The patterns nobody owns

Nothing here is as old as it sounds read the documentary record of paper folding as a table: what each claim is popularly dated to, and what the oldest surviving source for it actually says. A record is not a proof said what such a table can and cannot establish.

This is the same reading applied to this site’s own shelf — the eight crease patterns it prints at true scale, with a millimetre sheet each — and it produces a coincidence that is not one.

Three classes, read off rather than assigned #

Each pattern falls into exactly one class, and the classification is read from the provenance line the pattern already carries rather than from a second list kept beside it.

Traditional. A base with no attributable author — and the oldest book that shows one cuts the paper, which is a reminder of how little the record holds, arrived at repeatedly and recorded by nobody as an invention. The preliminary base and the waterbomb tessellation.

Published as mathematics. A pattern that entered the literature as a result rather than as a model, with a name and a date attached. The Miura fold, 1970, published as engineering; the Yoshimura pattern, 1955, published as a buckling mode.

Generated here. Produced here by code, from a stated rule, so its author is a construction. The square twist and the hexagon twist, both derived from Kawasaki’s condition; the fold-and-cut triangle, from a straight skeleton; the tapered corrugation.

Four generated, two traditional, two published. Nine thousand three hundred and seventy-nine millimetres of folding between them, of which the two published patterns hold three thousand four hundred — the Yoshimura alone is 2,380 mm on a sheet 170 mm across.

The class that is missing #

There is a fourth class and this shelf holds none of it: a designer’s model. A crease pattern with a named, living or recent author, published as a design.

The name is not the date makes the parallel point about theorems: a result’s eponym and its earliest proof are different dates, and the gap between them is what a record does rather than what happened. The site’s stated reason has always been the constraint: the folding community treats a designer’s crease pattern as theirs, and this site does not reproduce one. That reads as a rule applied at the end — choose patterns, then remove the ones that belong to somebody.

Read the record instead and the rule is redundant. A designed pattern is one whose origin the record knows, because it was published as an invention by somebody who wanted the credit. A traditional pattern is one whose origin the record does not know, because nobody claimed it. Attribution and ownership are the same fact seen twice.

What the dates do #

The strongest form of the observation is in the dates, and it needs no interpretation at all.

Every pattern here that carries a date was published as mathematics. 1955 and 1970. Every undated one is traditional or generated. Not one traditional pattern carries a date.

That is not a coincidence about these eight. Publishing is what a date is: a date is the moment a thing entered a record, and a pattern that entered no record has no date to carry. A traditional base was not invented on a Tuesday; it accumulated, and accumulation has no timestamp.

So the shelf divides on ownership and on datedness in exactly the same place, and it would divide there whatever eight patterns were chosen.

The generated class is the interesting one #

Half the shelf was produced here, and that class is worth separating from the other two because it is the one the record has never had before.

A generated pattern’s provenance is a program. The square twist is not a traditional object that this site happens to draw; it is the output of a construction that derives a twist unit from the two vertex conditions, and its author is that construction. Anybody with the rule gets the same pattern.

That gives it a property neither of the other classes has: it is reproducible from its own description, exactly. A traditional pattern is reproducible because it is simple and widely known; a published one because a paper describes it; a generated one because the rule is the pattern.

It also makes attribution unusually clean. The question “who made this” has an answer that is a paragraph of code rather than a person, which is a kind of authorship the folding record has no vocabulary for and which this site is going to keep producing.

Nine thousand millimetres in two patterns #

The shelf’s shape in folding rather than in counts is worth setting down, because it is lopsided in a way the class counts hide.

Half the shelf is generated and generated patterns are 2,935 millimetres of the 9,379 — 31.3 per cent. The two traditional patterns are 3,014 and 32.1 per cent. The two published are 3,429 and 36.6 per cent.

The three classes are almost equal, which is not what the class counts would predict: four generated patterns carry the same folding as two traditional ones and very nearly the same as two published ones. A class of four and a class of two hold the same amount of work, so the average published or traditional pattern is about twice the average generated one.

