Almost every pattern fails

Every crease pattern on this site folds flat. That is the site’s proposition and the reason a reader is invited to print one and try it.

It is also, read the wrong way round, a considerable claim about luck. Nothing about a drawing of lines makes it foldable. The question worth asking is how unlikely a foldable pattern is, and the answer is not “unlikely” — it is a good deal stronger than that.

One equation, and what that costs #

At an interior vertex of degree four, the sectors are three free numbers — the fourth is whatever is left of the full turn. Kawasaki’s condition is that the alternating sum of the sectors is zero, which is one equation in those three numbers.

One equation cuts a three-dimensional space down to a two-dimensional surface inside it. A surface has no volume. Pick a vertex at random by any reasonable rule and the probability that it lands on that surface is exactly zero.

That is what “measure zero” means and it is not a technicality. It says that no amount of care in drawing gets a pattern closer to foldable in the sense that matters, because the target has no width at all. A vertex either satisfies the equation or does not, and drawing is not a way of satisfying equations.

The measurement in the figure is the practical restatement. Ask instead for the fraction that come within a tolerance, and the answer is proportional to the tolerance: halve what counts as close enough and half of the near-misses stop qualifying. On log-log axes that is a straight line of slope one.

The surface, and a point beside it #

The two-dimensional surface inside the three-dimensional space of vertices is not an abstraction. It has a description a folder would recognise: the sectors come in supplementary pairs, so opposite sectors add to a straight angle, and the two free numbers are which pair of angles to use.

The second figure is the whole difficulty in one picture. Allow every sector of a random vertex to move by a tenth of a degree and almost none of them reaches the condition; allow ten degrees and most do. A vertex a hand would draw as though it were the one above differs from it by an amount no drawing records, and one of them folds and the other does not. There is no intermediate condition, no “nearly folds”, nothing that degrades gracefully. What degrades gracefully is the paper, and that is a separate subject discussed below.

Vertices multiply #

A pattern with more than one interior vertex has to satisfy the condition at every one of them, and the conditions are independent.

So the fraction falls as the tolerance to the power of the number of vertices. Two vertices, slope two. Three, slope three. The figure measures all three and the slopes come out within a hundredth of the integers, which is the check: the exponent is not fitted, it is predicted, and a measurement that disagreed would mean the conditions were not independent after all.

The consequence at realistic sizes is severe. A four-by-three Miura fold has six interior vertices, so a random drawing of that shape is within a tenth of a radian of foldable about one time in a million, and within a hundredth about one time in a million million. The waterbomb tessellation at four cells across has twenty-five.

The independence is worth defending rather than assuming, because it is the step that turns one exponent into many. Two vertices of a general pattern have their own sectors and share at most a crease, and a shared crease constrains a direction rather than an alternating sum — so knowing that one vertex satisfies its equation says nothing about whether its neighbour does. Where that stops being true is in a pattern with enough structure to couple them, and a corrugation is exactly such a pattern: its vertices are all copies of one vertex, so the conditions are not independent at all and satisfying one satisfies every one of them.

That is what a tessellation is, seen from this angle. It is a way of paying for one equation and receiving hundreds, which is why almost every pattern anybody folds repeatedly is periodic. A design with a hundred unrelated vertices would have to solve a hundred equations; a design with a hundred copies of one vertex solves one.

The other measurement: nudge a pattern that works #

The random experiment answers a question about drawings in general. The complementary question is about a pattern that does fold: how close to it does a pattern have to be before it stops folding?

The answer is that there is no such distance. Move the vertices by two thousandths of a cell width — a tenth of a millimetre on a sheet of ordinary paper — and not one of the six interior vertices satisfies the condition any more. The failure is small, growing in proportion to the disturbance, and it is not zero, which is the only thing that matters to a theorem.

This is the sense in which the drawings on this site are constructed. Every pattern is built by a rule that makes the condition hold identically — the Miura’s zigzag, the twist’s supplementary sectors, the waterbomb’s right angles and half-right angles — and then checked. None of them was drawn and then found to work.

