Nothing meets at three

Put a dozen crease patterns from this subject side by side and something about them is uniform in a way that is hard to name. They are grids, or webs, or radiating fans. What they never are is trees.

No crease trails off and stops in the middle of the paper. No three lines meet at a point. Every line that arrives at a vertex leaves it again on the other side, and the drawings are therefore all crossings and no stars.

That is a fact about pictures, and it is a consequence of a counting theorem.

What every crease pattern in the subject looks like #

The statement being made is that an interior vertex of a flat-foldable pattern has an even number of creases and at least four. Both halves are needed and they are forced by different arguments.

The consequences are the appearance of the drawing. Four or more means a crease cannot simply end in the middle of the sheet, because the end would be a vertex of degree one. Even means three creases cannot meet, and neither can five or seven. What is left is a picture in which every vertex is a place where lines cross.

The Miura is the extreme case: every interior vertex has degree exactly four, and every one is the same vertex, which is what makes the sheet behave like a material. But the visual signature survives at higher degrees. A degree-six vertex is three lines crossing; a degree-eight vertex is four. The pattern reads as an arrangement of lines because it is one.

Two creases is not a vertex at all #

The lower bound needs one sentence of care before it is worth anything, because it hides a definition.

A point where exactly two creases meet is a point on a line. If the two creases are collinear, the drawing contains a straight crease with a dot marked on it and nothing has happened. If they are not collinear, the paper has a bend that changes direction at a point, which is a corner in the crease and not a meeting of two creases — and it fails immediately anyway, since the two sectors would have to be 180° each for the sheet to fold onto a line and they are not.

So the interesting bound is really “not one, not two, not three”, of which the first two are near-tautologies and the third is a theorem. Degree one fails everything at once: a single crease divides the paper into one sector of a full turn, and there is nothing to alternate with.

The bound the minimum degree puts on a whole pattern #

The lower bound is stated as a fact about one vertex, and it has a consequence for the whole drawing that is worth extracting, because it says how dense a crease pattern can be.

Count the incidences between creases and interior vertices. Every crease has two ends, so the incidences number at most $2E$ for $E$ creases. Every interior vertex uses at least four of them. So

$$4V \le 2E \qquad\Longrightarrow\qquad V \le \tfrac{E}{2}$$

A flat-foldable pattern has at most one interior vertex for every two creases. That is a hard bound, it needs no angles and no letters, and it follows from the degree-four minimum alone.

It checks out on everything here. The square twist has four interior vertices and twelve creases; the rhombille patch has a hundred and twenty-six and two hundred and eighty-two; a sixteen-by-sixteen grid has two hundred and twenty-five and four hundred and eighty.

Which the grid saturates #

The last of those three is the interesting one, because it is nearly tight, and pushing it says something about why box pleating is what it is.

An $n$ by $n$ grid has $(n-1)^{2}$ interior vertices and $2n(n-1)$ creases, so the ratio is $(n-1)/2n$, which climbs toward one half and never reaches it. At sixteen divisions it is 0.469; at sixty-four, 0.492.

So a square grid is asymptotically the densest a flat-foldable pattern can be in vertices per crease, and it gets there by having every interior vertex at the minimum degree and almost every crease interior. Nothing can beat it, and only a pattern of all-degree-four vertices with a vanishing boundary can match it.

That is an argument for the grid nobody usually makes. Designing on a grid is defended on accuracy and on reference availability; this says it also extracts the most decision points from a given amount of folding. A crease is what a folder spends, a vertex is where the design does something, and the grid maximises the second per unit of the first.

It also says what a higher-degree pattern costs. A vertex of degree six uses six incidences rather than four, so a pattern built from them has at most $E/3$ interior vertices — two thirds as many places where anything happens, for the same length of crease.

The colouring says even, and so does the parity #

There are two independent arguments for evenness, and it is worth having both, because they fail in different places and one of them is not about folding at all.

