A near miss is nearly as rare

The genericity result is one of the cleanest things this site can say. Kawasaki’s condition is one equation per interior vertex; a drawing satisfies an equation with probability zero; therefore a crease pattern that folds flat is a coincidence, and every pattern here folds because it was constructed to.

The result is correct and it is usually where the argument stops. Stopping there leaves an obvious escape route open, and the route deserves closing, because it is the one a reader with paper in their hands will reach for.

The escape route #

The objection runs: measure zero is a statement about exact equality, and nothing physical is exact. A crease has a width, paper stretches slightly, and a fold made by hand lands within a degree of where it was meant to. So the theorem forbids a set of patterns that a real sheet cannot distinguish from the ones it permits, and the rarity is an artefact of demanding infinite precision.

It is a good objection. It is also testable, and the test is straightforward once the right quantity is identified.

Failing is a distance #

Kawasaki’s condition at a four-crease vertex says the alternating sum of the sectors is zero. Call the alternating sum R. A vertex with R = 0 folds; a vertex with R ≠ 0 does not; and R is a number, so the question “how badly does it fail” has an answer.

Better than an answer, it has a repair. Adjust the four sectors by −R/4, +R/4, −R/4, +R/4 and the alternating sum becomes exactly zero while the four still sum to a full turn. So every random vertex sits at a known distance from a flat-foldable one, that distance is R/4 per sector, and no smaller equal adjustment will do — moving each sector by less than R/4 cannot change an alternating sum by R.

That converts a yes-or-no question into a measurement, and the measurement is the point of this rung.

The answer is thirty-one degrees #

Over forty thousand random four-crease vertices — three cuts of a circle, sorted, which is the same generator the rung below uses — the median distance is 31.3° per sector and the mean is 33.9°.

That is not a near miss. It is most of a right angle.

The tail is what the objection needed and it is thin: 0.8% of random vertices are within half a degree per sector of folding, 1.6% within one degree, 3.2% within two, 8.2% within five, 16.4% within ten. Even at twenty degrees a sector — an error no folder would call an error, it is a visible misalignment — only a third of random vertices come inside.

So the flat-foldable set is not merely thin. Its neighbourhood is thin too. Fattening the condition by a tolerance a real sheet might have converts “probability zero” into “one chance in sixty”, which is a different number and the same conclusion.

What a perturbed pattern does instead #

The comparison that makes the result useful is with a pattern that was flat-foldable and has been disturbed.

Take a Miura fold and move every vertex by a small random amount. At a displacement of a fortieth of a cell — 0.04, a substantial visible distortion — the mean distance to foldable is 0.73° per sector. At a hundredth of a cell it is 0.18°, at a five-hundredth 0.036°.

So the two populations are separated by nearly two orders of magnitude. A disturbed working pattern is a fraction of a degree from folding. A random pattern is thirty degrees from it. Paper’s forgiveness lands squarely on the first and nowhere near the second.

That is the useful form of the whole genericity argument. Tolerance does not make flat-foldability common; it makes it robust. A pattern that was designed to fold will still fold when it is drawn imprecisely, printed imprecisely and folded by hand, and this is why the subject works at all in practice. A pattern that was not designed to fold gains nothing from the same forgiveness.

What a degree of slack is worth in practice #

It is worth putting a number on the tolerance the objection appeals to, since the whole argument is a comparison of two sizes.

A crease made by hand from a drawn line lands within perhaps half a degree of where it was meant to on a sheet the size of a page — that is the accuracy of aligning two edges by eye, and it is what the error analysis of a folding sequence takes as its input. A crease is not a line but a region with a radius, and the angular slack that radius provides at a vertex is of the same order. Wet-folding, which deliberately exploits the material’s give, buys a few percent of strain and not a great deal of angle.

So the tolerance is one or two degrees and the median random vertex is thirty-one. The comparison is not close, and it does not become close for any material anybody folds. A sheet forgiving enough to fold a random vertex flat would have to absorb a third of a right angle at every crossing, which is not a sheet — it is a cloth, and what a cloth does with a curved form is a different subject with a different mechanism.