That is a fact about what the record preserves rather than about what folding is. A pattern published as mathematics or handed down as a base is a pattern somebody wanted for its own sake, and such patterns are whole objects — a tessellation, a buckling mode, a base. A pattern generated from a rule is whatever the rule was asked for, and this site asked for small ones.

The two largest are one of each #

The class-level claim is worth pressing on, because the two patterns carrying most of the folding do not belong to the same class.

The Yoshimura is 2,380 millimetres of the shelf’s 9,379 and the waterbomb is 2,290 — twenty-five and twenty-four per cent, and between them half the shelf. One is published as mathematics, in 1955, as a buckling mode of a cylinder. The other is traditional and belongs to nobody. The two largest objects on the shelf are indistinguishable in size and are in different classes, which is as clean a refusal of a size-and-class correlation as eight patterns can give.

Normalising by sheet size says the same thing again. In sheet-widths of folding — crease length over the pattern’s own printed side — the shelf runs: fold-and-cut triangle 1.7, square twist 4.7, preliminary base 4.8, hexagon twist 6.1, Miura 6.2, tapered corrugation 6.6, Yoshimura 14.0, waterbomb tessellation 14.3.

Read that column and the classes do not separate at all. The two at the top are one published and one traditional; the four in the middle are two generated, one published and one traditional; the one at the bottom is generated. There is no ordering by class anywhere in it.

What separates is scale, and scale is not a class #

The one real structure in the column is a gap, and it is between the tessellations and everything else.

The waterbomb and the Yoshimura are both repeating patterns over the whole sheet, and both sit at fourteen sheet-widths. Everything else is a base, a twist or a corrugation with a few dozen creases, and everything else sits between 1.7 and 6.6. The gap between 6.6 and 14.0 has nothing in it.

So the shelf separates by what kind of object a pattern is and not at all by how it came to be recorded. A tessellation is large because it repeats; a base is small because it is one figure; and whether somebody published it, handed it down or generated it is independent of both.

That is a stronger version of this rung’s own claim rather than a qualification of it. The classification is about the record, the size is about the geometry, and the two turn out to be orthogonal on every measure available here — which is what one would want if the classification is to be read as a fact about documentation rather than about patterns.

It is worth saying because the size reading is the one place this rung reasons from the shelf to the record rather than the other way. The classification is mechanical and holds whatever the shelf contains; the independence of size and class is a finding about eight patterns, four of which this site made itself, and a shelf of twenty would say whether it holds.

Which theorem was checked, and how #

The classification is mechanical and the checks are about what it must refuse.

Every pattern’s class is read from the provenance the pattern already carries, not from a list maintained separately — so a pattern added later cannot quietly escape the census by being left off a list, and a provenance that matches none of the rules comes back unclassified rather than being filed under the nearest one. That is the check that keeps the census honest as the shelf grows.

A provenance with no text at all is refused. A pattern added without saying where it came from breaks the census rather than defaulting to the class with the fewest consequences.

The date is read out of the same string rather than kept in a second place that could disagree with it. And a traditional provenance must carry no date, which is the fact the whole rung turns on and is checked rather than observed.

The counts are then measured off the patterns themselves — crease length in millimetres at each pattern’s own stated sheet size, interior vertex counts, crease counts — so the shelf’s shape is described in the units a reader folding one would care about.

What the constraint costs, in what a reader can hold #

The shelf is what a reader can print, and it is worth saying what is not on it rather than leaving the absence abstract.

Every named base beyond the preliminary — the bird base, the frog base, the fish base — is traditional and could in principle be here; they are absent because nothing has needed them yet rather than because of the constraint. Every designed model of the last sixty years is absent because of it: the tessellations of the modern practice, the insects that made the tree method famous, the curved-crease sculpture, all of it.

That is most of what a person would call origami. The constraint removes the subject’s living body of work and leaves its traditional substrate, its theorems and its constructions — and the site’s own argument is that the substrate and the theorems are enough to explain the geometry, which is a claim this shelf is the evidence for rather than a consolation.

It also explains a shape the site has that would otherwise look like a preference. There are more tessellations and corrugations here than models, because tessellations are what got published as mathematics; there are more generated patterns than either, because generating one costs nothing and owes nobody.

