Who found it, and when

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.

The oldest book cuts the paper

The Hiden Senbazuru Orikata of 1797 is the earliest surviving book of recreational paper folding, and its famous connected cranes are made by slitting one sheet into a grid. The founding rule of the modern subject is younger than the tradition it claims to describe.

The name is not the date

Kawasaki's theorem is in Husimi's book ten years before Kawasaki's paper. Maekawa's is Justin's too. The mean gap between a result in this field and the name it is known by is twenty-two years, and it runs in one direction.

Fifty years in the wrong language

Margherita Beloch showed in 1936 that one fold solves a general cubic. The result was correct, published, and in a mathematics journal — and the subject that needed it did not find it until 1991. The cost of a paper nobody reads is measurable, and it is most of a century.

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.

The kindergarten was a geometry class

Froebel put paper folding into mass education in the 1830s, and did it as mathematics rather than as craft. His three categories — the folds of life, of beauty, and of knowledge — are the first systematic treatment of folding anybody wrote down, and the third one is a geometry syllabus.

A schoolteacher's theorem

Kazuo Haga folded a corner of a square to the midpoint of the far side and found exact thirds. The construction needs one fold, no measurement and no compass, the numbers that come out are exactly 3/8, 7/8 and 2/3, and it was found by a biology teacher looking for something to do with a classroom.

What a dashed line can say

Before the Yoshizawa–Randlett symbols a model could not be transmitted, and the subject was not cumulative. The basic notation says exactly one thing — fold this crease, this way, now — which is precisely a simple fold, and the share of flat foldings that simple folds reach collapses from 71% to 13% as a model grows.

Publishing the pattern instead of the sequence

A diagram sequence is one picture per step and a crease pattern is one picture. When designers began releasing patterns rather than diagrams, the cost of publishing a model fell by two orders of magnitude and the difficulty moved onto the reader — which is what made the complex era possible and what made most of it unfoldable.

Found before it was designed

Crush a thin cylinder and it falls into a diamond lattice. That pattern was published in aeronautics in 1951, twenty years before anybody designed with it — and what the buckling load chose was not only the creases but the mountain-and-valley assignment, which is the part a designer gets wrong.

The same vertex, found four times

A degree-four vertex with a three-to-one assignment turns up in a buckled cylinder, in a Miura fold, in a Resch tessellation and in a crumpled sheet. It is not a coincidence and it is not influence: the flat-folding conditions are restrictive enough that a small set of vertices is nearly all there is.

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.

The paper had to arrive first

A model with sixty-four layers at its thickest point, folded in ordinary copier paper, is six and a half millimetres of stack. The layer count a design can reach is fixed by the substrate, not by the folder — so the elaborate tradition is downstream of a manufacturing achievement with its own dates.

The star that was cut before it was proved

Fold a sheet into ten wedges, make one straight cut, and a regular five-pointed star falls out. The trick is at least two centuries old and the theorem that any straight-line drawing can be released by one cut is of 1998 — because the traditional method is not the theorem, and works only on shapes with the symmetry the folding imposes.

From a shell to a solar array

The Miura fold was published in 1970 and flew on a satellite in 1995. The gap is not ignorance — the pattern was known, understood and available the whole time — and the same twenty-five year lag appears between every folding result and the hardware that uses it.

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.

The half no notation records

Every notation this subject has invented writes down the crease pattern or the sequence of folds, and the crease pattern is the half that does not decide the folded object. The field's interchange format has a place for the other half and nothing fills it in — including the files published here, which carry every vertex, edge and letter of a Yoshimura and none of the three hundred bits that would say which of its layer orders the folded object is.

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.

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.

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.

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.

The file records no verdict

A crease pattern file records vertices, edges and letters. Every one of the square twist's two hundred and fifty-six admissible letterings makes a perfectly valid file, and two hundred and forty-eight of them describe an object that does not exist. The format has a field for the layer order — the one thing that would settle it — and nothing fills it in, so a file is a drawing rather than a claim, and the field exchanges them as though they were claims.

The first thing about layers

A folder is taught four conditions at a vertex, or is taught nothing at all, and neither one says anything about the layers — which is where most of what goes wrong actually goes wrong. There has never been a rule about layer order simple enough to teach, because the question is global and every answer to it was a search. A chain of panels whose arrows all point the same way is the first one that fits on a finger.

Half the recipe is decoration

Every account of the Miura fold gives its letters as two instructions: the rows go one way, and the columns change letter every time they cross a row. Enumerate all sixty-four repeating rules and the second instruction is the whole of the condition — all four ways of writing the rows appear among the sixteen that fold, in every combination. The first instruction has never constrained anything.

