The name is not the date

The two theorems this site runs on every crease pattern it draws are called Kawasaki’s and Maekawa’s. Both names are firmly attached, both appear that way in every textbook and every piece of software in the field, and both are, as dates, wrong by about a decade.

The pattern in that figure is the essay. It is not a list of grievances about credit; priority disputes are the least interesting thing that can be said about any of these people. It is that the name records when a result became visible, not when it became true, and visibility is a social event with its own causes.

The two conditions, and who has them #

Take the alternating-angle condition first, since it is the one every figure here depends on.

The statement is that at a flat-foldable interior vertex the sectors alternate and each alternating group sums to a straight angle. It is called Kawasaki’s theorem, after a paper of Toshikazu Kawasaki’s from 1989. It appears in Koji Husimi’s book on the geometry of origami about a decade earlier, and Jacques Justin published the local conditions in 1986 in a form that contains it. Three people, three routes, a decade apart, and one name.

The mountain-and-valley count is the same shape. Maekawa’s theorem says the two letters differ by exactly two, Jun Maekawa is credited, Husimi’s book has it, and Justin has it independently. In parts of the literature it is Justin’s theorem and in parts the Maekawa–Justin theorem, which is at least honest about the difficulty.

Why a field with no journal does this #

The reason is structural rather than personal, and it explains why this subject is worse for it than most.

Until the late 1980s there was no venue. There was no journal of origami mathematics, no conference, no reviewing community, and no index. Results were published in origami society newsletters, in books aimed at folders, in languages that the other half of the field did not read, and in personal correspondence. A proof written down in Tokyo in 1979 and a proof written down in Paris in 1986 had no mechanism by which either author could learn of the other.

So independent rediscovery was not an accident in this field. It was the normal mode of transmission, and the surprising thing is not that the same conditions were found three times but that they were found only three times.

What a name is actually recording #

Once that is clear, the interpretation of a name changes.

A name is attached at the moment a result enters general circulation — when it appears somewhere that the people who need it will find it, in a language they read, in a form they can cite. That is a real event and it is worth marking. It is simply not the same event as the proof.

The two get conflated because a name looks like a citation and a citation looks like a date. Nothing in the phrase “Kawasaki’s theorem” says this is when the subject learned it, and everybody reads it that way anyway, including people who would never make the equivalent mistake about Pythagoras.

What Husimi’s book actually is #

It is worth pausing on the source that keeps appearing, because treating it as a citation flattens something interesting.

Origami no kikagaku is a book about the geometry of paper folding written for people who fold paper. It is not a research paper, it makes no claims of priority, it has no theorem environments, and it was not indexed anywhere that a mathematician searching for results about flat-foldability would have looked in 1985. What it has is the mathematics — stated, argued, and correct.

That is a genuinely awkward object for a chronology. A result stated correctly in a hobby book is known in one sense and unknown in another, and which sense matters depends on the question. For who understood this first, the book settles it. For when could a researcher have built on it, the book is nearly invisible, because building on a result requires finding it and citing it and being able to assume a reader has seen it.

The subject’s two answers to “when was this known” are both defensible and they differ by ten years. That is not a defect in the record; it is a real ambiguity about what knowing means in a field without a literature.

The same shape outside this subject #

The pattern is not peculiar to folding, which is worth saying because it stops the essay from reading as a complaint about one small field.

Stigler’s law — that no scientific discovery is named after its original discoverer — is stated as a joke and is very close to a description. The mechanism proposed for it is usually sociological: names attach to whoever brought a result to the community that needed it, and that is rarely the first person to have it. Folding gives an unusually clean example because the field is small, the results are few and sharply stated, and the whole documented history fits on one chart.

What folding adds is the magnitude. In a subject with a three-hundred-year literature a twenty-year attribution lag is a rounding error. In one whose mathematical literature is sixty years old it is a third of the record, and it systematically shifts the apparent centre of gravity of the field forward into the period when the naming happened.

