Why the difference is two

Count the mountains and the valleys at any interior vertex of a crease pattern that folds flat. The difference is two. Not usually two, not two on average — exactly two, every time, at every such vertex, in every pattern anybody has ever folded.

A constant that specific is a sign that something is being counted rather than measured, and the something is a winding number.

The cross-section #

Take a flat-folded vertex and cut across it with a plane a little way out from the point, perpendicular to the sheet. What the cut reveals is a zig-zag: the paper doubling back on itself once at each crease.

That zig-zag is a closed path. It has to be, because the paper is continuous around the vertex — walk all the way round and the cross-section returns to where it began.

Now walk along it and keep track of direction. At each crease the path reverses: a mountain turns it one way through 180°, a valley the other way through 180°. Between creases it goes straight and turns not at all.

The walk closes, so the total turning is a whole number of full circles. For a simple cross-section that does not wind round more than once, the total is exactly $\pm 360°$.

$$180°\,M - 180°\,V = \pm 360° \quad\Longrightarrow\quad M - V = \pm 2.$$

That is the entire proof, and nothing in it mentions paper.

What makes the constant two #

The number comes from the turning being one full circle, and the half-turn per crease.

If a crease turned the path by some other amount the constant would change, and creases do not: a flat fold reverses direction completely, by definition of being flat. A fold to 90° would turn the path by 90°, and then the counts would be constrained differently — which is exactly why the theorem applies only to the fully flat state and says nothing about a partly folded one.

And if the cross-section wound round twice, the constant would be four. It does not, for a vertex in a sheet that folds flat without the paper passing through itself — which is a real assumption and is where the theorem quietly leans on something it does not check.

An immediate consequence: the degree is even #

Maekawa gives a parity result for free, and it is worth extracting.

$M + V$ is the total number of creases at the vertex, and $M - V = \pm 2$. Adding and subtracting, $M$ and $V$ are $(n \pm 2)/2$ — which are integers only if $n$ is even.

So every flat-foldable interior vertex has an even number of creases. A three-crease vertex is impossible, a five-crease vertex is impossible, and this can be seen without measuring a single angle.

Kawasaki’s condition gives the same parity result by a different route — an odd number of sectors cannot alternate consistently round a cycle. Two independent theorems agreeing on a corollary is mild evidence that both are right.

The smallest case #

A degree-four vertex has four creases, so Maekawa forces three of one kind and one of the other. That single fact does a surprising amount of work.

Three-and-one means there is a unique odd crease out — one that differs from its three neighbours. Which one it is is not free either: the big-little-big lemma forces it to bound the smallest sector.

So a flat-foldable degree-four vertex has two free angles and essentially no free assignment. The generator on this site builds them exactly that way: sectors $\alpha$, $\beta$, $\pi - \alpha$, $\pi - \beta$, then the odd crease placed at the smallest sector, and the result is checked.

The count, and what it says about rarity #

Maekawa is a strong filter, and it is possible to say how strong.

Of the $2^n$ assignments of $n$ creases, the number satisfying $|M - V| = 2$ is $\binom{n}{(n+2)/2} + \binom{n}{(n-2)/2}$. For $n = 4$ that is 8 of 16 — half. For $n = 8$ it is 112 of 256, or 44%. For $n = 20$ it is 335,920 of 1,048,576, or 32%.

The fraction falls slowly with degree at a single vertex, and it falls very fast across a pattern, because a pattern with $k$ interior vertices has to satisfy the condition at all of them and the creases are shared. That compounding is where the rarity comes from, not from any one vertex being hard to satisfy.

The share falls as one over the square root of the degree #

The sequence 50, 44, 32 has a closed form, and having it says how weak the filter is at a point and how strong it becomes across a pattern.

The admissible count is $2\binom{n}{n/2+1}$, which is the central binomial coefficient times $n/(n+2)$ and doubled. Using the standard approximation for the central coefficient, the share of all assignments that survive is

$$\frac{2}{2^{n}}\binom{n}{n/2+1} \;\approx\; \sqrt{\frac{8}{\pi n}}$$

At degree twenty that gives 0.32, against the exact 0.3204. At degree eighty it gives 0.18, and at degree two hundred, 0.11.

So the filter weakens as the degree grows, and it weakens slowly — halving the share takes a fourfold increase in degree. A vertex of any size lets through a substantial fraction of everything.

Which is why the compounding is the whole story #

Set that against what happens across a pattern and the contrast is the essay’s point with numbers on it.

