Thickness has a sign

Take any pattern on this site, with its assignment on it, and change every letter: every mountain becomes a valley and every valley a mountain. Nothing this site can measure changes.

Developability reads only angles and does not see the letters at all. Kawasaki reads only angles. Maekawa asks the mountains and valleys to differ by two, and after the flip they differ by two the other way, which is the same condition. The big-little-big lemma asks the two creases round a strictly smallest sector to be of opposite kind, and opposite is a symmetric relation. The layer rules produce the mirror stacking, sector for sector. The folded object comes out reflected, which is to say it comes out the same object, seen from underneath.

That symmetry is not a quirk of how the conditions are written. It is a statement about paper: a sheet has no preferred side, a mountain seen from below is a valley, and any theory that could tell them apart would be a theory about ink rather than about folding.

Give the paper a thickness and it is gone.

Where a hinge can be #

A panel with depth cannot hinge on a line inside itself. Two slabs meeting along an axis on their common mid-surface interfere immediately: turn one of them by any angle at all and material from below the axis sweeps into material above it. There is no small angle that is safe, and the failure is not a matter of clearances — it is that the two solids want the same space.

Move the axis to a face and one direction opens completely. With the hinge on the upper face, the moving panel swings up and over and comes to rest on top of the fixed one, surfaces in contact, at a straight angle. In the other direction it has nowhere to go: the material below the axis is immediately inside the panel it is folding against.

Move the axis to the opposite face and the two travels exchange, exactly.

That is a step function rather than a curve, and it is the whole of the finding. Which face a hinge sits on is decided by which way its crease folds, and there is no intermediate placement that hedges. A crease’s letter chooses its hinge’s face.

There is a way of stating that which makes the break obvious. The theorems are about a surface, and a surface has two sides and no inside. A panel has an inside, and the inside is what a hinge has to be attached to one side of. So the flip is a symmetry of the surface and not of the solid, and every theorem on this site is a theorem about the surface.

What that costs the pattern #

Maekawa is what turns this from a nuisance into a structural fact. Every flat-foldable interior vertex has both letters at it — three of one and one of the other at degree four, and never all of one kind — so at every vertex of every pattern here, some hinges go on the upper face and some on the lower.

That would be manageable if the choice could be made panel by panel. It cannot.

Between 57 and 92 per cent of the panels of the patterns this site prints carry both letters somewhere round their own boundary. Such a panel has to accept a hinge at the top on one edge and a hinge at the bottom on another, and the two are separated by the panel’s thickness. The panel is no longer a slab with hinges along its edges; it is a slab whose neutral surface is being asked to be in two places.

The arithmetic of how much that costs is worth doing once. A panel a tenth as thick as it is wide, tapered symmetrically so that its hinge can sit at the surface and the fold still close, loses twice the taper: about fourteen degrees of the hundred and eighty. On a Miura whose columns are eight panels deep, a fold that has to close fully at every crease therefore arrives at the last panel more than a hundred degrees short if the losses accumulate — and whether they accumulate depends on whether consecutive folds close in the same direction, which is a question about the assignment.

So the letters are not merely choosing faces. They are choosing whether a pattern’s thickness losses cancel or compound, and a pattern and its flip make the same choice mirrored — which cancels and compounds in exactly the same places. That is the one thing the symmetry does survive, and it is worth saying so: the flip is not a way of escaping the loss.

This is the reason the thickness literature is a literature of techniques rather than of a formula. Every published accommodation — offsetting the panels, tapering them, moving the axes, splitting a crease into two — is a way of buying back the travel the face-mounted hinge gives away, and each one gives away something else.

How much of a pattern is mixed, before anybody measures it #

The measured range invites a question the measurement cannot answer. Is a mixed panel a common accident of the patterns this site happens to print, or is it forced? Maekawa settles half of that without any pattern in hand.

Walk the creases round an interior vertex in cyclic order and read their letters off as a cyclic word. Maekawa says both letters appear in it — three of one and one of the other at degree four, and never all of one kind at any degree. A cyclic word over two letters in which both appear changes letter an even number of times, and at least twice. Each of those changes falls between two consecutive creases, which is to say inside one sector, and a sector belongs to a panel. So every interior vertex of every flat-foldable pattern hands at least two of its panels a corner with a mountain on one side and a valley on the other, and those panels are mixed by construction rather than by luck.