Read a crease pattern as a set of panels rather than a set of lines. Crossing any crease turns the sheet over, so the side facing the reader alternates from panel to panel, and the panels take two colours with no crease having the same colour on both sides. Now walk once around an interior vertex. The walk crosses one crease per sector and returns to where it started, so it must have turned the paper over an even number of times. An odd degree makes that impossible.

That argument never mentions Kawasaki, Maekawa or a folded state. It is about which side of the paper is showing, and it holds for any flat folding whatever.

The second argument is the parity condition itself, and this site has already proved it by winding, so it is quoted here rather than derived. At a flat-foldable interior vertex the number of mountains and the number of valleys differ by exactly two. Their sum is the degree, and two numbers differing by two have a sum of the same parity as either of them doubled — which is to say the sum is even. An odd degree cannot be split into two counts differing by two, whatever the letters are.

Both arguments give evenness. Only the second gives it as an arithmetic obstruction that can be checked assignment by assignment, and that is what the next section does.

Three creases, and what happens instead of folding #

The smallest case anybody would actually draw is three creases meeting at a point, and it is worth watching fail rather than being told that it does.

The right-hand half of that figure is the more useful one, because the residual is computed by walking the fold rather than by evaluating a formula. Every crease reverses the direction of travel, so the sectors are laid down alternately one way and the other, and the last one has to arrive back at the first crease. At 140°, 110° and 110° it arrives 140° short. Changing the angles moves that number and never removes it, because with an odd count the walk ends travelling in the wrong direction and no set of three angles summing to a full turn can fix a parity.

Eight assignments, eight failures, and the failures are not close. Three of one and none of the other differ by three; two of one and one of the other differ by one. Two is not among the values available, and it cannot be, because three splits into two counts of opposite parity.

Kawasaki does not even get a hearing here. Its alternating sums need an even number of sectors to alternate over; with three there is no cyclic alternation to evaluate, and the condition is refused at the door rather than failed.

The census over the library #

None of the above establishes that real patterns behave this way. Theorems are about ideal objects, and the honest check is to count what the site has actually drawn.

Over the whole library the answer is ninety-two interior vertices across eight patterns: fifty-nine at degree four, thirty-two at degree six, and exactly one at degree eight. Nothing odd. Nothing below four.

The distribution is itself worth a remark. Degree four dominates because a degree-four vertex is the cheapest thing that satisfies the conditions and because the same one keeps being arrived at independently. Degree six appears wherever a pattern has a diagonal structure over a grid.

The waterbomb tiled supplies most of the degree-six count on its own, and the single degree-eight vertex is a landmark rather than a population: it is the centre of the preliminary base, where eight creases meet at 45° apiece.

Which theorem was checked, and how #

The claim here is a negative one — that no vertex of a certain kind exists anywhere — and negatives are the easiest claim to make vacuously. A census that examined no vertices would report the same thing.

So the check is built in two halves that fail differently. Every pattern in the library is constructed, its interior vertices found geometrically, and the number of creases at each one counted; any vertex that came back odd, or below four, stops the figure and names the pattern and the vertex. That is the half that could catch a bad pattern, and it runs on all ninety-two vertices in every view of the figure rather than only in the one that draws them.

The second half runs the argument the other way. Every assignment of a three-crease vertex is built as an actual crease pattern and put to the same checker the library patterns go through. If any of them satisfied the parity condition the figure would stop and say which, because a survivor there would mean the checker was wrong rather than that a three-crease vertex folds.

What would have made the whole thing fail is worth naming, because it nearly did once. A vertex sitting on the sheet’s edge has fewer creases than a full turn would give it, so a census that misclassified boundary vertices as interior ones would report odd degrees and low degrees everywhere. That the count comes back clean is partly a statement about the patterns and partly a statement about the interior test underneath it, which had to be repaired before either could be trusted.

The idealisation should be said plainly too. Every crease here is a line of zero width meeting the others at a point, and the vertex is a point. On real paper a crossing of eight creases at 45° is a small crumpled disc, the creases have radii, and the paper near the centre is thicker than anywhere else — none of which the count knows about, and all of which is why a preliminary base collapses with a pop rather than continuously.

