Publishing the pattern instead of the sequence

Some time around the 1990s, designers of complicated origami began publishing their work as crease patterns rather than as folding sequences. A pattern is one picture; a sequence for the same model may be three hundred. The change looks like a formatting decision and it reorganised the subject.

The arithmetic of publishing #

The comparison is stark enough to be worth doing plainly.

A diagram sequence has one picture per step, and a complex model has hundreds of steps. Each picture must be drawn accurately, in the correct state, with the correct crease marked. Producing the sequence is often described by designers as taking longer than designing the model, and it is a skill of its own — a good diagrammer is a specialist.

A crease pattern is one drawing. For a designer working on a grid it is very nearly free, because the pattern is the thing they were working in anyway.

So the cost of publishing fell from hundreds of drawings and weeks of work to one drawing and an afternoon. Anything that changes a cost by that factor changes what gets made.

What the pattern keeps #

A crease pattern is not a lossy summary of a model. It is complete in a specific sense: it specifies the folded state entirely.

Every crease is there, with its assignment. Given the pattern and the assignment, the folded state is determined — the paper has nowhere else to go. That is why this site treats the pattern as the canonical artefact and the folded object as a second view of it.

Completeness is not the same as legibility #

A pattern being complete does not make it readable, and the two get conflated because “specifies the folded state” sounds like “says what it is”.

Handed a dense pattern with several hundred creases, a competent folder cannot generally say what the model is. The flaps are not visibly flaps; the structure is visible only to somebody who can read the circle packing underneath it and reconstruct the tree the design came from. A crease pattern is complete in the way a compiled binary is complete.

Which is why the shift produced its own literature of reading patterns — annotating them with the packing, marking the axial creases, identifying the rivers. That literature is somebody rebuilding, by inspection, the design information that was thrown away when the sequence was dropped.

What the pattern loses #

What is missing is the order, and the order is not a convenience.

A theorem saying a folded state exists says nothing about whether anybody can reach it, and a crease pattern is exactly a statement that a folded state exists. The reader is handed the destination and no route.

For a simple model that is fine, because the route is easy to find. For a complex one it is a search problem in a space that grows very fast, and it is a search problem the designer has already solved and thrown away.

The difficulty moved, it did not vanish #

That is the essay’s central point and it is worth stating as a conservation law.

Producing a foldable model requires somebody to solve the reachability problem. When designers published sequences, the designer solved it once and every reader inherited the solution. When designers publish patterns, every reader solves it independently, and most of them fail.

The total difficulty in the world went up: one hard problem became one hard problem per reader. What went down was the difficulty borne by the publisher.

The grid did half of it #

A precondition often left out: patterns became publishable partly because they became drawable.

A design whose creases fall at arbitrary angles is a pattern that has to be printed at high accuracy to be usable, and a reader folding from it must place each crease by measurement. A design built on a grid, where every crease runs along a grid line or a 45° diagonal, needs only that the reader divide the square into thirty-seconds — which is six easy folds — and then everything else is found by counting squares.

So box-pleating and the crease-pattern publication format are not independent developments. The grid is what makes a pattern something a reader can transfer to paper at all, and the two spread together.

Why designers did it anyway #

It would be unfair to read this as laziness, and the reasons are good ones.

The first is that diagramming a three-hundred-step model is not merely long, it is often impossible to do well. Many complex models have no clean sequence: the collapse happens more or less at once, with dozens of creases forming together, and there is no honest way to draw that as a series of single folds. A sequence for such a model is a fiction imposed for the reader’s benefit.

The second is that the pattern is what designers actually work in. A design produced by circle-packing and a grid exists as a pattern from the start; the sequence, if any, is reverse-engineered afterwards.

The third is that the audience changed. A reader who wants a crease pattern is a reader who wants to understand the design, and for that purpose the pattern is strictly better than the sequence.

What became publishable #

The effect on what got made is the part that shows up in the record.

Designs that could not have been diagrammed became publishable, and they are exactly the designs that are most interesting mathematically: high-flap-count bases, dense tessellations, patterns with hundreds of creases collapsing simultaneously. The tree method’s output is a pattern and not a sequence, and a design produced by an algorithm has no natural order at all.