The distinction shows up in how the patterns are parameterised. A Miura fold here takes a zigzag angle and a cell count, and every value of those produces a foldable pattern, because the construction places the vertices so that the collinearity that forces Kawasaki holds whatever the angle is. A generator that instead took four sector angles and drew whatever it was given would produce an unfoldable pattern for almost every input, and would need a search to find one that worked.

This is why identical satisfaction of a condition is worth watching for. When a condition holds for every value of a parameter, the parameter is free — and it is a standing trap on this site that a parameter which looks free may simply be one the condition does not contain. That mistake has been made three times here in different fields, most recently over a corrugation’s row heights and column widths, where the widths turned out to be genuinely free and the heights turned out not to be.

Rarity has a computational face as well as a geometric one.

What a hand can draw to, and what that costs #

The exponents can be run at the accuracy a person actually achieves, and the resulting number is the argument’s strongest form.

A line placed to a tenth of a millimetre on a 150 mm sheet fixes an angle at a vertex to something like a thousandth of a radian. So $\varepsilon \approx 10^{-3}$ is what careful drawing buys, and the essay’s own calibration — six vertices at a tenth of a radian giving one in a million — makes the probability $\varepsilon^{V}$.

For the four-by-three Miura’s six vertices that is one in $10^{18}$. For the waterbomb tessellation’s twenty-five it is one in $10^{75}$.

There is no useful comparison for the second number. It is not that a careful draughtsman would need many attempts; it is that no process of drawing and checking, run for any length of time by any number of people, produces such a pattern.

So “constructed rather than drawn” is not a methodological preference. It is the only available route, by a margin with no physical referent, and the essay’s word constructed is load-bearing in a way the prose understates.

Periodicity, priced #

That also puts a number on what a tessellation buys, which the essay identifies and does not quantify.

A pattern of $V$ unrelated vertices needs $V$ equations satisfied and succeeds with probability $\varepsilon^{V}$. A pattern of $V$ copies of one vertex needs one equation and succeeds with probability $\varepsilon$ — the conditions are not independent, they are identical, and satisfying one satisfies all.

At the waterbomb’s twenty-five vertices and a thousandth of a radian, that is $10^{-3}$ against $10^{-75}$.

Periodicity is worth seventy-two orders of magnitude on that pattern, and the saving grows as $\varepsilon^{V-1}$ — so it doubles in exponent every time the patch doubles in vertices.

That is the reason the subject’s repertoire looks the way it does. Almost every pattern anybody folds more than once is periodic, and the usual explanations — that repetition is pleasing, that it is easy to remember, that it scales — are all true and all beside the point. A periodic pattern is one equation and an aperiodic one is a hundred, and a hundred equations is not a harder problem than one. It is a different kind of problem, with no solution reachable by drawing.

What paper does about it, which is not nothing #

A reader who has folded a crease pattern printed slightly wrong knows that the paper does not usually refuse. It folds, a little grudgingly, and the model comes out with a small twist in it.

That is real and it is not a counterexample. Paper bends. A sheet asked to fold along a pattern that misses the condition by a fraction of a degree accommodates it by curving very slightly out of plane between the creases — which is to say the folded state is not flat, and the theorem was about flat states.

So the honest statement has two halves. The geometry is exact and the material is forgiving, and how forgiving is a question about the material rather than about the pattern. A few tenths of a degree disappears into ordinary paper. A few degrees does not, and shows up as a model that will not close or a tessellation that cups.

This is the same escape route that wet-folding uses deliberately and at a much larger scale: a few percent of strain buys about twenty degrees of sphere. What is bought here is smaller, unintentional, and mostly invisible.

It is the first of the four things that are not true doing the work, and noticing which one matters. The paper is being allowed to bend, not to stretch: the panels stay the size they were and simply decline to stay flat. A material that could not bend at all — a pattern of rigid panels on hinges — has no such tolerance, and a hinge line drawn a fraction of a degree out is a mechanism that jams.

Which theorem was checked, and how #

The claim under test is a power law, and a power law is easy to produce by accident from a badly written experiment. Three things guard against it.

The sampling rule is stated rather than convenient. The four sectors at a vertex are drawn uniformly from the set of four positive numbers summing to a full turn, by cutting a circle at three random points. No flat-foldable vertex is ever produced deliberately, and the rule has no preferred angles in it.

