThe conditions at a flat-foldable vertex32 leftThe dashed-and-dotted diagram notation21 leftThe Miura fold32 leftA five-pointed star from one straight cut1nothing attests itAny straight-line drawing, from one straight cut21 leftby kind of source: artefact 2 · manuscript 2 (1 single) · printed 9 (3 single) · secondary 2 (1 single)the average overrun falls from 357 years to 193 without them, and the median from 201.5 to 184.5 — the effect is two rows, not a tendencywitnesses counted from the record itself · Paper is folded for amusement in Japan and The thousand cranes are the two largest overruns and have one document each

One lost source and the story changes

Five of the record's fifteen claims rest on exactly one surviving document. Take those away and the field's most-quoted statistic — how far ahead of its evidence a popular date runs — falls from three hundred and fifty-seven years to a hundred and ninety-three, while the median hardly moves at all. The effect is not a tendency spread through the record; it is two documents, and both of them are single-witness.

history · Two traditions 200400600800100002004006008001000120014001600mean years ahead of the evidenceresamplingsquoted: 3575%: 15095%: 591spread 13320,000 resamplings of the fifteen entries · five per cent of them fall below 150 and five per cent above 591

The interval is wider than the number

The field's most-quoted statistic — how far ahead of its evidence a popular date runs — is the mean of eight positive gaps, and it is quoted as three hundred and fifty-seven years. Resampling the fifteen entries puts ninety per cent of its weight between a hundred and fifty and five hundred and ninety-one. The interval is wider than the number, the middle gap's interval is a fifth as wide, and the difference is the same two documents the leave-one-out found.

history · Two traditions how many independent sources survive for each claimthe record states this per entry and nothing has ever read the column as a distribution1 source5paper-japan, recreational-folding, senbazuru, beloch, one-cut-star2 sources5paper-china, pajarita, yoshimura, yoshizawa-notation, fold-and-cut3 sources3paper-europe, vertex-conditions, miura-ori4 sources2ceremonial-wrapping, froebela claim resting on one source is one document away from disappearing; five of the fifteen are in that position

A question the record is too small to answer

Five of the fifteen claims rest on one surviving source and five on two, and those two counts are exactly what an estimate of the claims that left no source at all is made of. Applied, it says two and a half are missing. Its ninety-five per cent interval runs from fifteen to thirty-one, re-reading a single entry's source count moves it by a fifth, and its independence assumption is false in the one way documents actually fail — which is what makes computing it worth more than declining to.

history · Two traditions what one document would dono date is invented here — each row asks what the arithmetic would say if one turned upclaimdatedevidenceits gapif a source 200 years earlier turned upceremonial-wrapping12001600400357 → 332 (-25)recreational-folding7001680980357 → 332 (-25)senbazuru9001797897357 → 332 (-25)pajarita15001793293357 → 332 (-25)paper-china1051050nothing changespaper-europe11501056-94nothing changesbeloch19911936-55nothing changesyoshimura19691951-18nothing changesvertex-conditions19891979-10nothing changesyoshizawa-notation19611954-7nothing changesmiura-ori19951970-25nothing changespaper-japan610720110357 → 392 (+35)one-cut-star1776187397357 → 394 (+37)fold-and-cut1922199876357 → 397 (+40)froebel183718381357 → 408 (+51)the statistic is 357 as the record stands; a positive number in the last column is a discovery that makes the field look worse

Some discoveries would make it worse

The field's headline statistic averages the claims dated ahead of their evidence. So a document found for a claim that was nearly right removes a small number from a mean of large ones and the average overrun goes up: finding a source two centuries earlier for the kindergarten entry would take the figure from 357 to 408. The four claims where a discovery helps can take twenty-five years off each, and no single document at any date can bring the number to 250.

history · Two traditions

Named alongside it

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

AttestationPrimary sourceAttributionCrease patternIndependent discoveryExpected-valueIdentifiabilityRediscoveryCrossingFroebelGluingHiden senbazuru orikata

All concepts