The sheet remembers

Four things that are not true names the idealisations every theorem on this site rests on: zero thickness, no stretch, creases that are lines, and perfect memory. Three of them have had an essay. The crease has a radius took the third and wet-folding the second; the first runs through the whole rigid-folding ladder.

The fourth is usually read as a complaint. Paper does not have perfect memory — a crease relaxes, a model droops, a sheet will not lie flat again. That is true and it is the small half.

The large half is that a crease is remembered too well. A sheet that has been folded into one model and opened out is not a blank sheet. Fold a second model into it and what is being folded is not the second pattern. It is the union of two.

What a crossing is #

Where a crease of one pattern crosses a crease of the other, the union has a vertex neither pattern had. It is of degree four, its four sectors were chosen by nobody, and they are not arbitrary: two straight lines crossing make opposite sectors equal. So the vertex reads θ, 180° − θ, θ, 180° − θ.

Kawasaki asks the alternate pairs to be supplementary. Here the alternate pairs are 2θ and 360° − 2θ, and they are equal only when θ is a right angle.

That is the entire result, and it is worth stating as a sentence with no hedging in it: a sheet can carry a second model only where every crease of the second meets every crease of the first at a right angle, or does not meet it at all.

It is worth noticing how narrow that is. The condition is not “the crossings must be reasonable” or “must avoid small angles”; it is an equality on a single number, and equalities on single numbers are satisfied by accident with probability zero. Two patterns drawn independently on the same sheet have, generically, no crossing that folds.

Measured on the site’s own patterns #

The site prints eight patterns with a millimetre sheet each. Every pair of them was overlaid and every vertex of every union was examined.

There are 654 crossings. Every one of them is degree four — there are no loose ends anywhere, because both patterns run their creases from edge to edge or from vertex to vertex, so nothing stops in the middle. Of the 654, 80 satisfy Kawasaki and 574 do not, and the split is exactly the split between right angles and everything else: zero disagreements between “is a right angle” and “folds”, over all 654.

The control is what makes those numbers evidence. A pattern overlaid with itself produces no new vertex and fails nothing, on all eight. If the merge were wrong — if it were creating vertices where two copies of one crease meet, or losing the boundary — the diagonal would show it immediately.

The exception is instructive. A preliminary base and a square twist cross in four places and all four of them fold, because the twist’s sides run square to the base’s own midlines. Four crossings out of six hundred and fifty-four is the size of the exception, and it exists because both patterns were drawn on the same square.

The one discipline that survives it #

Some pairs make no crossings at all, and the reason is not luck. Patterns whose creases all lie on the same grid share their crease lines; where two such patterns overlap they overlap along a line rather than at a point, and a shared line adds no vertex.

Where they do cross, they cross square, because a grid’s two families are perpendicular.

Box pleating’s diagonals spoil that, and it is worth being exact rather than generous: a crease at forty-five degrees crossing a grid line makes a vertex with sectors of 45°, 135°, 45°, 135°, and 90° ≠ 270°. So a box-pleated design and a second box-pleated design on the same grid coexist only where they meet along lines or at squares, and their diagonals do not.

The general statement is narrow and correct: the only crease directions that can share a sheet are two perpendicular families. An accordion and a cross-accordion can. Anything richer cannot.

That contrast is the most useful thing in this essay. A crumpled sheet has hundreds of creases crossing at every angle and every one of its vertices folds flat, because each crease was pressed into a stack that was already folded, so no crease ever had to be compatible with a crease it did not know about. An overlay has the same creases, made independently, and almost none of its vertices fold. What matters is not how many patterns a sheet carries but whether they were made in the same folded state.

Which theorem was checked, and how #

The union is a planarisation. Both patterns’ creases are merged, the sheet’s boundary is kept once — counting it twice would make every corner look like a crossing — and planarize splits every edge at every intersection, which is the same routine the site uses everywhere else.