The exponent is predicted before it is measured. One vertex means slope one, and the generator refuses to draw if any measured slope is more than about a tenth away from the integer it is supposed to be. A figure that could not fail would be a decoration.

And the two experiments are independent. The random sweep and the nudged Miura share no code, sample different things and answer different questions, and they agree: the set of foldable patterns has no thickness, approached from outside or from inside.

There is a second limit and it is more serious than the first.

Where this argument stops #

It is about the angles, and only about the angles.

The assignment is a different kind of object. Mountains and valleys are a finite choice, not a continuum, so there is no measure-zero argument to make about them — the right question is what fraction of the finitely many assignments survive, and that fraction has its own behaviour which is nothing like this one. A pattern’s angles are foldable or not with probability zero; its assignments are foldable in a proportion that can be counted.

The argument is also entirely local. Satisfying Kawasaki everywhere is necessary and nowhere near sufficient, and the gap is the whole of the hard part.

So the correct summary is uncomfortable in both directions at once. Almost no pattern satisfies the local conditions; and satisfying them is not enough; and deciding the remaining question is NP-hard. The set of patterns that actually fold is a thin subset of a thin set, and there is no efficient test for membership.

The patterns this site draws are all constructed the same way, and the twists make the construction visible.

Why found patterns are not a counterexample #

There is an apparent exception on this site and it is worth confronting, because it looks like one and is not.

A crushed cylinder folds into a diamond lattice that nobody designed, and it satisfies the flat-folding conditions. If foldable patterns are so rare, how did a buckling event find one?

Because the paper was not sampling from the space of drawings. The sheet was subject to its own constraints — it could not stretch, it could not pass through itself, and it settled into the lowest-energy configuration available. Every one of those constraints is a restriction to states the paper can actually be in, and a state the paper can actually be in is a folded state.

The material searched a space in which the conditions hold by construction, exactly as this site’s generators do. What is remarkable about the crushed cylinder is not that it found a rare pattern; it is that the pattern it found is a regular one. Rarity was never the difficulty.

The same reading applies to the patterns biology arrives at. A leaf packed in a bud is not a drawing that turned out to fold. It is a sheet that grew under constraints, and the geometry it ends in is one the constraints permitted.

And the case that looks like a counterexample is worth having in view while reading the history.

Who put it this way, and when #

The observation that flat-foldable patterns form a very thin set is old and largely unattributed, in the manner of things that everyone in a field notices at once. It becomes a usable statement with the arrival of the computational treatment in the 1990s and after, when the question stops being aesthetic and becomes a question about the dimension of a solution set.

Where the argument has real teeth is in design software. A program that lets a user drag a vertex around has to decide what happens to foldability, and the honest answer is that it is destroyed instantly and has to be restored by projecting back onto the constraint surface. Every interactive origami design tool is, underneath, a machine for staying on a measure-zero set while its user pushes it off.

That is a decent one-line description of the whole subject.

It also explains an experience that most people who try to design a crease pattern by drawing one have had. The drawing looks right, the angles were measured with a protractor, and the paper refuses. The instinct is to blame the folding, and the folding is rarely the problem: a protractor resolves about half a degree, the condition needs zero, and the gap between those two is where the model went. Designing by drawing does not work in this subject, and the reason is a statement about dimensions rather than about anybody’s care.

The complementary experience is worth naming too, since it is why the illusion survives. Draw a symmetric pattern — a rosette, a grid, anything with a mirror through every vertex — and it very often folds, because symmetry forces the alternating sums to be equal without anybody solving anything. Symmetry is the amateur’s accidental route onto the surface, and it is a narrow one: it produces the handful of patterns everybody rediscovers and nothing beyond them.

Where the ladder goes next #

Two directions, and they are opposite in spirit.

Outward: if foldability is this fragile, what does the edge of the sheet do to it? A vertex on the boundary carries no condition at all, so cutting a patch out of a pattern removes equations — and a small enough patch has almost none left, which makes it foldable for reasons that have nothing to do with the pattern.

Inward: the conditions are exact, and a real fold is not. An error of a fraction of a degree does not stay a fraction of a degree, because a folded position is a composition of reflections and every fold after the error carries it. The set is thin, and the ways of missing it are not symmetric.