How rare a band that folds is

Almost no crease pattern folds flat is one of the collection’s standing claims and it has been made two ways.

The first is a count. Draw a pattern, enumerate the assignments of mountain and valley to its creases, and see how many of them satisfy the vertex conditions everywhere. The answer is a small fraction of $2^n$ and it shrinks fast. That is rarity among finitely many discrete choices, and it is measured by dividing one whole number by another.

The second is a perturbation. Take a pattern that folds, move a vertex by a hair, and ask whether it still does. Usually it does not, because the conditions are equations and an equation broken by a hair stays broken. That is rarity in a continuous space, and it is measured by observing that the good set has no volume.

A glued band is the cleanest instance of the second kind the collection has, and it is clean because the whole condition is one equation in three numbers, with nothing discrete about it at all.

The equation #

The crease angles of a three-crease Möbius band have to satisfy

$$\phi_1 - \phi_2 + \phi_3 \equiv 0 \pmod{180^\circ}.$$

That is one equation among three real numbers. Its solution set is a surface in a three-dimensional space — two-dimensional, and therefore of no volume in that space.

The consequence is not that folding bands are uncommon. It is that if a person picks three crease angles by any process that does not know about the equation, the probability of landing on a band that folds is nought.

Two spaces, and which one the claim is about #

There is a trap here that is worth stepping into deliberately, because the picture above appears to say the opposite of the claim.

The picture is a plot of the first two angles, and it is dense: given any two, a third exists. So if the question is can this pair be completed, the answer is almost always yes.

The claim is about triples. A triple is a point in a three-dimensional space; the admissible triples form a surface; a surface has no volume. So if the question is is this triple admissible, the answer is almost always no.

Both are true and they are answers to different questions. The one a folder meets depends entirely on whether the three angles are chosen together or two-then-one.

That distinction is more general than it looks. Every condition in this subject that determines one quantity from the others has the same double aspect, and reporting the wrong one makes a constraint look either vacuous or impossible.

What a near miss costs #

A set with no volume raises an obvious practical question. If nothing exactly satisfies the equation, and paper is not exact anyway, what does a nearly-admissible band do?

The composition of the reflections misses the gluing map by an amount, and the amount is proportional to how far the alternating sum is from its target. A triple thirty degrees out misses by a rotation of thirty degrees, which on a strip of any reasonable length is a mismatch of a substantial fraction of the width — the two ends of the paper arrive at each other pointing in visibly different directions.

Real paper absorbs a little of that. A strip a tenth of a degree out will close, because paper stretches slightly, tape has slack and creases have width. A strip five degrees out will not, and the failure appears as a permanent buckle rather than as a fold that nearly works.

So the practical version of the claim is that the admissible set has no volume and a tolerance around it does — which is exactly the situation every construction in this subject is in, and the tolerance is what a folder is actually working inside.

The discrete rarity, for comparison #

The collection’s other kind of rarity is worth putting beside this one, because they behave differently under every operation.

On a loop of paper with radial creases, the condition is a parity and the admissible set is half of everything. That is not rare at all. Nothing is measure zero, nothing is delicate, and moving a crease changes nothing: the count is what matters and the count is stable under any perturbation.

Two sheets, both glued, both with one loop, and the rarity of a foldable drawing on them is one in two on one and nought on the other. The difference is that a loop of paper’s creases are concurrent — they all meet the hole — so the composition is a rotation whose amount the identification can absorb, and the geometry has nothing left to say. A band’s creases are not concurrent, so the geometry has a condition, and a condition on continuous data is an equation.

The three stages, and how much each one refuses #

Laying the conditions out in order shows where the rarity actually enters.

The parity refuses half the crease counts. That is discrete rarity: one in two, stable, and the same at every count.

The alternating sum refuses all but a measure-zero set of angle triples. That is continuous rarity, and it is where the almost never comes from.

The fitting refuses all lengths below a threshold and admits everything above. That is not rarity at all — it is a half-line, which has plenty of room in it.

