The reader decides the junction

A crease pattern is taught as a thing to be read directly: here are the lines, here is which way each one folds, collapse it. That description is nearly complete, and the part it leaves out is a decision the reader makes several dozen times per pattern without being told they are making it.

At any point where two drawn lines meet, the reader has to decide whether they meet — one ends where the other passes — or cross, and the ink is identical either way. The two readings are different objects: the first is an ordinary vertex, and the second cannot fold at any angle under any lettering.

Forty-six decisions on five patterns #

Counted over the printed shelf, the places where one crease ends on the interior of another are: four on the preliminary base, eight on the square twist, twelve on the hexagon twist, nineteen on the Yoshimura, and three on the fold-and-cut triangle. The Miura, the tapered corrugation and the waterbomb tessellation have none — their creases meet only at declared vertices.

Forty-six junctions on five patterns, then, and at each of them the drawing is ambiguous and the reader is not.

They are not decorative. On the Yoshimura, every course runs from one side of the sheet to the other and every diagonal ends on a course; on the twists, every pleat ends on a polygon’s side. Take those junctions the other way and the pattern acquires nineteen or eight or twelve crossings, and stops being foldable.

Why nobody gets it wrong #

A reader confronted with an ambiguity they cannot see should, on the face of it, get it wrong half the time. They do not, and there are three reasons — none of which is that the notation helped.

The paper refuses. A learner who reads a junction as a crossing and folds accordingly finds that the sheet will not go; the paper is the check, and it is immediate. This is the sense in which the subject is taught by the material, and it is very effective — it just leaves the learner without a name for what went wrong.

The lines have ends. A crease that stops has a visible end: a T rather than an X. That is a real cue and it is why the ambiguity is not usually noticed at all. What makes it an ambiguity is that the cue is about the drawing, so a slightly heavy line, a photocopy, or a pattern drawn with lines overshooting by a hair reverses it.

The reading is constrained. A folder who has read a hundred patterns knows that a pleat ends on a twist polygon, that a diagonal ends on a course, that a perpendicular ends on the outline. The decision is being made by the reader’s model of the construction rather than by the ink — which is exactly why it is invisible: the knowledge doing the work was acquired elsewhere.

Where the junctions are, pattern by pattern #

The distribution of the forty-six says something about which constructions ask most of a reader.

The Yoshimura’s nineteen are the most of any pattern here, and every one is the same event: a diagonal of the zigzag arriving at a horizontal course. The courses are drawn in segments between junctions rather than as single lines from side to side — because a course drawn as one edge passes through every junction without being seen there, and every one of those vertices would then be analysed as carrying four creases instead of six.

The hexagon twist’s twelve and the square twist’s eight are the pleats arriving at the twist polygon. Two per side of the polygon, which is what a pleat is, so the count is twice the number of sides.

The preliminary base’s four are the half-creases meeting the sheet’s own edges — which is the same event with the outline playing the part of the other line.

And the fold-and-cut triangle’s three are the perpendiculars landing on the outline, at feet the construction computed rather than chose.

The three patterns with none — the Miura, the tapered corrugation, the waterbomb — are the grids. A grid’s creases meet at corners of cells, which are declared vertices, so a reader is never asked. That is a genuine property of designing on a lattice and part of why lattice designs are easy to read.

The same mistake, in code #

The interesting part of this is not that readers cope. It is that a machine handed the same patterns did not have to make the decision, and therefore never noticed that there was one — and consequently could not detect the case where the decision goes the other way.

Every checker here is handed a crease pattern as a list: which vertices exist, which pairs of them an edge joins. In that form a junction has already been decided — the shared vertex index is the decision — and a crossing simply is not present. So the checkers were asking a question that a reader has to answer, in a form where it had already been answered, and they were blind to a pattern in which it had been answered wrongly.

Four patches drawn here had been answered wrongly, and every check passed for years.

That is a teaching failure of a peculiar kind: a piece of machinery that inherited a reader’s competence without inheriting the reader’s check. A person who mis-reads a junction folds the paper and finds out. The machinery had no paper.