Where the rule does not apply #

Three limits, and the third is the interesting one.

The rule is about interior vertices. At the sheet’s edge there is no full turn to go around, and none of the conditions has a hypothesis available, so a crease may arrive at the boundary alone and at any angle. A pattern’s edge is therefore full of degree-one and degree-three junctions and none of them is a counterexample to anything.

The rule is about flat folding. A pattern folded into a three-dimensional shape is subject to a closure condition at each vertex, but not to this one, and rigid folding asks a different question again.

And the rule is about a sheet that stays a sheet. Three creases meeting is forbidden on a flat-folded sheet; it is not forbidden on a piece of paper. Take a full turn of paper around a point, remove a wedge and join the cut edges, and the point now has less than a full turn around it. The paper is fine and it does not lie flat: it stands up into a cone, permanently, and no amount of folding will flatten it, because folding cannot change the angle at a point. That is what a cut buys, it is a different subject with its own arithmetic, and it is the price of the rule that keeps everything here to what a flat sheet can become.

What it does to the practice #

Two consequences reach the drawing board, and both are the sort of thing a designer knows without a name for it.

The first is a check anybody can run on a crease pattern in a second. Look for a loose end. Look for a Y. If either is present and is not at the paper’s edge, the drawing is wrong — not a difficult pattern, not one requiring a clever sequence, but a picture of something that does not fold. It is much faster than evaluating any condition, and it catches most of what a novice draws.

The second is why grid systems dominate designed patterns. Working on a grid means every crease runs along one of a small set of directions and every crossing is automatically a crossing of two lines through a point, so the degree condition is satisfied by construction and never has to be thought about. What the designer is choosing is which lines to draw and where each one begins and ends — and since the lengths carry no condition at all, the whole of that choice is free of the theorems.

Put the two together and a familiar working style falls out. Draw on a lattice, keep every crease running between boundaries or through crossings, and no local condition can be violated by accident. What remains difficult is entirely global, which is where almost every pattern still fails.

Who noticed it, and when #

The parity condition is old, contested in attribution, and stated for interior vertices from the beginning. It sits in Koji Husimi’s book, in Jacques Justin’s 1986 statement of the local conditions, and under Jun Maekawa’s name by eponym — a gap of the kind this site has measured across the subject and found to average about twenty-two years.

The corollary about degree is a different matter. It is an immediate consequence, it is usually mentioned in a subordinate clause if at all, and it has no name. What has never been written down anywhere obvious is the visual version: that this is why crease patterns look the way they do, and why a folding diagram and a crease pattern are such different kinds of picture.

The two-colouring argument arrived from graph theory rather than from paper and is the one that generalises furthest, since it holds for any flat folding of any surface that has two sides. Practitioners met it first as a fact about which panels can show the reverse of the paper, which is the colour-change problem and is a design concern rather than a theorem.

The traditional repertoire encodes all of it silently. Classical bases are built from a small number of vertices of degree four and eight, arrived at by folding paper rather than by drawing lines, and a fold that produced a three-way junction was simply never made — because paper does not do it, and a folder discovers that in about four seconds without knowing there is anything to prove.

Where the ladder goes next #

The immediate continuation is the other direction of the same statement. The degree count says what a vertex may be; it does not say how many vertices of each kind exist, and the answer is that there are very few flat-foldable degree-four vertices and they keep turning up independently in buckled cylinders, crumpled sheets and designed patterns.

The other direction is the sheet itself. Everything here assumes a full turn of paper at every interior point, which is what makes the sheet a sheet — and the moment that assumption is relaxed the whole census stops applying at once. What a wedge of paper removed or inserted buys is the next rung of the material ladder, and it is bought at exactly the price of the rule this site otherwise keeps.

And there is a nearer neighbour. The degree condition fixes what the drawing may look like; it does not fix what the drawing is a picture of, because one marked pattern can have several folded states and the count is its own quantity.