Higher degrees do not help either #

Everything above is about degree-four vertices, which is the overwhelming majority of what patterns actually carry. The natural hope is that a vertex with more creases has more freedom and so sits closer to foldable.

It has more freedom and it also has a longer sum. Kawasaki at a degree-2k vertex asks two alternating sums of k sectors each to be equal, and the repair distributes the residual over 2k sectors instead of four — so the per-sector distance falls, and the number of sectors that have to move rises to match. The total movement is unchanged, and the total is what a sheet has to absorb.

Why the distances are so large #

The reason is worth a paragraph because it is not obvious that random vertices should be so far away.

The four sectors of a random vertex are three cuts of a circle, sorted — a uniform point on the simplex — and the alternating sum of such a quadruple is a difference of two sums, each of which is a random half of a full turn. The distribution of that difference has a spread on the order of the full turn itself. There is nothing pulling it toward zero, so its typical size is a substantial fraction of 360°, and a quarter of that is a substantial fraction of 90°.

Put the other way round: the condition asks for two independent-looking sums to be exactly equal, and two quantities that have no reason to be equal are typically not nearly equal either. Near-equality is a coincidence of nearly the same kind as equality.

The distribution has a closed form, and it agrees #

The forty thousand vertices are a measurement, and a measurement of a quantity this simple ought to be checkable against arithmetic. It is, exactly, and the arithmetic is worth having because it turns the headline number from a sample statistic into a constant.

The generator cuts a circle at three uniform points and sorts, so the four sectors are the spacings of three uniform points on a circle of $360°$. The alternating sum $R$ is $(a_1 + a_3) - (a_2 + a_4)$, and since the four sum to a full turn, $R = 2S - 360°$ where $S$ is the total of two of them. That total, as a fraction of the turn, has the density $6x(1-x)$ on the unit interval — the symmetric distribution that two spacings out of four always have.

So the distance to foldable, which is $|R|/4$, is $90°$ times $|2X - 1|$ for $X$ with that density, and both the numbers this essay quotes come out of it.

The mean of $|2X-1|$ against $6x(1-x)$ is $3/8$, so the mean distance is $90° \times 3/8 = 33.75°$. The measurement reported 33.9°, which is that number with forty thousand samples’ worth of noise on it.

The median needs the root of a cubic. The chance that the distance is under $d$ works out as $(3t - t^3)/2$ with $t = d/90°$, and setting that to a half gives $t^3 - 3t + 1 = 0$, whose root between nought and one is $0.34730$. Multiply by $90°$: 31.26°. The measurement reported 31.3°.

One in sixty, per degree #

The same expression gives the tail directly, and it gives it a shape worth remembering.

For a small tolerance the cubic term is negligible and the chance of landing within $d$ degrees per sector is very close to $3t/2$, which is $d/60$. A degree of slack buys one vertex in sixty, and the buying is linear — two degrees buys one in thirty, five degrees one in twelve.

Every figure in the tail above is that formula: 0.83 per cent at half a degree, 1.67 at one, 3.33 at two, 8.32 at five, 16.60 at ten, 32.78 at twenty. The sampled values are 0.8, 1.6, 3.2, 8.2, 16.4 and about a third. The curve in the first figure is not a fit to the samples; the samples are a check on the curve.

That linearity is the sharpest way to state what tolerance does and does not buy. There is no threshold, no knee, and no width of forgiveness at which random vertices suddenly start folding — the share grows in proportion to the slack, from a base of one in sixty per degree, all the way out to where the cubic starts to bend it down. A material ten times more forgiving than paper would admit ten times as many random vertices, which is still one in six.

And it explains why the median is where it is rather than anywhere else. The distance is a fixed multiple of a quantity whose distribution is symmetric, spread across its whole range and heaviest in the middle — so the typical value is a substantial fraction of the largest possible one, and the largest possible one is $90°$. Thirty-one degrees is not a surprising answer to that question. It is close to the only answer such a distribution could give.

Where the model stops #

This is one vertex. A pattern has many, and each has its own distance. Repairing a pattern is not repairing its vertices independently — moving a sector at one vertex moves a crease that other vertices share — so the distance from a random pattern to a flat-foldable one is not the largest of its vertex distances and is not computed here at all. It is a harder question and a more interesting one.