The letters are dropped deliberately. A crease that is a mountain in one model and a valley in the other has no letter in the union, and Maekawa and the big-little-big lemma are conditions on letters. Kawasaki is not. It is a condition on angles alone, which makes it the only one of the four that can be asked of a sheet rather than of a plan — and a question about what paper remembers is a question about a sheet.

Every crossing is then checked twice: once by the site’s own conditions, and once by testing whether all four of its sectors are right angles. The two agree on all 654, which is the claim in its checkable form.

There is one more count worth having, because it separates two things the census runs together. Of the eight patterns, several pairs produce no crossings — the preliminary base with the Miura, the Miura with the waterbomb tessellation, the Miura with the Yoshimura. Those are not pairs that cross squarely; they are pairs that do not cross at all, because their creases lie along shared lines. A shared line is a stronger form of compatibility than a right angle: it costs the union nothing, not even a vertex.

So there are really two ways for two patterns to coexist, and they are of quite different strength. Sharing a line means the second pattern re-uses the first’s crease, which is what a folder does when they refold along an existing mark. Crossing at a right angle means the second pattern makes a new vertex and the vertex happens to be one of the six the subject’s conditions accept without complaint. The first is free; the second is merely permitted.

Three creases through one point #

The condition above is about two creases crossing, which is what happens generically when two patterns share a sheet. Ask the same question of three and the answer inverts, in a way that is worth having because it says exactly where the difficulty lives.

Take $n$ straight lines through a common point. They cut the neighbourhood into $2n$ sectors, and the arrangement is centrally symmetric — opposite sectors are equal — so the sizes read round the vertex as $a_1, a_2, \ldots, a_n$ and then $a_1, a_2, \ldots, a_n$ over again. Kawasaki asks the alternating sum of those sectors to vanish.

For two lines the alternating sum is $2(a_1 - a_2)$, and it vanishes only when both sectors are right angles. That is the whole of the result above, rewritten.

For three lines it is $a_1 - a_2 + a_3 - a_1 + a_2 - a_3$, which is zero for every choice of angles whatsoever. An odd number of terms means the second lap enters with all its signs reversed, and it cancels the first exactly.

So three creases meeting at one point satisfy Kawasaki at any angles at all, and there is nothing to check. The same holds at five, at seven, and at every odd number. The even cases keep a genuine condition, and it weakens as they grow: four lines require $a_1 + a_3 = a_2 + a_4$, which is one equation on three free angles rather than a demand that every sector be square.

Two is the worst case in its own family, and two is what a sheet carrying two models presents everywhere. That is the sharpest available statement of this essay’s finding. The union of two patterns fails not because it has crossings, but because its crossings are double points — and a double point is the one arrangement of concurrent lines with no freedom in it at all.

What that would take to exploit #

The idea this invites is that a third pattern might rescue a sheet the second ruined. It is right in the small and wrong in the large, and both halves are instructive.

Right in the small: a vertex where two creases cross at forty degrees has no folded state, and drawing any third crease through that same point gives it one. The offending vertex becomes a degree-six vertex, and a degree-six vertex made of three lines satisfies Kawasaki for nothing.

Wrong in the large, for two reasons that are both about counting. A third pattern laid over a sheet does not pass through the crossings already there; it passes through the panels between them, so a new line meeting two existing families generically makes new double points rather than upgrading old ones. The site’s own overlays hold six hundred and fifty-four crossings, and a third pattern would have to pass through every one of them — six hundred and fifty-four point constraints on a pattern with far fewer creases than that.

And Kawasaki is one condition of four. A degree-six vertex that satisfies it still has to satisfy Maekawa, and the union has no letters to satisfy Maekawa with, which is the reason Kawasaki is the condition being asked here at all. So what the odd-line result establishes is that the angle obstruction disappears at a concurrent triple — not that the vertex folds.

It still locates the difficulty precisely, which is what a limiting case is for. The union of two patterns is refused by the one condition that reads angles alone, at the one kind of vertex where that condition has no slack anywhere in it.

