What a dashed line can say

A dashed line means fold this toward the reader. A dash-dot line means fold it away. An arrow says do it now, and in this direction. That is the whole of the basic notation, it was settled in the middle of the twentieth century, and it is the reason this subject has a literature.

The claim sounds inflated and it is not. Before it, a model existed in the hands of whoever could fold it and in nobody else’s, and a book about folding had to describe each step in sentences. After it, a model is an artefact that travels.

What the notation is #

Akira Yoshizawa developed a system of symbols over the 1940s and 1950s and published it in Atarashii Origami Geijutsu in 1954. Samuel Randlett and Robert Harbin brought it into English-language books around 1960, adding to it and regularising it, and the resulting convention is universal.

Its core is small: two line styles for the two fold directions, an arrow for the action, a hollow arrow for turning the model over, and a convention that the diagram shows the state before the operation named on it.

That last part is the subtle one. A diagram is not a picture of a fold; it is a picture of a state with an instruction written on it, and the sequence of diagrams is a sequence of states. The notation is imperative.

Why it made the subject cumulative #

The consequence is a matter of what can be built on.

A craft in which each practitioner learns by watching another is bounded by how many people any one person can teach and by what any one person can remember. Nothing accumulates: a model is lost when its folder stops folding.

A craft with a notation is bounded by nothing similar. A model published in 1962 is available in 2026 to somebody who never met its designer, can be checked, improved, referenced, and combined with another. That is the difference between a tradition and a field, and it is the same transition that music went through with staff notation and that chemistry went through with structural formulae.

What the reach costs per crease #

The figure’s fall — seventy-one per cent at three creases to thirteen at six — has a rate in it, and the rate says what the notation can and cannot carry.

Three extra creases take the reach from 0.71 to 0.13, a factor of 5.5, so each crease costs a factor of about 1.76: the reach is multiplied by roughly 0.57 every time a crease is added.

Extrapolate. At ten creases that is about one and a half per cent. At twenty it is under one part in ten thousand. A strip with twenty creases has essentially none of its flat foldings reachable by any sequence of the basic symbols, and twenty creases is a modest pattern.

That is the quantitative form of the notation’s ceiling, and it explains a change in publishing practice that is usually attributed to effort. Complex designs came to be published as crease patterns rather than as diagram sequences, and the usual reason given is that diagramming a three-hundred-step model is enormous work. The reach curve says something stronger: for most such models there is no sequence to diagram, and the missing diagrams are missing because the notation cannot express the fold.

The symbol that changes the answer #

There is one addition to the basic set that moves the reach from a seventh to everything, and its size is worth appreciating.

The symbols as counted above make a fold that takes every layer at the crease line. The convention fold the near layer only — one extra mark on a diagram — lets a fold take a chosen block of layers instead.

That is exactly the difference between two machine models this collection has measured: a machine forced to take every layer loses states rapidly, and a machine that may take any contiguous block from the top or the bottom reaches every flat folding, on every spacing and every assignment tested.

So one convention takes the notation from thirteen per cent to a hundred. Not an improvement in coverage — a change in kind, from a restricted machine to an unrestricted one.

That is a remarkable amount of work for a single symbol, and it is why the layer conventions are the part of the notation a diagrammer is most careful about. The line styles and the arrows make the notation legible; the layer marks are what make it complete.

What it replaced #

To see the size of the change it helps to look at what a folding instruction was before.

The 1797 book’s plates are pictures of finished arrangements with a cutting diagram, and the folding is left to the reader on the grounds that a reader already knows how to fold a crane. That works because the crane is common knowledge in the culture the book was printed in. It transmits nothing to anybody outside it.

European instruction books of the nineteenth century use prose: bring the lower right corner up to meet the upper left, crease firmly, turn the paper over, and so on. Prose is unambiguous about a single fold and becomes unreadable after about six of them, because the reader has to hold the current state in their head and the text never shows it.

The diagram convention fixes both at once. The state is drawn, so nothing has to be remembered; the instruction is a mark on the drawing, so nothing has to be parsed. That is why the format has not changed in seventy years.