The tail was named somewhere else

The search for a mountain-valley labelling of a tessellation patch costs eighty-four steps at best and does not finish at all two runs in five, and the cure is to stop and start again rather than to wait. None of that was discovered here. The distribution was described in the study of satisfiability solvers in the nineteen-nineties, the restart arithmetic is older still, and what a crease pattern contributes is one more instance.

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.

A test imported without its hypothesis

The rule that a loop in a folded sheet's layer relations proves the pattern cannot fold arrives from the layer-ordering literature, where the sheet is a disc and the panels are finitely many. This collection took the rule and not the sentence that says which sheets it is about, then applied it for years to patterns whose whole interest is that they repeat.

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.

A theorem with an unstated hypothesis

Maekawa's and Kawasaki's conditions are quoted everywhere without saying which sheet they are about, and they do not need to be — they are conditions at a point and every point is the same. The two-colouring is quoted the same way and it is not a condition at a point, and the omission there is not harmless.

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.

A sheet has a size as well

The layer count a design reaches is fixed by how thin the paper is. What size the finished thing comes out at is fixed by how large the sheet is, through a factor the pattern decides: the folded footprint times the mean layer count is the area of the paper, so the linear shrink is the square root of the layer count and a sixty-four-layer model finished at a hand's width wants more than a metre of sheet.

Which ceiling is binding

Two constraints hold a design's layer count down and both are ceilings on the same number. The stack gets better as the paper thins; the grid gets better as the sheet grows, because piling layers needs divisions and a division cannot be finer than a folder can place it. They cross at a sheet size that rises as the paper thins — so on the papers a classical folder had, the substrate really is the limit, and only at tissue weights does the hand take over.

How many wedges the paper allows

A k-pointed folk star is folded into 2k equal wedges and cut once, so the scissors go through 2k thicknesses of the sheet. The geometry is indifferent to k and the paper is not: at a stack a pair of scissors will shear cleanly in one pass, ordinary copier paper takes a three-pointed star and nothing more, and the five-pointed one everybody knows needs washi or thinner.

Which cranes can stay joined

The 1797 book slits a square into a grid and leaves the cranes attached at the interior lattice points, each of which holds four of them at once. Of the sixteen ways to choose which of a three-by-three's four points to leave joined, exactly one leaves the piece in a single object — and it is the one that uses all four. The cutting is very nearly forced rather than chosen.

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.

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.

What the hindsight was worth

The best restart cutoff for the one tessellation search with a heavy tail was read off a hundred and twenty measured runs, which nobody running the search could have done in advance. The universal schedule needs no such knowledge, and on the same runs it costs 3,222 nodes in expectation against 512 for the cutoff chosen by looking — a factor of 6.3, which is close to the base-two logarithm of that cutoff, as the theory of the schedule says it should be. A larger unit brings the schedule within a few per cent of the hindsight, and choosing the unit is choosing the scale the schedule was meant not to need.

Nearly every cutting fails at one crane

Six by six connected cranes have twenty-five joins and thirty-three million ways to keep some of them, and an exhaustion over all of them takes a fifth of a second. Of the 33,412,811 that fail, 99.64 per cent fail at a single crane — one left holding none of the joins at its corners — which a maker can check by looking at each crane in turn. The arrangements that pass that check hold together less often as the grid grows: all of them at three by three, 54 per cent at six by six.

A recipe needs degree four

The Miura's letters are taught as a recipe of same-and-differ clauses, and the recipe is exact: the sixteen repeating rules that fold are precisely the rules two clauses allow. The Yoshimura's twenty-six cannot have such a recipe, because twenty-six is not a power of two. The waterbomb's thirty-two is a power of two and still has none — its clauses allow sixty-four rules and half of them fail. The difference is one vertex: at degree four the counting theorem leaves a parity, and at degree six it leaves 'not all alike', which no clause of that kind can say.

The border is where the cranes come apart

Counted one join at a time rather than one subset at a time, the slit grid of connected cranes runs to twelve by twelve, where there are 2¹²¹ ways to keep some of the joins. Among the cuttings that hold every crane by something, the share that also hold together keeps falling — 54.3 per cent at six by six, 35.1 at twelve — and from eight by eight on it falls by the same factor at every size. A constant factor is the signature of the border: at six by six, 99.5 per cent of the sound cuttings that come apart do so through a stray piece touching the outermost ring of joins.

Two sentences for the Yoshimura

No recipe made only of same-and-differ clauses picks out the Yoshimura pattern's twenty-six folding rules, because its vertices have six creases. A recipe allowed one other kind of sentence does, and it is short: an even number of the four zigzag classes are mountains, and no course carries the letter all four zigzags share. That is three clauses, one parity and a prohibition for each of the pattern's two kinds of vertex, and it allows exactly the twenty-six. The waterbomb tessellation, with four kinds of six-crease vertex, needs three parities and only two prohibitions, because its parities do half the prohibiting.

