Where the lemma says nothing

The big-little-big lemma is one sentence long and every word of it has been paid for. A sector strictly smaller than both its neighbours must have creases of opposite assignment on either side of it, because the paper in it has to escape somewhere and the only way out is between two creases that go opposite ways.

The word strictly is not decoration. It is the hypothesis, and it fails at the vertices this subject was built on.

The vertices everybody folds #

The preliminary base is where most people meet this subject, usually before they meet any of its mathematics. Fold a square in half both ways, fold it in half both diagonals, and collapse. The centre of that pattern is an interior vertex with eight creases meeting at it, and its eight sectors are all forty-five degrees.

The waterbomb base is the same lines with the letters exchanged. The bird base and the frog base contain the same vertex. A folder who has never heard of a sector angle has nevertheless spent a great deal of time at vertices where every sector is equal to every other, because equal sectors are what folding a square in half repeatedly produces.

At such a vertex there is no strictly smallest sector. There is no sector smaller than both of its neighbours, because every sector is exactly the same size as both of its neighbours. The lemma applies to nothing and forbids nothing.

What the silence is worth, counted #

The lemma’s job is to cut down the markings, so the price of its silence can be measured by counting the markings that survive.

Take a four-crease vertex with sectors a, b, 180° − a, 180° − b, which is the general flat-foldable four-crease vertex: Kawasaki forces opposite sectors to be supplementary and leaves two free numbers. Enumerate all sixteen markings and test each against developability, Kawasaki, Maekawa and the lemma.

With a and b different, four markings survive. With a and b equal — which puts the two smallest sectors adjacent and equal — eight survive. Not four and a bit; eight, exactly, and at every value of the shared angle from thirty degrees to eighty-nine and nine tenths.

Everything in this essay is a count of markings that satisfy the four conditions, which is what the lemma is a condition of. That is the right count for the question being asked here, and it is not the same as the number of markings that fold: at the tied vertex, six of the eight have a flat folded state and two do not, which is a separate discovery about what the conditions decide and is made in its own essay. The step from four to eight is unaffected — the untied vertex’s four all fold — and every sentence below is about the conditions rather than about the paper.

And at the base itself the factor is seven #

Four against eight is the four-crease case, and the vertex this essay opened with has eight creases — where the same count is worth doing, because the silence is much more expensive there.

At a degree-eight vertex with all its sectors distinct, the reduction is forced at every step and the surviving markings number $2^{4} = 16$ of 256, which is 6.25%.

At the preliminary base’s vertex every sector is forty-five degrees, so the lemma is silent at all eight positions at once and the conditions reduce to Maekawa alone. That admits the splits differing by two:

$$\binom{8}{3} + \binom{8}{5} = 112 \ \text{of}\ 256, \quad\text{or } 43.75\%.$$

A factor of seven rather than a factor of two, and nearly half of all markings surviving where a generic vertex of the same degree leaves one in sixteen.

So the doubling measured above is the mildest instance of the effect, not a representative one. The lemma removes one constraint per strictly-smallest sector, a vertex of higher degree has more of them to lose, and a vertex of equal sectors loses every one at once.

Which is worth carrying because of where those vertices are. Halving a square produces equal sectors by construction, so the vertices with the most markings and the least constraint are precisely the ones at the centre of the first pattern anybody folds.

A step function of the angles #

This is the part that is worth being careful about, because the conditions themselves look so thoroughly continuous.

Developability is a sum of angles equalling a full turn. Kawasaki is two alternating sums being equal. Both are equations, both move smoothly as the angles move, and both have solution sets that are surfaces in the space of vertices. Nothing about either suggests that anything discontinuous is going on.

Maekawa is a statement about counts and it does not involve the angles at all. And the lemma is a statement about angles whose hypothesis is an inequality — and an inequality is exactly the kind of thing that switches on and off.

So the number of markings a vertex admits is a step function of its angles, and the step sits on the set where two sectors are equal. That set has measure zero. Almost every pattern fails to fold at all for a similar reason and the arithmetic there runs the same way: the interesting configurations are the rare ones, and the rare ones are the ones people draw.