So the shift did not merely change how things were published. It changed which things could exist as published objects at all, and the complex era of the 1990s and 2000s is downstream of that.

What gets lost between designer and reader #

There is a specific and avoidable loss in the format as usually practised, and it is worth naming because it is not inherent.

A published pattern is normally the crease pattern alone: lines, sometimes with the assignment, sometimes without. The information the designer had and did not include is substantial — the packing the design came from, the tree it realises, which creases are structural and which are shaping, and which of the flaps corresponds to which feature of the subject.

None of that is expensive to include and most of it is not included. So the format’s economy is partly real and partly an artefact of convention: the cheap thing to publish would be the pattern with its design data attached, and what is actually published is the pattern stripped.

The reader’s side of the bargain #

A subculture grew up around the difficulty, which is itself evidence for how large it is.

Folding from a crease pattern became a recognised skill with its own name and its own conventions — working out the collapse, finding the order, identifying the pre-creases that have to be made before the model can be brought together. There are folders known for being able to do it and models known for being nearly impossible.

None of that vocabulary would exist if the pattern carried the route.

A pattern is a claim, and claims can be wrong #

One consequence of the format that the community discovered the hard way.

A published diagram sequence has been folded, by definition — the diagrammer folded it to draw it. A published crease pattern has not necessarily been folded by anybody, including its designer, because a pattern can be produced by an algorithm and printed without ever being collapsed.

So the format admits a failure mode the older one did not: a pattern that does not fold. It may violate the theorems at some vertex; it may satisfy them everywhere and fail globally, which the local conditions cannot detect; or it may fold in principle and be unreachable by any human sequence.

That is precisely why this site runs every pattern it draws past four conditions before allowing it onto a page. The check is cheap, it is necessary rather than sufficient, and the format it defends against is the one that made it necessary.

What a machine would need #

Reading the shift through this site’s machine models makes the size of the gap precise.

Given a pattern, deciding whether it folds flat at all is NP-hard. Given that it does, finding a sequence of simple folds that reaches the state is a further question, and the simple-fold machines reach a strictly and rapidly shrinking fraction of flat foldings. So a program that turns a pattern into a diagram sequence would have to solve a hard problem and would frequently correctly answer that no sequence exists.

That is a real result about a practical annoyance: for many published patterns there is no sequence of simple folds, so the missing diagrams are not missing through neglect.

The economics, stated as a rule #

The general form is worth extracting because it applies well beyond folding.

A publication format has two costs — what it costs the author to produce, and what it costs each reader to use — and the total borne by a community is the first plus the second times the number of readers. A format that shifts work from author to reader looks efficient to the author and is efficient for the community only when readership is small or when the reader’s cost is low.

Origami’s shift to patterns was made by authors, on author’s economics, at a moment when the readership was growing. That is the configuration in which the trade is worst, and it is why the complaint about missing diagrams has never gone away.

It was still the right move, for the reason above: without it, most of the work would not have been published at all, and a hard-to-fold model in circulation beats a diagrammed model that was never finished.

Where the break-even actually is #

The rule stated above has a break-even in it, and putting numbers to it changes the verdict from an indictment into something closer to a draw.

Write $A$ for what a format costs its author, $R$ for what it costs each reader, and $N$ for how many readers fold the model. The pattern format is the better deal for the community when

$$A_{p} + N R_{p} \;<\; A_{s} + N R_{s} \qquad\Longrightarrow\qquad N \;<\; \frac{A_{s} - A_{p}}{R_{p} - R_{s}}$$

The numerator is what the author saves, and the essay’s own revised figure puts it at eighty drawings against one — call it three weeks of work against a day, so twenty author-days. The denominator is what each reader pays extra: working out the collapse, which for a complex model is an evening at best and several at worst.

At one extra reader-day the break-even is about twenty readers. At three, it is seven.