The symbols are a machine model #

Here is where the historical claim becomes a checkable one, and it is the reason this essay sits on a figure-first site.

A dashed line with an arrow says: take the model in its current state, fold along this line, and do it now. That is exactly one operation, applied to the whole stack of paper in the current state. It is, in this site’s own vocabulary, a simple fold — the move a press brake makes, the move a folding machine makes, and the move a diagram can ask for.

So the basic notation is not a general-purpose language. It is a language for a particular machine, and the machine is the one this site has spent a whole field characterising.

Why the correspondence is exact rather than loose #

It would be easy to overstate the match, so it is worth checking the fit in both directions.

A simple fold, as this site defines it, takes a flat state, chooses a line, and folds everything on one side of that line through 180° — either all the layers, or exactly one. A basic diagram step gives a line and a direction and applies it to the model as it stands. Those are the same operation.

The correspondence fails in one direction only, and instructively. A diagram can ask for a fold that does not go all the way — a partial fold, a fold to a point, a fold left standing at an angle — and these have their own symbols. Every one of them produces a state that is not flat, which is exactly the state the simple-fold model has no representation for.

So the notation’s basic layer matches the machine exactly, and its first extension is in the direction the machine model cannot go at all.

What that machine cannot reach #

Once the correspondence is made the measurement follows, and it is not kind to the notation.

Take a strip with a few creases. Enumerate every mountain-and-valley assignment, ask which of them fold flat at all, and then ask which of those can be reached by some sequence of simple folds. At three creases, seventy-one per cent. At four, forty. At five, seventeen. At six, thirteen.

The share is collapsing, and it is collapsing because the constraint of folding exactly one layer or exactly all of them removes reachable states faster than the state space grows.

So the vocabulary had to grow #

This is exactly what happened, and knowing why makes the history legible.

Diagram notation did not stay at three symbols. It acquired the reverse fold, the squash, the sink, the petal, the rabbit ear, the crimp — each a named, illustrated compound with its own symbol or its own recognisable diagram pair. Every one of them is a move that is not a simple fold.

A reverse fold changes the direction of an existing crease inside a stack; there is no single line and single arrow that asks for it. A sink turns a point inside out; the operation cannot be described as folding along one line at all. These are not abbreviations for convenience. They are the vocabulary extending into the region the basic symbols cannot address.

The notation grew where the machine stopped #

That is a striking alignment and it is the essay’s central claim.

The vocabulary’s named moves are not distributed randomly across the space of things folders do. They cluster precisely where the simple-fold model runs out — at operations involving layers moving differently, at operations that reverse existing creases, at operations with no single fold line.

Nobody designed it that way. Folders added symbols when the existing ones could not express what they needed, one at a time, over decades, with no theory of machine models available to any of them. The resulting vocabulary is a map of the boundary of simple foldability drawn from the outside, by people who did not know the boundary existed.

Counting the extensions #

A rough census makes the clustering concrete.

The symbol set in general use has of the order of a dozen marks beyond the two line styles and the fold arrow. Sorting them by what they do: two are about viewing — turn over, rotate — and carry no fold at all; two are about incomplete folds, which leave a non-flat state; and the remainder are named compound moves that rearrange layers.

The last group is the interesting one and it is the largest. Reverse, squash, sink, petal, rabbit ear, crimp, swivel: seven or so operations, each with a name a folder says aloud, each requiring more than one crease to change at once, and none expressible as a sequence of the basic mark.

Seven named exceptions is a lot for a notation with three primitives. It is what a language looks like when its primitives are a strict subset of what its users need.

What a diagram cannot say #

There is a second limitation, independent of the first, and it is about order rather than about moves.

A diagram sequence is totally ordered: step 4 comes after step 3. But a great many models have steps that are genuinely independent — fold the left flap and the right flap, in either order — and the notation has no way to say so. It picks one and pretends the choice was necessary.

That matters when a sequence is being checked or automated, because a reader cannot tell a forced order from an arbitrary one. The and-or graph that a machine-model search builds carries exactly this information and a diagram sequence throws it away.