That converts into a floor. A quadrilateral panel has four corners and can absorb at most four such demands, so a pattern with $V$ interior vertices and quadrilateral panels carries at least $V/2$ mixed panels; on a grid, where the panel count and the interior-vertex count differ by a row and a column, that is a floor near a half. The measured 57 to 92 per cent sits above it everywhere, which is the right relationship — the bound counts only the mixing Maekawa forces, and a real assignment mixes more than it has to.

It also says where a pattern would have to look to do better. The floor is loosest where panels have many corners, so a pattern of large panels meeting at few vertices carries proportionally less forced mixing than a fine grid does. That is an argument for coarse patterns in thick material which has nothing to do with stiffness or with part count, and it survives every refinement of the hinge model, because it is arithmetic on the assignment rather than a fact about hardware.

And it will not go to zero. A pattern with no mixed panel at all would need every panel’s whole boundary in a single letter — but a crease has one letter and two panels, so the two panels either side of it agree, and agreement propagates across every shared crease until the letter is constant on the whole pattern. Maekawa forbids that at the first interior vertex. Unmixed is available only to patterns that have no interior vertex: a fan, a pleat, a strip creased across. Anything with a vertex in it pays.

That also answers, in the direction that matters, the graph question the closing section leaves open. Assigning hinges to faces is not a choice — the letter fixes it, so there is nothing to make consistent and nothing that can fail to be satisfiable. What the panel adjacency decides is not whether an assignment exists but what it costs, and the count of mixed panels is the count of splits, offsets or tapers a build has to buy. The floor above is the part of that sum no design can negotiate away.

The symmetry, checked rather than argued #

The flip invariance is easy to argue and easy to argue wrongly, so it is measured. Every printed pattern is run through the site’s own checker twice — once as drawn, once with every letter swapped — and the four conditions are compared vertex by vertex rather than in summary. A pattern that failed for one reason before and another reason after would be an interesting thing to have missed.

Nothing differs anywhere.

There is a corollary about counting that falls straight out of the same invariance, and it is exact rather than empirical. The flip sends a valid assignment to a valid assignment, it is its own inverse, and it fixes nothing at all — an assignment equal to its own flip would have to give some crease both letters, and a pattern with a crease in it has no such assignment. An involution with no fixed point pairs its domain off two by two, so the number of flat-foldable assignments of any creased pattern is even. The census of what folds reports four at a degree-four vertex, eight surviving Maekawa alone, and a hundred and twelve at the eight-crease centre of the preliminary base; all of them are even, and they are even for this reason rather than by any property of the search that found them. A count that comes back odd is reporting a fault in the search, and that is the cheapest test the symmetry buys.

The flip is worth separating from the other two-valued things on the sheet, because they are easy to run together. It is not the two-colouring, which is about which face of the paper each panel shows and is a property of the pattern’s angles rather than its letters. It is not the direction a geared vertex turns, which is a fact about the folded state at a given angle. It is an operation on the assignment alone, and the reason it commutes with every theorem here is that no theorem here reads an assignment except to compare letters with each other.

The travel measurement is a collision test rather than a claim. Two slabs, an axis at a stated depth, a sampled sweep of the fold angle, and the largest angle at which no sampled point of the turned slab lies inside the fixed one. It comes back as a step at the surface: 180 degrees one way and zero the other with the hinge at a face, zero both ways anywhere inside, and the two directions exchanged exactly when the axis moves to the other face. The zero-thickness case is run as a control and gives 180 degrees in both directions, which is the case every theorem on this site is about.

Where the model stops #

The slab model is crude on purpose. Real hinges are not lines: they are films, tapes, living hinges with a bend radius, or pin joints with a diameter, and every one of those has its own accommodation and its own limits. What the model establishes is not how far a particular hinge closes but that the two directions are not equivalent, and that conclusion survives any of those refinements because it comes from which side of the material the axis is on.