Which is roughly the actual readership #

That is the part worth sitting with, because twenty is not a number far from the truth. A super-complex model published as a crease pattern is attempted by dozens of people worldwide, not thousands, and completed by rather fewer.

So the trade is not obviously wrong; it is close. And a trade that is close is exactly what produces the pattern this essay notices — a complaint that never goes away and never wins. A format clearly worse for the community would have been abandoned; one clearly better would have stopped being argued about. Thirty years of unresolved grumbling is what a near-tie looks like from inside.

The feedback nobody accounts for #

There is one term the formula leaves out, and it runs the wrong way.

$N$ is not independent of the format. A reader who fails to fold a pattern has spent their evening and produced nothing, and readers who fail repeatedly stop attempting. So a format with a high failure rate reduces its own readership, which reduces $N$, which by the inequality above makes the format look better.

That is a self-justifying arrangement, and it is the sort that is very hard to argue against from inside. Every measurement taken after the shift is taken on the audience the shift left behind: the people still folding from patterns are the ones for whom $R_{p}$ is small, so the observed reader cost is low, so the format appears vindicated.

The honest comparison would need the readers who left, and nothing counts those. That is the same shape of absence the essay’s own conclusion is about — the sequence that was never drawn leaves no record, and neither does the folder who gave up.

A comparison worth making #

The same trade appears in other fields and looking at one sharpens what is peculiar here.

A chemical structure diagram specifies a molecule completely and says nothing about how to synthesise it; the synthesis is a separate, harder publication. A circuit schematic specifies behaviour and not layout. In both cases the community keeps the two apart deliberately and values them separately.

Origami arrived at the same split by accident, in one direction, over about a decade, and without ever agreeing that it had. The result is a literature where the what is abundant and the how is scarce, and where the scarcity is not acknowledged as a structural feature.

What this site does with the choice #

This site publishes patterns and is explicit about what that means, which is the only defensible position for it.

Every printable figure here is a pattern, at true scale, with a FOLD export beside it, and each is checked against the theorems before it is drawn. What is not offered is a folding sequence, and the reason is exactly the one above: the site can verify that a pattern folds flat and it cannot verify that a reader can get there.

Offering a sequence it had not checked would be a claim of a kind nothing here can support. Offering the pattern and saying so is smaller and true.

What the format did to attribution #

A last effect, and it connects the ladder back to the field it sits in.

A diagram sequence is unmistakably somebody’s work: it is hundreds of drawings in an identifiable hand, and copying it is obvious. A crease pattern is a set of lines, and two designers arriving at similar packings for similar subjects will produce similar patterns — not because either copied but because the packing is largely determined by the flap lengths.

So the shift made the boundary between independent design and derivation harder to see, at exactly the moment the volume of published work rose. The community’s norms around attribution date from that period and are stricter than they look from outside, and the strictness is a response to a genuine ambiguity the format introduced.

The idealisation, named #

The arithmetic at the top of this essay treats a diagram sequence as one picture per fold, and real sequences are more efficient than that.

A good diagrammer combines steps, uses repeat marks, shows several folds at once where they are independent, and relies on the reader’s knowledge of standard bases to skip whole regions — “begin with a bird base” replaces about a dozen steps. So the true ratio is not three hundred to one but perhaps eighty to one.

Eighty to one is still two orders of magnitude and the argument is unaffected. The idealisation is named because the exact number matters to nothing here and an unstated approximation is a bad habit.

There is a smaller version of the same trade inside this site. Every essay here publishes a figure and the code that generated it, and the code is the analogue of the sequence — complete, checkable, and read by almost nobody. What a reader takes away is the picture, which is the pattern.

Where this goes next #

This closes the notation ladder. Sideways, designing on a grid is what a published pattern usually is, and a state no motion reaches is the reachability problem in its own terms.

The surprising connection: the shift to patterns is the same event as the shift from craft to a cumulative field, running one step further and in the opposite direction. A notation made models transmissible by encoding the route; publishing patterns made more models transmissible by dropping it. Both increased what the subject could hold, and the second one did it by making the individual reader’s job harder — which is a trade a field can only make once it has readers willing to take it.