The one that is not close #

Two of the six gaps are of a different order, and they are worth separating from the rest.

The result that one fold solves a general cubic is attached to Huzita and to 1991. Margherita Piazzolla Beloch proved it in 1936, in Italian, in a mathematics journal, fifty-five years earlier. That is not a decade of parallel work in an unindexed field; it is a published result in a findable venue that the subject failed to find. It has an essay of its own because the failure mode is different and worth isolating.

The Miura fold is the other. It is dated to 1995 — the year a satellite flew a solar array folded that way — and Koryo Miura described the pattern in 1970, in work about the buckling of shells that was not about satellites at all. That gap is twenty-five years and it is a gap between the mathematics and the application, which is a third distinct mechanism.

Three mechanisms, not one #

So the single line in the first figure is hiding three different things, and untangling them is the useful part.

The first is parallel discovery in an unindexed field: Husimi, Justin, Kawasaki, Maekawa. Nobody failed at anything. The gap is the time it took for a community to form around results that already existed in several places.

The second is a result in the wrong place: Beloch. The venue was fine, the mathematics was fine, and the audience that needed it was not reading that journal. The gap is a failure of search.

The third is the application arriving later than the idea: Miura. The pattern was published, understood, and available; what took twenty-five years was somebody needing to fold a solar array. The gap is not about knowledge at all.

Naming and the shape of a syllabus #

There is a downstream effect that matters more than the bookkeeping, and it shows up in how the subject is taught.

A course in the mathematics of folding, assembled from the standard names, produces a chronology that starts around 1986 with Justin, gathers pace with Kawasaki in 1989 and Huzita in 1991, becomes a real field with Bern and Hayes in 1996 and Lang’s tree method in the same decade, and arrives at the present. That story is coherent, it is roughly what the literature looks like from the inside, and it has a whole prehistory missing from the front of it.

Put the proof dates back and the shape changes. Beloch is at 1936, contemporary with the classical algebra she was applying. Yoshimura is at 1951, in aeronautics. Husimi is at 1979 and Miura at 1970, both in Japanese, neither in a mathematics venue. What emerges is not a field that began in the late 1980s but one that existed in several disconnected places for fifty years and acquired a literature in the late 1980s.

Those are different subjects to teach, and the second one is the true one.

What this does to a citation #

The practical consequence for anybody writing about this subject is a rule that is easy to state and irritating to follow: cite the result and the name separately.

Saying “Kawasaki’s theorem (1989)” makes a claim about 1989 that is false. Saying “the alternating-angle condition, which appears in Husimi (1979) and is generally called Kawasaki’s theorem” is longer, correct, and preserves the useful thing the name does, which is to tell a reader which theorem is meant.

This site does the second, everywhere, and it is more effort than it looks.

The cost of getting it wrong is not zero #

It would be possible to read all of this as harmless. Names are labels, everybody knows a label is not a date, and the mathematics is unaffected by who is credited.

The mathematics is unaffected. Two other things are not.

The first is search. Somebody looking for prior work on a folding problem searches the names they know, in the language they read, in the venues they have access to. A field whose apparent history begins in 1986 is a field where nobody searches 1936, and the fifty-five-year gap in the axiom result is precisely what that costs. The attribution problem and the rediscovery problem are the same problem seen twice.

The second is the sense of what the subject is. A discipline that believes itself thirty years old behaves differently from one that believes itself ninety years old: it is more willing to think a question is open, less willing to think an obvious approach has been tried, and much more likely to reprove things. Every one of those is visible in this field.

What the figure asserts, and what it cannot #

The table is checked in one respect and only one. Every row carries a proof year and an eponym year, and the generator refuses to draw if any eponym precedes its proof — which would mean a name arriving before the thing it names, and would mean one of the two dates was entered wrong.

That check is real and it is narrow. It cannot tell whether Husimi’s 1979 is right; it cannot tell whether some earlier manuscript exists that would move a bar; it cannot weigh a proof of a special case against a proof of the general one, which is the substance of most genuine priority questions. What it catches is a typo and an inversion, which is worth catching and is not the same as verifying a history.