The information a sequence carries #

A comparison with the alternative sharpens what the notation is for.

A diagram sequence of n steps is n pictures. The crease pattern for the same model is one picture. The pattern is smaller, it is complete, and it is what this site treats as the canonical object.

What the pattern does not carry is the order, and the order is what a person needs to actually make the thing. So the two notations divide along a real seam: the pattern says what the model is, the sequence says how to get there, and the second is precisely the reachability question that the theorems do not answer.

A notation is a claim about what matters #

The choice of what to give a symbol to is a claim about what the important operations are, and diagram notation makes one that is worth noticing.

It has symbols for actions and none for structure. There is no mark meaning “these two flaps are the same flap reflected”, no mark for a repeating unit, no mark for a symmetry. A model with sixteen identical points is diagrammed by folding one and writing “repeat fifteen times”, which is a note in the margin rather than part of the language.

That is the opposite emphasis from the crease pattern, where structure is all that is visible and action is absent entirely. The two notations are near-exact complements, and a designer working in one is systematically blind to what the other shows.

Yoshizawa’s other contribution #

The symbols are the transmissible part, and there is a second thing that is harder to credit precisely.

Yoshizawa also changed what the folded objects were. Wet-folding — dampening the paper so it takes and holds curved forms — is his, and it moves origami away from the polyhedral, creased-everywhere look that the flat-folding theorems describe so well. That whole area of the subject is about the idealisations failing, and it is a folder’s contribution rather than a mathematician’s.

So the same person is responsible for the notation that made the discrete subject cumulative and for the technique that moves furthest away from discreteness. That is not a contradiction; it is what somebody does who is interested in what paper can be made to do.

What the measurement does not settle #

The figure measures simple-fold coverage on strips, which is a one-dimensional problem, and models are two-dimensional.

That is a real gap. The one-dimensional case is chosen because it is exactly enumerable — every assignment of six creases is sixty-four cases, and the layer solver can decide each — where the two-dimensional case is NP-hard and not enumerable at any useful size. So the numbers are about strips and the argument is extended to models by analogy.

The direction of the analogy is defensible: two dimensions give a machine more ways to be blocked, not fewer, so the true coverage for models is very unlikely to be higher. It is still an extrapolation and is marked as one.

What a symbol costs to add #

There is an economics to a notation that explains why the extensions took decades rather than months.

A new symbol is only useful if readers know it, and readers learn it from books that use it, and authors use it only if they expect readers to know it. So the barrier to adding one is not invention but coordination, and in a field with no standards body the coordination happened through a small number of widely-read books — Harbin’s and Randlett’s in English, Yoshizawa’s own in Japanese — which is why the vocabulary is as uniform as it is.

That also explains a feature of the record. The notation’s growth is concentrated in the two decades after 1960, when a handful of books were doing most of the transmitting, and it has been nearly static since — not because folders stopped needing new operations but because the coordination mechanism that added symbols no longer exists in the same form.

The idealisation, named #

The simple-fold model treats a diagram’s instruction as an idealised operation on an idealised stack, and a real folder is more capable than that.

A person folding from a diagram routinely does things the model forbids: holding some layers back with a finger, easing a fold that does not quite lie flat, working a stubborn point by feel. The gap between what the notation says and what a competent folder does with it is large, and it is filled by tacit knowledge that no diagram carries.

Which is the honest version of the essay’s claim. The notation made the subject cumulative; it did not make it complete, and every diagrammed model still requires a reader who knows things the diagram does not say.

Where this goes next #

Publishing the pattern instead of the sequence is the next rung: what changed when designers began releasing crease patterns rather than diagrams, and why that is a statement about the economics of publishing as much as about notation.

The surprising connection to end on. A notation designed by folders for folders, with no mathematics anywhere near it, turns out to be an exact description of a restricted machine — and the places where folders had to invent new symbols are exactly the places where that machine fails. The vocabulary is an empirical map of a theoretical boundary, drawn thirty years before anybody stated the theory.