At one vertex the survival share is $n^{-1/2}$ — algebraic, and gentle. Across a pattern, each interior vertex costs a factor of two: a pattern with $E$ creases and $V$ interior vertices of degree four has about $2^{E-V}$ admissible assignments out of $2^{E}$, so the share is $2^{-V}$.

Exponential in the vertex count, algebraic in the degree. A Miura with fifteen interior vertices admits one assignment in thirty-two thousand; a tessellation patch with a hundred and twenty-six admits one in $2^{126}$.

So the rarity is not produced by the theorem being strict. It is produced by the theorem being applied many times, and the two behaviours are qualitatively different: doubling a vertex’s degree barely moves the share, while adding one vertex halves it.

That also says where a designer’s difficulty is. Nothing is gained by avoiding high-degree vertices — a degree-eight vertex is only slightly more constrained than a degree-four one — and everything is decided by how many vertices the pattern has. A pattern’s assignment problem is hard because it is large, not because any part of it is tight, which is the same conclusion the search essays reach from the other direction and is unusual for a subject whose conditions are all local.

What the theorem does not see #

Maekawa counts. It does not look at anything else, and the list of what it ignores is instructive.

Angles. A vertex can have a perfect three-and-one assignment and sectors that fail Kawasaki completely. The two conditions are independent, which is why both are needed.

Which crease is which. Maekawa says three of one and one of the other; it does not say where the odd one goes. Putting it in the wrong place gives a vertex that satisfies Maekawa and cannot fold.

Layers. The winding argument assumes the cross-section does not pass through itself. It does not check that, and self-intersection is precisely what makes the global problem hard.

The boundary. A vertex on the edge of the sheet has creases with free ends, the cross-section is not closed, and the theorem simply does not apply.

The same argument elsewhere #

The structure of the proof — a closed walk, a fixed turn at each event, total turning a whole number of circles — is one of the most reusable arguments in geometry, and recognising it here is worth more than the theorem.

The exterior angles of any simple polygon sum to 360°, by the same walk. The winding number of a closed curve about a point is the same integer. The turning number of a smooth closed curve is that integer again, and the Whitney–Graustein theorem says it classifies such curves completely.

Maekawa’s theorem is that argument applied to a cross-section of paper. Once the cross-section is seen as a closed polygonal path, the result is not surprising at all — the work is all in noticing that the cross-section is a closed path, which takes about one drawing.

Justin, Maekawa, and the naming #

The attribution is worth a paragraph because it is characteristic of the field.

Jun Maekawa, a Japanese folder and physicist, stated the result in the 1980s in the Japanese origami literature. Jacques Justin, a French mathematician, had it independently at about the same time and published in a European origami newsletter. Neither venue was indexed anywhere a mathematician would look.

The result is now usually called Maekawa’s theorem and sometimes the Maekawa–Justin theorem, and the same pair of names attaches to Kawasaki’s condition with the order changed.

This is what happens to a field whose practitioners are enthusiasts and whose venues are club publications. The mathematics of origami was worked out substantially in newsletters between 1985 and 1995, and the process of getting it into journals took another decade.

Counting in the other direction #

The theorem is usually used as a filter — reject any assignment where the difference is not two. It can also be used constructively, and that is how the patterns on this site get their assignments.

Given the crease lines at a vertex, the assignments satisfying Maekawa are exactly the three-and-one splits, or five-and-three, or whatever the degree allows. Those are easy to enumerate: choose which creases are mountains, subject to the count. Filter that list by Kawasaki — which depends only on the angles and so either passes everything or nothing — and by big-little-big, and what remains is the set of assignments that fold.

That is the procedure the preliminary base on this site uses. The geometry is laid down, the search runs over all 256 assignments of the eight half-creases, and one of the 112 survivors is drawn. It is a small computation and it removes an entire class of error: nothing was remembered, so nothing could be misremembered.

The theorem holds for patterns nobody designed #

A pleasing consequence of the argument being about winding rather than about paper is that it applies to things that were never folded on purpose.

Crumple a sheet of paper and flatten it. The result is a crease pattern — a messy one, with many vertices — and it folds flat, because it just did. So Maekawa holds at every interior vertex of it. Count them and the difference is two, every time.

The same is true of a crushed drink can, of the buckling pattern in a compressed cylinder, of a rumpled shirt, and of a leaf emerging from a bud. None of those was designed and all of them satisfy a theorem that was written down in the 1980s.

That is the strongest possible evidence that the constant is structural. A rule that holds for deliberate folding might be a fact about folders; one that holds for crumpling is a fact about sheets.