The direction, again #

The gaps in the first figure all have the same sign, and a table in which every error ran one way would ordinarily be a table describing its compiler.

Here the sign is forced, and that is worth stating plainly rather than treating as a finding. A name cannot precede the result it names — the generator asserts exactly this and would refuse the figure otherwise — so every bar points forward by construction. The content is not the direction but the magnitude: a mean of twenty and a half years, in a subject whose entire mathematical literature is about sixty years old, means roughly a third of the field’s history is inside the gap between when things were known and when they were named.

That is the number worth carrying away. Not that the names are wrong, but that the subject’s felt chronology is compressed into the last three decades by an artefact of how it labels things.

The mean is in a place where nothing sits #

That mean is worth handling carefully, because it is a summary of a distribution that has no middle.

The six gaps are 7, 8, 10, 18, 25 and 55 years. Three of them sit between seven and ten; the other three sit between eighteen and fifty-five. Nothing sits between ten and eighteen, and the mean of twenty and a half falls in that empty region — which is the standard signature of a mixture of two populations rather than one spread.

So the single figure is describing a subject none of these results belongs to. Quoting it invites the reader to picture a typical twenty-year lag, and there is no typical case: there is a decade and there are decades.

Which separates the mechanisms by their size #

Split the six along that gap and the three mechanisms the essay names sort themselves without being told to.

The three vertex conditions — 7, 8 and 10 years — span three years between them. Those are the results found and refound inside one small community, and the lag is how long it took that community to acquire a venue. Three independent results with essentially the same gap is what one mechanism operating on one population looks like.

The three others — 18, 25 and 55 — span thirty-seven years. Every one of them had to cross a boundary: a language and a discipline for Beloch, aeronautics into folding for Yoshimura, mathematics into hardware for Miura. Crossings are slow and they are slow by wildly different amounts, because what is being waited for differs each time.

So the magnitude alone identifies the mechanism. A gap under about a decade is a community forming around results it already had. A gap over about fifteen years is a result crossing a boundary. The mean of the first group is 8.3 years and of the second 32.7 — a factor of four, and the tightness of the first is what makes the reading more than a partition drawn after the fact.

That also says which number to quote. For the mathematics of a vertex, the subject’s felt chronology is late by about eight years, which is annoying and not structural. For anything that arrived from outside, it is late by decades, and there the lateness is the story rather than a correction to it.

The idealisation, named #

The assumption underneath the whole table is that “the earliest proof anybody can point at” is a well-defined object. It is not, quite.

Some of these results appear in a form that a modern reader recognises immediately and that the original author did not state as a theorem. Some appear as an observation inside a longer argument about something else. Husimi’s book is a book for folders with mathematics in it, not a paper making claims of priority, and reading it as the latter is a modern imposition.

So each proof date has a judgement inside it, made once, by hand, and not re-checked on any build. That is the standing weakness of this whole field and it applies here as much as anywhere.

A small defence of the names #

None of this is an argument for renaming anything, and the case against renaming is stronger than it first appears.

A name’s job is to identify a theorem uniquely and briefly among people who already know several theorems. “Kawasaki’s theorem” does that perfectly. “The alternating-angle condition at a flat-foldable interior vertex, Husimi 1979” does it too, at four times the length, and any attempt to redistribute credit by renaming produces hyphenated compounds that nobody says out loud and that go stale the moment somebody finds an earlier source.

The fix is not in the names. It is in refusing to read a date out of one — which costs a clause, and which this site pays on every mention.

Where this goes next #

The ladder continues into the two anomalous gaps. Fifty years in the wrong language takes Beloch’s paper and asks what a published, correct, findable result costs when nobody finds it. Found before it was designed takes the Miura gap and shows that the pattern was produced by a buckling shell before any person chose it — including its mountain-and-valley assignment, which is the part a designer gets wrong.

And a record is not a proof closes the field by admitting what none of this can do.