There is a second way of putting the same thing that makes the discontinuity feel less like an artefact. Consider a sequence of vertices approaching a tie — the smaller sector at 89.9°, then 89.99°, then 89.999°, with its neighbour fixed at 90°. Every one of them admits four markings. The limit admits eight. So the count is not merely discontinuous; it is lower semicontinuous in the wrong direction, in the sense that the limit has strictly more than any of the terms approaching it. Four of the eight markings at the tie exist only there.

Which pair of creases the lemma was going to constrain #

The count is a summary. The mechanism is worth having in the open, because it says which markings the tie lets back in.

At a vertex with a strictly smallest sector, the lemma names two creases — the ones on either side of that sector — and forbids them from carrying the same letter. Half the markings of those two creases go, and since the rest of the vertex’s markings are already pinned down by Maekawa, half the vertex’s surviving markings go with them.

At a tie there are two candidate sectors of equal smallest size, and each of them wants a demand made about a different pair of creases. The lemma refuses to make either, and it is right to refuse: the argument behind it is that the paper in the small sector has nowhere to go except between two creases that turn opposite ways, and when the neighbour is exactly as small the paper has somewhere else to go — into the neighbour.

It is worth writing out which four markings come back. At an untied vertex the surviving four are: the odd crease may be any of the four, and Maekawa’s three-to-one split then fixes the rest — except that the lemma forbids the odd crease from being either of the two bounding the smallest sector. Two positions for the odd crease, two global inversions, four markings. Untie nothing and the odd crease may sit at any of the four positions, which doubles it.

So the extra markings are not obscure ones. They are the ones in which the crease that goes the other way is one of the two the lemma would have ruled out — and in a preliminary base, which of the eight creases is the odd one out is precisely the choice a folder is making when they decide which way the base collapses.

That is the honest form of the statement. The lemma is not wrong at a tie; it is inapplicable, which is a different thing, and the underlying stacking rules it is a shortcut for are still there and still have to be checked by other means.

The ties are not an accident #

It would be easy to treat this as a curiosity about a measure-zero set. It is not, because of how origami patterns are made.

A folder produces angles by halving. Fold a square edge to edge and the angle is ninety degrees; fold corner to corner and it is forty-five; bisect again and it is twenty-two and a half. Every one of those operations produces two equal angles by construction, because bisecting is what the second and third axioms do and a bisector makes two of something.

So the vertices a folder reaches are systematically degenerate. A vertex whose sectors are all distinct is a vertex somebody had to work for — by an exact-division construction, or by starting from a rectangle that is not a square, or by deliberately choosing an angle that no sequence of halvings produces.

What the extra markings are for #

Eight markings rather than four is not merely a larger number; the extra four are usable, and folders use them.

The preliminary and waterbomb bases are the standing example. They have the same crease lines and different letters, and both fold flat, and they fold into visibly different objects — one closes into a point and the other opens into a bowl. That is two markings of one vertex being genuinely different folds, and it is possible because the vertex is degenerate enough to permit both.

There is a further consequence worth stating, because it turns the count into something a folder can feel. A vertex with four markings has, in effect, two decisions available once the sheet is folded: which crease is the odd one out, and which side of the paper ends up outermost. A vertex with eight has three. Every extra factor of two is a fork in what the same set of lines can become, and the standard bases have been standard for a century largely because they fork so freely.

A folder would put this the other way round: the standard bases are standard because they are versatile, and they are versatile because their vertices leave choices open. The lemma’s silence is the formal statement of that versatility.

What the tie is worth at every degree #

The count of eight is for a four-crease vertex whose two smallest sectors are equal, and the same arithmetic runs at any degree — which answers the question the closing section leaves open about how the doubling behaves as creases are added.

Maekawa admits the markings whose two counts differ by two, so at degree $d$ it admits $\binom{d}{d/2-1} + \binom{d}{d/2+1}$: eight at degree four, thirty at six, a hundred and twelve at eight. The last is the preliminary base’s own vertex, and it is the number this collection quotes for that pattern elsewhere.

Now break the tie. The lemma names the two creases flanking the strictly smallest sector and forbids them from agreeing, and among markings with $m$ mountains the ones where those two disagree number $2\binom{d-2}{m-1}$. Summing over the two admissible values of $m$: four of eight at degree four, sixteen of thirty at six, sixty of a hundred and twelve at eight.