Both misreadings are caught by one theorem #

The claim that a crossing cannot fold at any angle under any lettering is stated here as a fact to be looked up. It is two lines, and doing them shows that the same theorem catches the other misreading too — which is what makes the decision testable rather than a matter of taste.

A crossing. Two lines passing through a point give four creases, but they are two collinear pairs, and a collinear pair is one straight crease carrying one letter. So the four letters round the point read $h, v, h, v$, and the counts are four-and-nought when $h = v$ and two-and-two when they differ. Maekawa wants a difference of two. Neither reading supplies it, and no choice of angles enters the argument anywhere.

An odd degree. If a misreading leaves a vertex with three creases — one line stopping on another in the sheet’s interior — then $M + V = 3$ while $M - V = \pm 2$, which needs $2M = 5$. Every odd-degree interior vertex is refused by parity alone, before any angle is measured.

So the two ways a junction can be got wrong land on the same theorem by two routes: collinearity forces the letters in one case, and parity forbids the count in the other.

Which is why the check is a count #

That matters for what a reader can do about it, because both refusals need only the crease list’s shape.

Nothing above measures a sector, tests developability or folds anything. A reader — or a checker — resolving a drawing into a graph can put every vertex past two questions: is its degree even, and are its creases collinear in pairs. A yes to the first and a no to the second is required at every interior vertex of every foldable pattern.

That is the check the machinery was missing. Not a harder test than the ones it already ran, and not a geometric one: the cheapest test in the subject, applied to the one step nobody had noticed was a step.

What a teaching text should say #

Three sentences would cover it, and none of them appears in any account of crease patterns this collection has read.

A junction is a decision. Where one line stops on another, the pattern has a vertex; where two lines pass, it has none, and the difference is not in the ink.

A crossing cannot fold. Maekawa’s count refuses it at every angle, so any reading that produces one is the wrong reading — which makes the decision testable rather than a matter of judgement.

And a drawing is not a list. A pattern published as a picture has to be read into a list before anything can be computed about it, and the reading is where this decision lives.

The third is the one worth the most and is the least often said. Every diagram in every book on this subject is ink, and everything the subject knows how to compute is about a graph. The step between them is performed silently, by the reader, every time.

The photocopy test #

There is a practical version of all this that anybody can run, and it is the reason the ambiguity is not merely theoretical.

Take a printed crease pattern and reduce it. At a half the junctions are still junctions; at a quarter the lines thicken relative to their length and a T begins to look like an X; at an eighth the whole pattern is a grey mesh and every junction reads whichever way the ink fell. The same happens with a fax, a low-resolution scan, a pattern drawn with a thick pen, or a pattern reproduced from a photograph of a folded model.

That is how patterns actually circulated for most of the period in which they circulated at all — photocopied, faxed, scanned, redrawn from memory — and it is why a reader’s model of the construction is doing so much of the work. The ink degrades and the reading survives, because the reading was never really coming from the ink.

The same test applied to this collection’s own figures says where the limit is: at the size these patterns are drawn, a junction is a few pixels of difference. The shallowest crossing this collection has produced is 0.16 mm deep at printed size, which is thinner than a drawn line — so at that depth, a crossing and a junction are the same mark on paper, and only the coordinates distinguish them.

Why the problem is recent #

The ambiguity did not exist for most of the subject’s history, for a reason that is about what was published rather than about anybody’s care.

Diagrams record a sequence — fold this crease, this way, now — and a sequence has no junctions, because it shows one fold at a time on a sheet that already has the earlier folds in it. A learner following a diagram never sees a whole pattern and never has to resolve anything.

The crease pattern as a published artefact is recent: it belongs to the era when designers began publishing patterns rather than sequences, which is roughly the last four decades. That change is what made the design methods legible and it also handed the reader a new job — reading a static picture of a completed pattern, junctions and all.

So the reading skill this rung is about is younger than the subject and younger than most of its literature. It is taught by exposure and by paper, and nowhere else.