The prediction held at eight and broke at ten

A join in a slit grid of cranes merges at most four pieces, so n² cranes need at least ⌈(n² − 1)⁄3⌉ joins, and exhaustion found four and six by six meeting that floor in exactly one way. The guess was that eight by eight would too, with twenty-one joins. Counted a join at a time, it does — twenty-one, one way. Ten by ten does not: it needs thirty-four against a floor of thirty-three, and has 7,076 ways to spend them. The grids that meet the floor with no merge to spare are four and eight by eight among every size to fourteen, and sixteen by sixteen by construction, because a perfect tree of joins on a grid twice as wide is four perfect trees and one join in the middle. The even grids were never the pattern; the doublings are.

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.

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.

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.

A sheet is as large as two arms

A model's finished size is its sheet divided by the square root of its layer count, so a sixty-four-layer model at a hand's width wants more than a metre of paper. A hand-made sheet is formed on a mould somebody lifts out of a vat and shakes, and what that bounds is an area rather than a thickness: over the whole plausible range of mould weights and what arms can do repeatedly, the largest square sheet runs from about half a metre to about two. The demand and the bound are the same sizes, which is the one thing about them nobody has to know the constants to see.

The paper that will not hold a crease

Every constraint these essays have found improves as the paper gets thinner: the stack, the size, the layer count. A crease does not. A crease is a plastic hinge in the fibres at the fold, and a sheet one fibre thick has nothing through its thickness to hinge — so there is a floor under the thickness that no manufacturing skill moves, because the fibre diameter is a constant of the plant. The papers a tradition folds sit between one and a half fibres and four, and the two below that in this collection's own shelf are tissues, which are backed with foil before anybody creases them.

Eighty layers and the sheet decides the rest

Five of these essays each bound one thing and none of them bounds a design. Put together they close. The crease floor fixes the thinnest usable paper at about a fibre and a half; that paper's stack runs out at eighty layers; the largest sheet two arms can make falls away as the square of the finished size. The region under both is every model anybody can fold, and it has a ceiling at eighty layers and a corner at about a hand's width — above which the paper is no longer the limit and the vat is.

A failure teaches a schedule nothing

The universal restart schedule costs 6.3 times the cutoff chosen by hindsight on the one folding search with a heavy tail, and the obvious repair is a schedule that learns its scale from the attempts it has already made. It cannot. A failed attempt costs exactly its cutoff and reports only that the run needed more, so every rule that chooses the next cutoff from its own failures writes down the same list whatever happens — a fixed schedule in disguise. On the measured runs, doubling after every failure costs at least 3.6 times the hindsight, and growing by half at least 2.4. What does come near is information from outside the run: the universal schedule given the longest search on four other patches as its unit costs 1.7 times the hindsight. The field that supplied the schedule reached the same conclusion, and answered it by watching runs from the inside.

The field is empty where it would say nothing

The interchange format for crease patterns has a field for the layer order and nothing ever fills it in. Filling it in where the folded states can be listed — four of the eight printed patterns, and Miura patches to twelve panels — finds that the preliminary base and both twists have exactly one folded state, so every one of the field's signs follows from the crease pattern and the field would record nothing a reader could not compute. The fold-and-cut triangle has two states. The Miura is different: every patch with three or more columns has several — three, six and eleven on the three-by-two, three-by-three and four-by-three — so on the pattern that gets built the field carries information from six panels up, and the field's size had been measured as log₂ of the panels' orderings, which on the preliminary base is fifteen bits for an object that has zero.

One choice with eleven answers

A folded state was proposed as a short list of free choices — which way a flap lies, where a rim panel sits — with the layer-order field's signs following from them. Listed exhaustively on every Miura patch small enough, the choices are never independent: every sign that varies is tied to every other through a shared panel, so the states are one choice with many answers. And there are more answers than the record said. The overlap test had a blind spot a third of a panel wide, and with it corrected the three-by-three Miura has six folded states, not one, and the four-by-three eleven, not five.

The deepest point pays for the paper

A folded design uses its sheet according to its mean layer count and its paper according to its deepest point, and the ratio of the two is a property of the crease pattern. Measured on every printed pattern it runs from exactly one to nearly six — and it does not split tessellations from bases, as expected. It splits patterns whose every panel lies over every point from patterns that keep a footprint with structure in it. The ratio moves the corner of the substrate map by its square root, so the fold-and-cut triangle can reach the paper's eighty layers at 326 millimetres where the preliminary base must stop at 134.