The sampling is a real limit. The travel is measured by sweeping the fold angle in fixed steps and testing a grid of points, so a very thin sliver of interference between samples would be missed. At the resolution used the step is sharp to within a degree, which is enough for the claim and would not be enough for a tolerance.

And the whole essay is about a single crease. What a sheet of thick panels does — whether the face assignments can be made consistently across a pattern at all, and what happens where three hinge planes meet — is the offset-panel question and is not settled by anything here.

One further limit is worth naming because it is the sort a reader will reach for. The whole analysis assumes the two panels either side of a crease have the same thickness. They need not: a design can taper a sheet across its area, and where a thin panel meets a thick one the axis can sit at the thin panel’s surface and inside the thick one, which changes the arithmetic without changing the conclusion. The direction is still chosen and there is still no placement that opens both ways.

What the picture cannot show #

The section figures are two-dimensional, and a crease is a line in a plane. At a vertex, four hinge axes meet, on faces chosen independently by four letters, and the figure of that is a solid-modelling problem rather than a drawing. Nothing here shows it, and the honest statement is that the vertex case is where every thickness technique earns its keep.

Nor does anything show the flip. Two patterns that differ by swapping every letter produce identical drawings under this site’s conventions except for the colours and dashes, and the folded objects are mirror images — which photograph identically. The symmetry is precisely the thing that cannot be drawn, which is part of why it went so long without being stated.

The generalisation #

The pattern here has a name in other subjects: a symmetry of the model that the world does not have. Every theory carries some, and the interesting ones are those where the broken symmetry costs something specific.

Flat-folding’s version is unusually clean. The theory is invariant under swapping the two letters because it is a theory about a surface with no thickness, and the invariance is exact — not approximate, not up to a small term. So the theory cannot distinguish a pattern from its flip, and any question whose answer differs between them is a question the theory is not equipped to ask. Handedness is such a question. So is which face carries which hinge, and so, further down the ladder, is which way a mechanism jams when it is over-driven.

The lesson generalises to the other idealisations this site rests on. Zero thickness buys the flip. No stretch buys the isometry, and everything that follows from it. Creases as lines buys the vertex as a point, which is what makes the local theorems local at all. Each one buys a symmetry, and each symmetry is a set of questions the theory has silently declined to answer.

Who found it, and when #

The thickness problem is as old as the first attempt to build a folding mechanism out of anything stiffer than paper, and the modern accommodations are recent: Tomohiro Tachi’s offset-panel technique dates from 2011, tapered-panel methods from the same period, and the Hoberman and Trautz–Künstler treatments from before them. The handedness is not usually stated as a theorem because it is obvious to anybody holding two pieces of card — which is exactly the kind of fact that never gets written down and never gets connected to the theory it contradicts.

The connection is the point. Maekawa’s theorem, proved in 1979 as a statement about winding, guarantees that every vertex has both letters. Read as a fact about hardware it says that no sheet of thick panels can put all its hinges on one face, and that is not a corollary anybody drew at the time because there were no thick panels to draw it about.

There is one more consequence, and it is about printing. Every crease pattern this site publishes is printable, and a printed pattern is folded by hand out of paper thin enough that all of this is negligible. A reader who takes one of those sheets to a laser cutter and a sheet of acrylic is not making the same object: the pattern says where the creases go and says nothing about which side of the material each hinge belongs on, and that information is not recoverable from the drawing without the assignment — which the drawing does carry, in its colours and its dashes, and which is therefore doing more work than it looks as though it is doing.

Where the ladder goes next #

Two things follow. The first is a design question with a definite answer somewhere: given a pattern, is there an assignment of hinges to faces that is consistent across every panel, and if not, how many panels have to be split? That is a graph question on the panel adjacency, and it is answerable.

The second is the mirror image of this essay. A pattern and its flip are the same object for every theorem here and different machines once built — so a thick mechanism has a handedness that its crease pattern does not record, and a manufacturer working from a pattern is working from an under-specification. That is the same complaint the folded object makes about its own crease pattern, arriving from the hardware end.