What the eight patterns ask of a beginner #

Ranking the shelf by how much silent reading it demands gives an order that is not the order of difficulty, and the mismatch is worth teaching from.

The waterbomb tessellation, the Miura and the tapered corrugation ask nothing: every meeting is a declared vertex, so a beginner can read them line by line. They are also three of the harder patterns to actually fold, because a grid of ninety creases has to be collapsed all at once.

The Yoshimura asks the most — nineteen junctions — and is among the easier collapses, because its courses fold in order.

The preliminary base asks four and is the easiest fold on the shelf. The twists ask eight and twelve and are the ones a beginner most often fails at, for reasons that have nothing to do with reading: the pleats have to turn together and the paper resists.

So reading difficulty and folding difficulty are close to independent. A teaching order chosen for one is not chosen for the other, and the traditional order — bases first, tessellations later — happens to start with the patterns that ask least of both.

The three explanations this collection has had to repair #

Taught with a wrong reason collected three explanations here that were right in their conclusions and wrong in their reasons — a ring read as one letter, an assignment the symmetry suggested, a tessellation verified because its unit was. This is a fourth of a slightly different kind: not a wrong reason but an unnamed step.

The distinction matters for what to do about it. A wrong reason is repaired by replacing the reason. An unnamed step is repaired by naming it, and the naming is cheap — three sentences above — but it can only happen once somebody has noticed that a step was being taken.

What made it noticeable here was the machine failing where a reader would not have. That is worth recording as a method rather than as an anecdote: the way to find the steps a reader takes silently is to hand the same task to something that cannot take them. A checker with no paper and no expectations is exactly such a thing, and its failures map the reader’s competence.

What the check costs a reader who wants one #

A folder holding a printed pattern can run the machine’s check by hand, and it is quick enough to be worth describing.

Look at every junction and ask whether the line stops. A T is a junction; an X where four lines leave a point is either four creases meeting or two crossing, and the difference is whether the letters change across the point. Four creases meeting alternate their letters in the way Maekawa’s count requires — three of one and one of the other at a degree-four vertex; two lines crossing keep their letters straight through, which is two and two or four and none.

So the letters resolve it. At any X on a printed pattern, follow each line through the point: if the letter changes on either line, the point is a vertex. If both lines keep their letters, the pattern cannot fold, and the reader has found a mistake in the drawing rather than in themselves.

That is a genuinely useful thing for a learner to know and it takes one sentence to teach. It also inverts the usual relationship between the reader and the printed pattern: a reader who can check a pattern is a reader who can find a fault in a book, which is not what a diagram-following tradition prepares anybody for.

The rule has one case that looks like an exception and is not. Two lines meeting at a right angle satisfy the angle condition — they are the only crossing that does — so a reader testing the geometry alone would let them through. The letters still refuse them: a line carrying its letter through the point contributes two of that letter, so the four spokes come to four and none, or two and two, and neither of those differs by the two the count requires. Swept over eight angles and every lettering of the four spokes, the readings that fold are the eight in which a letter changes across the point, and a letter changing across a point is four creases meeting there. So the reader’s rule is the right one and it is the letters carrying it, at every angle including the square.

What this does not say #

Not that readers are unreliable. The evidence is that they are extremely reliable at this: forty-six junctions on the printed shelf and no reported confusion in the history of the subject. The claim is that the reliability comes from the paper and from experience rather than from the notation.

Not that the notation should change. A convention already exists and works — a line that stops is drawn stopping — and it fails only under reproduction, at small scale, or when a construction produces something nobody drew. Adding a mark at every junction would clutter every pattern to guard against a case that a reader almost never meets.

Not that the count is the whole of the reading. A junction is the ambiguity that has a wrong answer with a consequence. A reader also decides which lines are creases and which are guides, which letter a faint line carries, and where the sheet’s edge is — none of which has been counted here, and all of which the same drawing leaves open.

And not that this is the only silent step. It is the one that was found because a construction fell into it. A population cannot test for a fault none of its members has, and the same is true of a reader: the steps a person takes without noticing are found when something else takes them wrongly, which is not a systematic search.