The repair is equal adjustment and not the nearest repair. Moving all four sectors by R/4 is the smallest change if the four are weighted equally. A different measure — move as few sectors as possible, or move the largest sector only — gives a different distance, and the honest statement is that no equal adjustment smaller than R/4 works.

Sectors are not the only thing that can move. A real pattern is repaired by moving vertices, and moving a vertex changes the sectors at every vertex its creases reach. The jitter figure works that way round and the random figure does not, which is why the two are compared as populations rather than matched case by case.

Nothing here is about layer order. Every distance above is about angles, and a pattern whose angles are perfect can still fail to fold for reasons no vertex condition reaches.

Why this changes how the result should be quoted #

“Flat-foldable patterns have measure zero” is true and it invites a wrong picture: a thin set, densely surrounded by patterns that nearly work, so that any pattern is a small nudge away from folding.

The measurement says the opposite. The set is thin and isolated: a random pattern is not a small nudge away, it is a large one. What sits near a flat-foldable pattern is other flat-foldable patterns and their small perturbations, which is a neighbourhood the construction methods on this site never leave.

That has a practical consequence for anybody trying to design by adjustment. Taking an arbitrary crease pattern and nudging it toward foldability is not a small optimisation; the target is thirty degrees away in a space where every vertex is pulling in its own direction. Every design method the subject has — circle packing, molecules, the twist constructions, the axioms — starts from a construction that satisfies the condition by build and never asks a pattern to be repaired into satisfying it. That is not a stylistic preference. It is the only approach the geometry leaves.

The same measurement on a whole pattern #

One thing the vertex-level measurement can be pushed to say about patterns, without solving the hard version, is what happens when a working pattern is disturbed by a known amount.

The jitter figure does exactly that, and its shape is the informative part: the mean residual is proportional to the displacement — 0.145°, 0.364°, 0.727°, 1.453°, 2.900° at displacements of 0.002, 0.005, 0.01, 0.02 and 0.04 of a cell. Doubling the disturbance doubles the residual, with no threshold and no plateau.

That linearity is the reason a designed pattern is usable. It also means there is no safe distance: a pattern disturbed by a hundredth of a cell fails Kawasaki by 0.18° per sector, which is a failure, and it folds anyway because the paper absorbs 0.18°. The condition is violated everywhere in practice and satisfied nowhere, and what makes the subject work is not that real patterns satisfy the theorem but that they miss it by less than the material notices.

What the objection gets right #

One part of the escape route survives and it is worth conceding clearly.

Real paper does absorb error, and this is why folding works. A sheet folded by hand from a printed pattern accumulates error at every step, the creases land a degree or two from where the drawing said, and the model still folds. The forgiveness is real and it is doing exactly the job the objection claims.

What it cannot do is manufacture flat-foldability where none was designed. The forgiveness is a small ball around a working pattern, and the measurement above is a statement about how much of the space that ball covers: essentially none of it. Both halves are true at once — the theorem is not fragile in practice, and it is not weak in principle either.

The number that should be quoted #

If the genericity result is to be carried around as a single fact, this is a better one to carry than “measure zero”.

A random four-crease vertex is a median of 31° per sector from folding flat, and fewer than one in fifty is within a degree. It says the same thing, it survives contact with a physical sheet, and it forecloses the objection rather than inviting it. The measure-zero statement is stronger mathematically and weaker rhetorically, which is an unusual combination and worth noticing when choosing which to state.

Where the ladder goes next #

The obvious continuation is the pattern-level distance. Given a crease pattern that does not fold, what is the smallest movement of its vertices that makes every one of its vertex conditions hold at once? That is a constrained optimisation with the pattern’s own connectivity in it, the answer is not the sum of anything, and it would say what “nearly foldable” means for an object rather than for a point.

The other direction is the population that has neither of these distributions. A crumpled sheet’s vertices sit at distance exactly zero, in numbers, having been produced by folding rather than by drawing — so there are at least three populations of vertex in this subject and their distances are 0, a fraction of a degree, and thirty-one degrees. What produced a vertex turns out to matter far more than what it looks like.