Where an error goes

This subject has two accounts of what happens to a misplaced crease and they are both about accumulation.

A folded position is a composition of reflections, so a reflection in a line that is slightly off turns everything beyond it by twice as much; the error does not stay where it was made, and whether it grows with the crease count or with its square root depends on whether it is the same error every time. And paper spreads a misplaced crease along its length as a curvature nobody notices, while a rigid panel cannot, so the error arrives at the hinge and the hinge has to be given room.

Both are about a quantity being carried somewhere. There is a third case in which nothing is carried anywhere, and it is worse.

The displacement that changes nothing local #

Take a quadrilateral mesh built the way the family the Miura belongs to is built: a fan of straight lines for the columns, and rows drawn across them that reflect in every line they cross. Every vertex is developable and satisfies Kawasaki, by construction rather than by search.

Now move one row. Change where it crosses the second column, and let the reflection rule carry the change along the rest of that row — so every vertex on the moved row goes on reflecting, exactly.

The worst departure from Kawasaki over the whole sheet, before the displacement, is 5.1 × 10⁻¹⁴ degrees. After it, 5.1 × 10⁻¹⁴ degrees. The same number in the same decimal place. Every vertex condition the subject has is exactly as satisfied as it was, and every checker that reads vertices says the sheet is unchanged.

The sheet no longer folds.

Where the failure is, and where it is not #

Fold it anyway — drive the motion with one angle, as the family is driven — and the panels come apart. At a two per cent displacement two panels leave a shared corner 6.6 × 10⁻⁴ of a sheet-width apart; at twelve per cent, 7.8 × 10⁻³, which on a sheet the size of a book is a couple of millimetres of daylight.

The interesting question is where, and the answer needs care because the obvious measurement gives a misleading one.

Measuring the gap panel by panel requires placing the panels, and placing them means starting somewhere and walking outward — so which corner the disagreement turns up at depends on where the walk began. That is bookkeeping and not physics, and a map drawn from it would be a map of the bookkeeping.

The quantity that does not depend on the walk is the one the condition is stated in. A mesh of this kind folds when the table of cosines at its vertices is a column of numbers times a row of numbers — equivalently, when the ratio between any two columns is the same in every row. Before the displacement those ratios agree between rows to 10⁻¹⁶. After it they differ by 0.028.

And the difference is not attached to a place. It is a disagreement between rows about what the columns should be doing, and a disagreement between rows lives in the pairing rather than in either party.

Why the row cannot absorb it #

Here is the part that overturns the intuition.

A row that has been moved is still a perfectly good row. Every one of its vertices reflects; its own creases are consistent with one another; folded in isolation, it would be a strip of paper that folds. What it can no longer do is agree with the other rows about the columns they share.

The column creases are the shared quantity. A column crease has one fold angle down its whole length, and every row it passes through has an opinion about what that angle should be — an opinion set by the cosine at the vertex where they cross. Move one row and its opinions all change; the other rows’ do not; and there is now no set of column angles that satisfies everybody.

So the failure is between rows, and it is global in one direction and local in the other. Every row is implicated because every row shares the columns. That is the anisotropy: a mistake in a row is felt by the columns, and through them by every other row on the sheet.

What that does to a repair #

The practical consequence is a rule that is easy to state and unpleasant to obey.

Nudging the displaced row back is a repair, obviously. Nudging any other row is not: it changes that row’s opinions too and does not restore agreement. Adjusting a single vertex is not, because the row’s cosines are set by the reflection rule and one vertex cannot be adjusted alone without breaking Kawasaki at its neighbours.

And there is no way to trade the error away. In the accumulating cases the subject already knows about, an error can be distributed: given room at the hinges, a rigid mechanism will take a small misplacement and pay for it in fold angle, and the room needed can be computed and provided. Here the error is not a quantity to be shared out; the sheet has one motion or none, and the pattern either satisfies the ratio condition or it does not.