So the phrase almost no band folds is carried entirely by the middle stage, and the middle stage is invisible to any measurement that counts things.

Measure zero, said carefully #

The phrase carries a good deal of weight above and it is worth being precise about what it does and does not assert, because it is a phrase that invites over-reading.

A subset of a space has measure zero when it can be covered by regions of arbitrarily small total volume. A surface inside a three-dimensional space has that property: cover it with a thin slab and the slab’s volume goes to nought as the slab thins. So does a curve, and so does a point.

What follows from it is a statement about probability with respect to a chosen distribution. If three angles are drawn from any distribution that spreads its weight smoothly over the space of triples — uniform, normal, whatever — the probability of landing on the surface is nought. That is a mathematical fact and not an approximation.

What does not follow is that the surface is small in any everyday sense. It contains infinitely many points, it is two-dimensional, it is connected, and it passes through every neighbourhood of the space. A person walking the space at random will never step on it and can reach it from anywhere in one deliberate step.

And what also does not follow is that a physical band chosen carelessly will not fold. Physical angles are not real numbers; they are real numbers with a tolerance, and a tolerance turns the surface into a slab. Somebody cutting by eye is sampling from a distribution whose support is a slab around whatever they were aiming at, and if they were aiming at sixty and a hundred and twenty degrees, most of that slab folds.

The distribution nobody has, and the one everybody uses #

There is a second layer of care to take here, and it is the one that makes measure-zero arguments awkward in practice.

Probability nought is a statement relative to a distribution over the space of triples. Nothing in the world hands out crease angles according to such a distribution, and asking what fraction of crease patterns fold flat is not a well-posed question until somebody says which patterns are being drawn from and how.

The collection has met this before. Four different ways of drawing a pattern give four populations with different statistics, and choosing between them changes what typical means. The measure-zero argument is stronger than any of those, because it holds for every distribution that is smooth: no choice of population changes the answer from nought.

That robustness is the argument’s main virtue. It is also its main limitation: an answer that is nought for every smooth distribution says nothing about which nearly-admissible bands are common, which is the question a folder actually has.

A tolerance, and what it is worth #

Since a slab is what a person is really working in, it is worth knowing how thick it is.

Suppose the angles are each out by a degree. The alternating sum is then out by up to three degrees, and the composition’s linear part is a reflection three degrees away from the one the gluing wants. On a strip of width one, closing the band forces the ends together anyway, and the paper takes up the discrepancy by twisting slightly along its length — which real paper does very willingly over thirty centimetres.

Out by five degrees each, and the alternating sum can be fifteen out, and the twist required is visible: the band lies flat with a ripple, and one of the three panels does not sit down properly.

Out by fifteen, and it buckles.

Those are not computed numbers and they are not offered as ones — they are what happens with ordinary paper, and the reason to state them is that the mathematics gives a threshold of exactly nought and the material gives a threshold of a few degrees. Both facts are true and the second is the one that governs whether an evening’s folding works.

The near miss, and why it is nearly as rare #

There is a version of this argument that has been made about ordinary patterns and it applies here with a twist worth noting.

A near miss is nearly as rare as an exact hit is the claim that the set of patterns close to folding is also small — not measure zero, but small in a way that scales badly. For a band the corresponding statement is easy to check: the set of triples whose alternating sum is within $\epsilon$ of a multiple of a straight angle is a slab of thickness proportional to $\epsilon$, and its share of the whole space is proportional to $\epsilon$ as well.

So near misses are linearly rare rather than measure zero, which is the mildest possible failure. A band is an unusually forgiving object in this respect: halve the tolerance and halve the chance, with no exponent anywhere.

The reason is that there is exactly one condition. Patterns with $v$ vertices have conditions at all of them, the tolerances multiply, and the share of near misses falls like $\epsilon^v$ — which at any realistic vertex count is indistinguishable from nought. A band’s single loop is the whole of why its arithmetic is friendly.

Rarity does not mean hard to find #

A set of measure zero can be perfectly easy to construct, and this one is.