Where the model stops #

Fully flat only. The half-turn per crease is what makes the constant two, and it holds only in the completely flat state. A rigidly folded intermediate position obeys no such rule.

No self-intersection. The winding argument assumes the cross-section is a simple closed path. Paper that would have to pass through itself breaks the assumption and the theorem does not notice.

Interior vertices. Boundary vertices are unconstrained, and a crease pattern’s edge carries freedom the theorems say nothing about.

Zero thickness. The cross-section is drawn as a path of zero width. A real one has a stack whose height is the layer count times the paper thickness, and the turns are radii rather than points.

The figure slows the walk down. The cross-section drawn here has four creases and wide spacing so the turns are legible. A real vertex’s cross-section at a millimetre from the point is a compressed zig-zag a few paper-thicknesses across, and nothing about it is easy to see.

Two, and the exceptions that prove it #

The constant survives every case anybody has tried, and the near-exceptions are informative.

A vertex on the boundary has creases with free ends, so the cross-section is an open path rather than a closed one, the turning need not be a whole circle, and the theorem does not apply. Boundary vertices are genuinely unconstrained by Maekawa.

A vertex where the paper passes through itself would have a self-intersecting cross-section, which can wind more than once, and the constant would be four or six. No physical sheet does this, which is why the theorem holds — but the proof assumes it rather than proving it, and that assumption is exactly the one the global problem cannot make.

A cone rather than a sheet — angles summing to something other than 360° — is not a flat sheet at all, and the whole framework goes with it. Developability is checked first for that reason.

Each exception fails a hypothesis rather than the conclusion, which is what one wants from a theorem. The constant is two whenever the setup is what it claims to be.

The theorem as a tool #

Maekawa’s theorem is usually presented as a fact and is more useful as an instrument, and folders use it constantly without naming it.

Diagnosing a stuck fold. A vertex that will not collapse is very often a vertex where the count is wrong. Counting takes two seconds and identifies the problem immediately, where staring at the paper does not.

Completing an assignment. Given a crease pattern with most of the assignment specified, the remaining creases are often forced: the count at each vertex must come out right, and that propagates.

Checking a transcription. A crease pattern copied by hand or read from an image accumulates errors. Running the count at every vertex catches most of them, cheaply, without folding anything.

Generating patterns. This site’s preliminary base has its assignment found by enumeration under exactly this constraint, rather than remembered.

None of that requires understanding the winding argument. The theorem is a two-second check that catches a large fraction of the errors people actually make, which is a good deal more valuable than its proof.

Where the theorem does not reach #

A short list of adjacent questions Maekawa says nothing about, because knowing the boundary is most of knowing the theorem.

It does not say which creases are mountains — only how many. Getting the count right and the positions wrong gives a vertex that fails the third condition.

It does not apply to a partly folded state. The constant is two because each crease contributes a half-turn, which requires the fold to be complete.

It does not apply at the boundary of the sheet, where the cross-section is not a closed path.

And it does not scale to the whole pattern: satisfying it at every vertex is necessary and not sufficient.

What it does say, it says exactly and cheaply, for every flat-foldable vertex that has ever existed. That is a good ratio of scope to certainty, and it is why the theorem is used constantly despite covering so little.

A theorem about sheets, not about folders #

The most useful way to hold this result is as a fact about a physical situation rather than a rule of a craft.

Anything that is a sheet, and reaches a flat state, and does so without passing through itself, satisfies it. Deliberate folding qualifies. So does a crushed can, a crumpled ball of paper, a rumpled shirt, and a leaf that was packed in a bud.

None of those consulted anything. The theorem holds because it is a consequence of the geometry of a closed walk, and the objects are doing geometry whether or not anybody is watching.

That framing is worth adopting early, because it is what separates the parts of this subject that are about paper from the parts that are about mathematics. Kawasaki and Maekawa are the second kind. Crease radius, thickness and memory are the first, and they are where paper reasserts itself.

The ladder from here #

Later rungs: the winding argument in full. Parity, and the even-degree corollary from both directions. Counting assignments satisfying Maekawa. The interaction with Kawasaki, and why the two are independent. Big-little-big as the third condition. Self-intersection, and the assumption the proof hides. Boundary vertices. Maekawa’s theorem for non-flat folded states, which does not exist and probably cannot. And the generalisation to higher dimensions, where the cross-section becomes a surface.

The proof fits in a paragraph and produces a constant that holds for every flat fold anybody has ever made, including every one made before the theorem existed. Several million paper cranes were folded correctly before anybody could say why the count came out that way.