No odd grid holds a perfect tree

A perfect tree of joins holds a slit grid of cranes with every join merging four pieces that were separate. Two joins side by side share two cranes and close a loop, so a perfect tree never has two — and so all its joins lie on one colour of the checkerboard the lattice points make. The four corner cranes each touch one lattice point, which must be a join; on a grid with an odd side those four points are two colours. So no odd grid holds a perfect tree, and seventeen by seventeen, the size the count was pointing at, is settled without counting. On one colour the search is small enough to finish: sixteen by sixteen has exactly one perfect tree, the doubled one, and twenty by twenty has none.

A perfect tree is the joins it leaves out

On the colour a perfect tree lives on, the lattice points and the cranes they share form a graph full of loops, and a perfect tree is that graph with some points cut out. Euler's count fixes how many: every cut removes three loops' worth, and the loops always number exactly three times the cuts a perfect tree needs — so a set of that many cuts, no two sharing a crane, leaves a perfect tree exactly when what remains is one piece. Searched that way, ten by ten fails in seven cases that fit on one page, fourteen in 761, and sixteen's one tree is four copies of eight's diamond of cuts with a cross of cuts along the middle.

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.

A resampled maximum can only fall

The joint date of paper folding's five lineages is 1838, and resampling the record twenty thousand times returns 1838 in two draws of three and an earlier year in the rest — never a later one. That is not firmness. It is the chance that a draw of five from five includes one named entry, and it is the reason a resampled interval cannot be put on a maximum at all. Read over all 105 pairs in the record, a joint date inherits the later half of its pair, not the better-attested one: the better in 46, the worse in 35, and a single surviving witness in 28.

A line in the columns, a binomial in the rows

A Miura strip two rows high was found to have 2c − 3 folded states for c columns, from three columns to seven, and the search that found it ran out at eight. Counted instead from the pairs of panels that actually overlap, the count runs to twenty columns and nine rows, and 2c − 3 turns out to be one row of a formula: a patch c columns by r rows has 1 + (c − 2)(B − 1) folded states, with B the central binomial coefficient of r + 1. The line in the columns has a reason that can be drawn. The binomial in the rows is measured and not explained, and it holds only while the folded columns two apart still overlap, which is the same as the pile being more than two deep.

A long run shows itself early

A restart schedule that learns only from failures was shown to be a fixed list in disguise, unable to beat the cutoff hindsight would choose. A run reports more than whether it has finished, though, and the search for a lettering of the rhombille patch already counts the one thing worth reading: how often its loop test has refused it. Read at node fifty, the count separates the runs. Of fifty runs refused eight times or more, one finishes within a hundred nodes; of seventy refused fewer times, twenty-two do. Restarting on that count costs 387 nodes in expectation against 483 for the best fixed cutoff — and chosen on half the runs and priced on the other half, 437 and 435 against 541 and 579, which on one half is cheaper than that half's own hindsight. The search literature's answer was right, and on a crease pattern it needs one number.

Two stacks shuffled in glued pairs

A Miura patch three columns wide and r rows high has a central binomial number of folded states — 2, 3, 6, 10, 20 and on to 252 at nine rows — and the count was measured without a reason. The reason is a shuffle. In every state the middle column lies at the bottom and the two end columns lie above it, interleaved like two halves of a deck. Some pairs of an end column's panels can never be parted, because the row crease joining them lies inside the other column's folded footprint, and the two columns take those welds in turn, one at every row crease. That leaves r + 1 unbreakable blocks, and the number of ways to interleave them is the binomial.

A trapped run is holding four letters

Thirty of the hundred and twenty searches for a lettering of the rhombille patch are refused sixteen times in their first fifty nodes, and the proposal was to find the crease that traps them. No crease can: turning every letter over turns a folded state upside down, so no single letter is ever wrong. What the thirty share is a relation among four creases, mountain, valley, mountain, valley, and the proof that it has no lettering takes seven nodes. The runs cannot find that proof, because they change their most recent decision whether or not it had anything to do with the refusal. Change instead the most recent decision the refusal names, an idea from the 1970s, and the tail is gone: all 120 finish, the longest in 462 nodes, at an average of 148 against 387 for the best restart policy, and no restart helps at all.

Half the waterbomb's rules have no stack

The recipes for the waterbomb and the Yoshimura were written for the repeating rules that pass the vertex conditions, and nobody had asked those rules for a folded state. Asked, sixteen of the waterbomb's thirty-two have none, and four of the Yoshimura's twenty-six. Four panels are enough to refuse a waterbomb rule, in five lines, by forcing two creases on one folded line to interleave. And the waterbomb's surviving rules turn out to be exactly what a recipe of its kind was proved unable to describe: five same-and-differ clauses and no prohibition. Counting the layers shortens the waterbomb's recipe and lengthens the Yoshimura's, whose refusals are not local at all.