How much room a maker actually has #

It would be reasonable to conclude from the above that these meshes cannot be manufactured, and that is not the finding.

Everything above is exact. A real assembly has clearance at every hinge, panels that bend a little, and a tolerance budget; what the measurement gives is the size of the discrepancy the budget has to cover, as a function of how far the pattern is off the family. At a two per cent displacement the gap is 6.6 × 10⁻⁴ of a sheet-width — a fifth of a millimetre on a 300 mm panel array, which is inside the clearance of most hinges. At twelve per cent it is 7.8 × 10⁻³, over two millimetres, which is not.

So the rule for a maker is a threshold rather than a prohibition: the mesh has to be inside the family to within whatever the hinges will forgive, and the amount it is outside can be computed from the pattern before anything is cut. That is a considerably more useful statement than rigid foldability is not generic, which is true and gives no number.

There is a second reason the threshold is the right form of the answer. The displacement measured here is a design error — a row drawn in the wrong place — and design errors are exact and repeatable. A manufacturing error is a scatter: every panel slightly off, in a different direction each time. Those two have different arithmetic and the second is the one the growth law was written for. What the ratio condition adds is that a design error of this kind cannot be averaged out by anything, however many panels there are, because it is not a random quantity being summed.

The gap grows faster than the mistake #

The two measured points say something a threshold needs and the essay does not draw out. Six times the displacement — two per cent to twelve — gives not six times the gap but very nearly twelve times it: 6.6 × 10⁻⁴ becomes 7.8 × 10⁻³.

So the discrepancy is superlinear in the design error. That runs the useful way for a maker and the unhelpful way for anybody extrapolating from a large error to a small one. Halving a displacement buys rather more than half the gap back, so a design that is nearly inside the family is much better than proportionally better; and a design far outside it is much worse than a linear reading suggests, which is why the twelve-per-cent case is over two millimetres rather than the millimetre and a bit that scaling from two per cent would give.

Two points do not fit a law and no exponent should be quoted from them. What they do establish is the sign of the curvature, and the sign is what a tolerance argument needs: extrapolating linearly from a large measured error underestimates a small one’s improvement and overestimates a larger one’s damage, so the honest way to set a threshold is to measure at the displacement that matters rather than to scale from a convenient one.

And it can be diagnosed on the drawing #

The section above says the third kind of failure is the only one that can be diagnosed before anything is built, and leaves the diagnosis as a comparison of ratios between rows. There is a cheaper form of the same test, and it is cheap enough to be a habit.

The ratio condition holds exactly when every two-by-two block of the table has equal diagonal products: the cosine at one corner times the cosine at the opposite corner equals the product of the other two. Four vertices, four cosines, two multiplications and a comparison — per cell of the mesh, on the flat drawing, with a protractor and no folding at all.

That has two advantages over reading the ratios row against row. It needs no reference row, so nothing has to be chosen and no result depends on the choice. And it names the cell: a block whose products differ is a cell the pattern is wrong at, which is a location on the sheet rather than a disagreement between two abstractions.

The location is worth having even though the essay is right that the failure is not attached to a place in the folded object. The fault has a place in the drawing even where the symptom does not have one in the assembly, and a designer repairing a mesh wants the first. A row displaced by two per cent shows up as every cell along that row failing the block test by a small amount, and the band of failing cells is the row that moved — which is exactly the information the gap measurement cannot supply, because the gap depends on where the placement walk began.

The two questions a tolerance has to answer #

It is worth separating what has now been established, because the three rungs of this ladder answer three different questions and they are easy to run together.

How large is the error when it arrives? The reflection account answers that: a misplaced crease turns everything beyond it, and the growth law depends on whether the mistakes are correlated.

Where does the error end up? The hinge account answers that: paper spreads it as curvature, a rigid panel cannot, so it goes to the hinge and the hinge needs room.