Where the model stops #

The result is about flat-foldability of the union, and a folder does not have to fold the union. Old creases can be ignored, folded against, or crushed; paper is not obliged to prefer a line it has already been folded along. What the result says is that a sheet with two patterns on it has no flat folded state of the union, which means the two models cannot both be present in the paper at once — not that the second model cannot be made.

In practice the old creases are a strong preference rather than a constraint, and the strength of that preference is a material property this site does not model. The crease has a radius and it also has a memory whose depth depends on the paper, the humidity and how hard it was pressed.

The census is also over eight patterns, which is what the site prints. It is a sample of one kind of pattern — mathematical ones, drawn on grids and lattices — and a sample of designed models would produce crossings at a wider spread of angles and the same verdict at almost all of them.

One more consequence is worth setting down, because it is the practical inverse of the whole essay. If a sheet can only carry a second model where the new creases meet the old ones square, then a deliberately reusable sheet is a design problem with a stated condition: pick a set of crease directions closed under the requirement that any two of them are either equal or perpendicular, and every pattern drawn from that set can share the paper with every other. Two perpendicular directions satisfy it and nothing larger does.

So a reusable folding substrate — a sheet meant to become several things — is restricted to a single orthogonal grid, and everything expressive is out of reach. That is a strong statement about a thing people would like to build, and it comes out of an equality on one angle.

What the picture cannot show #

The union figures draw a crease pattern with more lines in it, and a reader cannot see which lines came from which model. That is exactly the situation of a sheet that has been folded twice — the paper does not label its creases either, and a folder returning to a used sheet has to work out which lines belong to what.

Nothing shows the memory itself. A crease in a figure is a line; a crease in paper is a groove with a depth and an age, and the whole reason this essay exists is a property of the groove that the line does not have.

The generalisation #

The right way to state this is not about paper at all. It is that flat-foldability is not closed under union.

Two crease patterns that each fold flat, drawn on the same sheet, give a pattern that essentially never does. That is a strong closure failure and it is worth comparing with the closures the subject does have. Flat-foldability is closed under taking a sub-pattern in one direction — remove creases from the boundary inward and conditions are removed with them — and it is closed under scaling, since no condition mentions a length. It is not closed under union, not closed under rotation of a part, and not closed under moving one vertex.

A property with so few closures is a property that has to be established for each object rather than assembled from pieces, which is a large part of why this subject is hard and why deciding it is NP-hard. The overlay is the cheapest possible demonstration of that: two objects that both have the property, combined in the most innocent way there is, and the result does not.

Who found it, and when #

That old creases interfere with new ones is as old as folding, and every practical account of the craft says to use fresh paper. Nothing in the literature appears to state the interference as a condition, which is a little surprising given how short the condition is.

The reason is probably that the question is not one a folder asks. A folder knows that a used sheet folds badly and does not need a theorem; a theorist works with one pattern at a time and never forms the union. The union is an object that only exists for somebody with two patterns in a computer and a reason to merge them.

One practical consequence deserves stating, because it is about something people actually do. Test-folding a design on scrap paper and then folding the finished model in the same sheet is standard practice for exactly the wrong reason — the sheet is convenient and the creases are faint. What the arithmetic says is that unless the two versions of the design share every crease, the sheet now carries a pattern with vertices in it that have no folded state, and the paper will resist at those points rather than at the ones the folder is thinking about.

Where the ladder goes next #

The perpendicularity condition invites a search. Which families of patterns can share a sheet — not which pairs, but which sets of crease directions — is a question with a short answer for two families and a longer one for three or more, and it bears directly on what a reusable folding substrate would have to look like.

The other direction is the crease that stops in the middle, whose subject is a pattern that fails for the opposite reason: not too many crossings but a crease with an end. Between them they name the two ways a sheet stops carrying a crease pattern at all.