The equation determines the third angle from the first two. So a person who wants a band that folds picks two angles freely, computes the third, solves two linear equations for the positions, and cuts a strip above the threshold length. Every step is arithmetic, none of it is a search, and the result folds.

That is worth separating from the rarity, because the two are usually conflated. Rare is a statement about what a random choice produces. Hard is a statement about what a deliberate one costs. For flat-foldability in general the two coincide and the problem is hard; for a band they come apart completely, and the reason is that the condition is an equation rather than a constraint satisfaction problem.

Where the same distinction bites elsewhere #

The two rarities recur throughout the subject and it is useful to know which is which.

Discrete: how many assignments of a pattern fold, how many stackings a folded state has, how many rules a corrugation admits. All counted, all stable under small perturbations of the geometry, all reported as a proportion.

Continuous: whether a vertex satisfies Kawasaki, whether a pattern’s panels close up, whether a twist tessellation’s distances are consistent. All equations, all measure zero, all destroyed by a perturbation and all approached through a tolerance instead.

The subject’s headline claim — flat is rare — is usually argued from the first kind, because a proportion is a number one can print. The second kind is the stronger claim and the harder one to state, since its content is that a probability is exactly nought rather than merely small.

A band is a good place to see it because there is nothing else in the way: no letters to count, no vertices to check, no layers to order. One equation, three numbers, and no volume.

Why the loop of paper is the misleading case #

It is worth spending a paragraph on why the annulus behaves so differently, because it is the case a reader is most likely to have in mind.

On a loop of paper with radial creases, every crease passes through the same place — the hole is at the centre and the creases are rays out of it. Composing reflections in concurrent lines gives a rotation about the common point, and the amount of that rotation is twice the alternating sum of the angles between them.

The identification a loop of paper needs is also a rotation about that point: going once round the annulus and coming back is a rotation by a full turn, which is the identity. So the condition reads twice the alternating sum is a multiple of a full turn, and the sectors between the creases add to a full turn by construction, which pins the condition to a parity and leaves the individual angles free.

That is why an annulus with four rays folds however the four rays are placed, and why the count is the whole answer there. The geometry has been used up by the concurrency.

A band’s creases are not concurrent and cannot be made so — they cross the strip, and two of them meeting inside it would not divide it into panels. So the composition can be a translation, a rotation or a glide depending on the angles, the identification is one specific glide, and matching them is a condition with content.

What the reader should take from the two rarities #

The subject says flat is rare and means two different things by it, and a reader who holds only one of them will be surprised regularly.

Discrete rarity is a proportion of finitely many things. It is what a table can show, it is stable under small changes, and it is usually what a figure in this collection is reporting.

Continuous rarity is an equation’s solution set having no volume. No table shows it, it is destroyed by any perturbation, and it is why the subject is about constructions rather than about searches: the good objects are found by solving rather than by looking.

A band has one of each, stacked. The parity is the first, the alternating sum is the second, and the second is where the almost in almost no band folds is doing its work.

The other rarity results, placed #

Three of this collection’s results are about how rare a folding pattern is, and it is worth saying which kind each is.

Almost every pattern fails is continuous rarity: perturb a folding pattern and it stops folding, because the conditions are equations.

How many assignments fold is discrete rarity: a proportion of finitely many letterings, counted.

A near miss is nearly as rare is about the tolerance round the continuous case, and it is the one that decides whether real paper can find the good set at all.

The band has all three in miniature, which is why it is a useful object to hold them against.

What the equation does not make rare #

Two things survive the measure-zero verdict and both matter.

The existence of foldable bands is not in question. There are infinitely many, they form a two-dimensional family, and every one of them can be built to order.

The physical realisability is not in question either. Paper has tolerance, and a tolerance around a surface is a slab with volume. Whether a person can fold one is a question about that slab, and the answer is that a band cut to within a degree or two of the equation folds without complaint.

What is rare is a band that a person did not aim at. That is the whole content of the claim, and it is worth stating in that form, because almost no band folds is otherwise easily heard as bands hardly ever fold, which would be a discouraging and false thing to say about an object anybody can make in two minutes.