Can it be absorbed at all? That is the question here, and the answer is that it depends on what kind of thing was got wrong. A misplaced crease in a strip is a length error and lengths can be traded. A displaced row in a mesh is a violation of a condition between rows, and conditions are not traded.

What a folder in paper sees #

None of this is visible in paper, and it is worth saying why, because the contrast is the reason rigid folding is a separate subject at all.

Paper is not made of rigid panels. A sheet whose rows disagree will fold perfectly well, because the panels between the creases bend by a fraction of a degree each and absorb the whole discrepancy — which is exactly what a flat sheet does with any error it is given, and which is why hand-folding is so forgiving. A folder can make one of these patterns, fold it, and see nothing at all.

The discrepancy is real and the paper is paying for it in a currency the paper has plenty of. Build the same pattern out of panels and hinges and the currency is gone, and the arithmetic arrives all at once. That is the whole of why a pattern that folds in the hand can fail on a bench, and it is a considerably more specific answer than “paper is forgiving”.

The three failures, side by side #

It is worth setting the three kinds of tolerance failure out together, because a maker meets all of them and they want different responses.

An error that accumulates. Each crease slightly misplaced, the errors compounding along a sequence. The response is to control the process — better registration, fewer folds, correlated rather than independent mistakes — and the growth law says how much that buys.

An error that has to be housed. A rigid panel cannot bend, so a misplacement arrives at the hinge. The response is clearance: work out how much room the hinge needs and provide it, at a price in thickness.

An error that violates a condition. The pattern is off a compatibility requirement, and there is no motion at all. The response is not clearance and not process control; it is to change the pattern, because nothing else touches it.

The three are easy to confuse in a finished assembly, where all of them present as a mechanism that binds. Telling them apart needs the pattern rather than the object: only the third can be diagnosed before anything is built, and it is the only one that has to be.

Where the model stops #

One family of meshes. Everything measured here is on quadrilateral meshes with straight column creases. The general quadrilateral mesh has a more complicated compatibility condition and there is no reason to expect the anisotropy to have the same shape.

One kind of displacement. Moving a row’s crossing of the second column is a displacement that preserves flat-foldability exactly, which is what makes the contrast clean. Other mistakes — a crease at slightly the wrong angle, a vertex nudged off its column — break Kawasaki too, and then the local checkers do report something.

No material and no stiffness. Panels are rigid by declaration and hinges are ideal. What a real assembly does with an incompatibility of this size is a question about stiffness and about where the strain goes, and that belongs to structures rather than to geometry; the numbers above are geometric discrepancies and are named as such.

The gap is measured at one point of the motion. It is reported at a fixed driving angle. How it behaves through the motion — whether there is a fold angle at which the mismatch vanishes — is not measured, and there is no reason to expect one.

A rule of thumb, and where it comes from #

The whole of the above collapses to one instruction, and it is worth having in the form a person would use.

In a mesh of this kind, the rows must be copies of one another. Scaled about the point where the columns meet if they meet; translated if they are parallel. A row adjusted on its own — for local reasons, to fit a boundary, to accommodate a fixing — has left the family, and the sheet will not move.

That is a strange rule by the standards of the rest of this subject, where nearly everything is a condition at a point and can be checked by looking at a neighbourhood. It cannot be checked by looking at a neighbourhood, and there is no local symptom to look for. The reason it exists is the ratio condition, and the reason the ratio condition exists is that a column crease has one fold angle and several rows have opinions about it.

Where the ladder goes next #

The obvious next measurement is the cheapest repair. Given a mesh off the ratio condition, what is the smallest set of changes to the pattern that brings it back, and does that set look like anything a designer would accept? The condition is a rank condition and the question is a projection onto a rank-one table, which is a well-posed problem with a definite answer.

The other direction is the one a manufacturer would ask first. The threshold above was quoted against a hinge clearance, which is a number from somebody else’s discipline. What would make it a proper engineering result is the relation between the geometric discrepancy and the force required to close it — and that is a question about stiffness, which is named here and derived nowhere.