So the tie is worth a factor of 2.00 at degree four, 1.88 at six and 1.87 at eight — and carrying the arithmetic further gives 1.88 at ten and 1.89 at twelve, drifting slowly back toward two. The factor never leaves the range between 1.86 and 2 at any degree.

Which separates degeneracy from degree #

That settles the decomposition the essay asks for. The count at a vertex grows steeply with the degree — eight, thirty, a hundred and twelve, four hundred and twenty — and that growth is Maekawa’s binomial coefficients doing what binomial coefficients do.

The tie contributes a constant factor of very nearly two, at every degree, and nothing more. It does not compound, it does not grow, and it does not fade: a tied vertex of any degree admits about twice what the same vertex admits with its smallest sector made strictly smallest.

So the preliminary base’s hundred and twelve is not evidence that degenerate vertices become more permissive as they grow. It is a large number because eight creases is a lot of creases, and the tie is worth the same one bit there as it is at degree four. Degree buys the exponential and the tie buys a factor of two, and separating them is what turns a striking count into two ordinary ones.

Where the model stops #

Eight and four are counts of markings, not of folded objects. A marking that passes every local condition may still fail to stack, and the number of ways a marked pattern can lie flat is a further question that these counts do not touch.

The tie is exact and paper is not. A real vertex folded from real paper has angles that are equal to within whatever the folder achieved, and a vertex a hundredth of a degree from a tie is, to this analysis, an untied vertex admitting four markings. What that means physically is the question of how much slack the material has, and the answer there was that paper’s tolerance rescues perturbed patterns and does nothing for drawn ones. The same applies in reverse here: a drawn tie is a tie and a folded one is a near-tie, and they are not the same object.

The count is of what the conditions admit, not of what folds. At a tied four-crease vertex two of the eight have no flat folded state at all, so the number of objects a folder can reach is six rather than eight. The doubling is real and the destination is not what it appears; the reduction that decides the difference is a later rung.

Only the four-crease case is counted. Higher degrees have more ways for sectors to tie — two equal, three equal, all equal — and the arithmetic of which demands the lemma makes at each is not done here. The eight-crease vertex of the preliminary base is the extreme case and is quoted rather than analysed.

Nothing here decides whether a sheet folds. Every condition used is local, and deciding the global question is NP-hard. What is counted is what the local conditions permit, which is an upper bound on what folds.

What a folder should take from it #

Two things, and they pull in opposite directions.

The first is a warning about software. A program that decides flat-foldability by applying the lemma will, at a tied vertex, correctly decline to apply it and then have to fall back on the stacking rules — which are more expensive and which some implementations quietly skip, because the lemma is presented in the literature as the condition on a vertex and the fallback is presented as a detail. A pattern built by halving is a pattern made entirely of the case the shortcut does not cover.

The second is an invitation. The extra markings at a degenerate vertex are real folds, and the two bases that share the preliminary base’s lines are the famous instance of a family that is much larger than two. A designer who wants a vertex to be able to collapse several ways should make its sectors equal on purpose, and that is a design rule derived from an inequality failing.

Who noticed, and when #

The big-little-big lemma is usually credited to the collection of results that Justin, Kawasaki and Hull assembled between the late nineteen-seventies and the nineteen-nineties, and it has always been stated with the strict inequality in it. Nobody has ever got it wrong.

What has happened instead is that the degenerate case has been treated as an edge case in a proof rather than as the main case in practice, which is a reasonable division of labour for a mathematician and a misleading one for a folder. The literature’s examples are generic vertices because generic vertices are what the theorems are about; the paper on the table is a preliminary base, because that is what people fold.

Where the ladder goes next #

The obvious next question is what the lengths do, since the lengths of the creases are free as far as flat-foldability is concerned. That freedom is exact at a generic vertex; whether it survives a tie, when the stacking rules rather than the lemma are doing the work, is not settled by anything above.

The other direction is a count. Eight markings at a tied four-crease vertex, four at an untied one — and at the eight-crease vertex of the preliminary base the count is much larger and has never been written down here. The question of how the count grows with the degree, and how much of that growth is degeneracy rather than degree, is a piece of arithmetic waiting to be done.