Where the model stops #

Eight patterns is a shelf and not a survey. Nothing here says that the three classes exhaust the possibilities in general — only that they exhaust this shelf, and that the shelf has no fourth kind by construction rather than by luck.

The classification is also legal in the shallowest sense, which is to say not legal at all. Whether a crease pattern is a copyrightable work, in which jurisdiction, and on what theory, is a question this site has no standing on and does not answer. What is described here is the practice of a community and the shape of a record, and the constraint the site works under is a decision made out of respect for that practice rather than a reading of any statute.

And “traditional” is a claim about the record that an earlier rung is careful about. A base recorded as traditional may have had an inventor whose name did not survive, and calling it unowned is a statement about what can be established rather than about what is true.

What the picture cannot show #

A table of provenances shows what the record says and cannot show what is true. Two of these eight are called traditional because nobody claimed them, and a figure has no way to mark the difference between “nobody made this” and “the record lost who did”.

Nothing shows the missing class. The most informative thing about this shelf is the kind of pattern that is not on it, and an absence cannot be drawn — which is why the argument has to be made in dates, where the absence leaves a mark.

Rediscovery, which the classes handle badly #

There is a case the three classes do not cleanly hold, and it is one this site has already written about.

The Yoshimura pattern was published in 1955 as a buckling mode — the shape a thin cylinder takes when it is crushed, arrived at by analysis rather than by folding. It is also a traditional origami pattern, folded for a long time before that, and found before it was designed is the essay about exactly that collision.

So which class is it in? The census puts it under published as mathematics, because that is what its provenance line says and because a date exists. That is the right answer for the census’s purposes — the date is real and the publication is what makes the pattern citable — and it is not the whole truth about the object.

The general shape of the difficulty is that a class is a property of a record and an object can be in more than one record. A pattern that was traditional and then published is dated by its publication and owned by nobody, which is the one combination the essay’s neat coincidence does not predict.

That is one case out of eight and it is worth flagging rather than smoothing. It is also the case that most rewards attention, because it is where the craft and the literature touch.

The generalisation #

The useful form of this is about what a record selects for, and it applies to any archive.

A record does not contain a random sample of what existed. It contains what somebody had a reason to write down, and the reasons are not evenly distributed: an invention is recorded because somebody wanted credit, a tradition is not recorded because nobody thought it needed saying, and the two are systematically different classes of object.

So a constraint of the form “use only what nobody owns” and a constraint of the form “use only what is undocumented” pick out the same set, not because ownership and documentation are logically linked but because the same act — claiming something — produces both.

That has an uncomfortable consequence for a site like this one, and it is worth stating plainly. The patterns this site can print are the patterns nobody wrote about, plus the ones published as theorems and the ones it made itself. The living, documented, attributed body of work that constitutes most of the subject is exactly the part a figure-first site cannot show, and the reason is not squeamishness — it is that being documented and being owned are the same condition.

Who found it, and when #

Nothing in the classification is a discovery. That traditional bases are unattributed is obvious, that Miura’s fold has a date is on the paper, and that a generated pattern’s author is a program is a description rather than a finding.

What is not usually said is that the three coincide with what can be reproduced, and the reason is that the two halves are normally discussed by different people. The provenance of traditional models is a question for historians of the craft; what may be reproduced is a question for the community’s own conventions; and neither has a reason to notice that the answers line up.

They line up because both are answers to “did somebody claim this”, and that question is asked once and read twice.

Where the ladder goes next #

The immediate continuation is the one this rung’s own limits point at: whether the coincidence holds on a larger shelf. Twenty patterns rather than eight, chosen for coverage rather than for what this site happened to need, would say whether the three classes are exhaustive or whether the shelf is small enough to be tidy by accident.

The other direction is the generated class. It is half this shelf and it is growing, and a pattern whose provenance is a program raises a question the record has not had to answer before: what it would mean for such a pattern to be rediscovered, and whether two constructions that emit the same pattern are one object or two. The same vertex, found four times is the version of that question for vertices, and the version for constructions is not written.