Error is folded too

A crease pattern is a set of exact lines and a folded sheet is not. Somewhere between the two there is a printer, a scoring tool, a pair of hands or a press, and every one of them puts the crease very slightly somewhere other than where the drawing said.

The interesting question is not how large that error is. It is what happens to it afterwards.

The exchange rate between bias and scatter #

The two curves differ by a square root, and dividing one by the other turns that into a rule a toolmaker can use.

A consistent error of $\delta$ at every crease accumulates as $n\delta$; a random one of the same size accumulates as $\delta\sqrt{n}$. So the systematic case is worse by a factor of

$$\sqrt{n}$$

At four creases that is two. At thirty-two it is 5.7. At a hundred, ten. The penalty for bias grows without bound relative to the penalty for scatter, which means the two are not comparable quantities and cannot be traded at a fixed rate.

Which inverts the obvious preference #

Run it the other way and it gives a threshold with a number on it.

Suppose one tool places creases to half a degree but always the same way — a scoring head mounted a hair out of true, or a folder who consistently overshoots. Suppose another scatters by two degrees with no bias. The first is four times more precise by any single-crease measurement.

Their accumulated errors are $0.5n$ and $2\sqrt{n}$, equal when $\sqrt{n} = 4$, which is

$$n = 16$$

Past sixteen creases the sloppy unbiased tool beats the precise biased one, and by thirty-two creases it is better by forty per cent. At a hundred creases — an ordinary count for a tessellation — the biased tool is two and a half times worse despite being four times more accurate per fold.

That is the practical content of the square root, and it is the opposite of what a specification sheet encourages. A tool is sold on its per-operation precision, which is the quantity that does not accumulate; what accumulates is the part nobody quotes.

So the first thing to do with a folding machine is not to make it more precise. It is to find its bias and remove it, after which its precision matters at half the rate. And it explains a habit good folders have that looks superstitious: alternating which way a sheet is turned between folds, which converts a consistent hand bias into a scattered one and buys the square root.

Why an error travels #

Folding flat about a crease is reflection in that crease. A panel’s position in the folded state is therefore a composition of reflections — one for every crease crossed on the way to it from wherever the walk started.

Now suppose one of those creases is at an angle δ from where it should be. Reflection in a line turned by δ differs from reflection in the correct line by a rotation of 2δ, and that rotation is applied to everything beyond the crease. Not to the crease itself, not to the panel it borders: to the whole remainder of the sheet.

Two consequences follow and both are the essay.

The angular error is doubled and then stays put. Composing further reflections does not amplify it further, because rotations compose additively and each subsequent fold contributes its own error and nothing more.

The positional error is not bounded at all, because a rotation of 2δ moves a point by 2δ times its distance from the centre. So the error at a given place is the angular error times the lever arm, and the lever arm is however much sheet is left.

That is the whole mechanism. It has nothing in it about material, force or mechanism, and it would be the same on paper, on steel and on a drawing.

The two laws #

The strip used here is a sheet n + 1 panels long and one panel high, with n creases that should run straight across it and instead are tilted by a small angle each. Nothing about the pattern becomes invalid: every crease still runs from one raw edge to the other, so there is no interior vertex and no theorem to violate.

A systematic error grows with the crease count. If every crease is out by the same angle in the same direction, every fold turns the remainder of the sheet the same way, and the turns add. Half a degree per crease puts the far end 0.070 panel widths adrift after eight creases, 0.140 after sixteen, 0.279 after thirty-two and 0.419 after forty-eight. Doubling the error to a degree doubles all of it: 0.559 after thirty-two, and 0.838 — most of a whole panel — after forty-eight.

A random error grows as the square root of the crease count. If each crease is out by an independent amount, half the turns go one way and half the other, and they partly cancel. The same half-degree scattered rather than repeated gives 0.024 after eight creases, 0.029 after sixteen, 0.049 after thirty-two and 0.065 after forty-eight.

The ratio between the two is what a designer should carry: at eight creases the systematic error is three times worse, at thirty-two it is nearly six times worse, and the gap keeps widening because one grows like n and the other like √n.

What it looks like in the stack #

The numbers are abstract and the picture is not.

A perfectly folded accordion puts every panel exactly on top of every other. What a systematic error produces is a fan: each panel turned a little further than the last, so the stack opens out like a deck of cards pushed sideways.

What a random error produces is a blur: the panels roughly on top of one another, none of them quite aligned, no overall direction to the disagreement.

The two are easy to tell apart by eye and they mean different things. A fan says the process has a bias — the tool is set wrong, the paper is being fed at an angle, the fold is consistently being made a little short. A blur says the process is imprecise and unbiased, which is the ordinary condition of anything done by hand.

They also need different fixes. A biased process can be corrected once, at the tool, and the correction is worth n times its size. An imprecise one cannot be corrected at all; it can only be made more precise, and every halving of the error buys a halving of the drift rather than the factor of n the calibration bought.

What the drift is measured against #

A drift needs a reference, and the reference here is the folded state of the same pattern with every crease exactly where it belongs.

Both folded states are computed the same way and differ only in the crease angles, so the comparison is between two exact computations rather than between a computation and a measurement. That removes the usual worry about a small difference being an artefact of how the two things were produced.

It also fixes what “the far end” means. The quantity reported is the largest distance any corner of the last panel has moved, which is the strictest reasonable choice: the last panel is the one with the most creases behind it, and its corners are the points furthest from the folds that moved them. A gentler measure — the centre of the last panel, or the average over all panels — gives smaller numbers and the same two exponents, since the exponents are properties of how the errors combine rather than of which point is watched.

There is one thing this choice hides and it is worth naming. In a flat folded state the panels all end up in more or less the same place, so the lever arms are short and the drifts are modest. A partly folded array — which is what a deployable spends its working life as — has its panels spread across a much longer arm, and the same angular errors put its tip proportionally further out. The numbers below are therefore the mild case, and the pattern of the argument, not its magnitudes, is what transfers.

Which theorem was checked, and how #

The two laws are asserted rather than described, and each against its own shape.

For the systematic case the quantity checked is the drift per crease, which must be constant over the second half of the range. If it varies by more than six per cent the generator refuses to draw, because a drift that was not proportional to the count would mean the mechanism proposed above is not the mechanism operating.

For the random case the quantity is the drift per root crease, allowed a wider spread — thirty-five per cent — because a mean over forty samples is itself a noisy number and demanding tightness of it would be demanding the wrong thing.

Both are measured on folded states computed by composing reflections, and the folded states are themselves checked: every panel keeps its area exactly, and the layer count integrates back to the sheet. A drift measured on a folded state that had quietly stretched would be measuring the stretch.

There is a third thing the experiment establishes by not failing. Every one of these patterns is valid. A tilted crease that runs from one edge of the sheet to the other creates no interior vertex, so Kawasaki has nothing to evaluate, Maekawa has nothing to count, and the pattern passes every check this site has. The error is not a mistake any theorem can catch — it is a correct pattern that is not the intended one, which is exactly what a tolerance is.

The systematic case has a mechanical counterpart worth naming.

Where this sits, and what it is not #

The mechanism here is a composition of reflections on a sheet. It is not a mechanism in the engineering sense: nothing is counted, no mobility is computed, no linkage is analysed, and no claim is made about whether anything moves. A folded state is a destination, and this essay is about arriving at the wrong one.

That boundary is deliberate. Whether a rigid-panel version of a pattern can move at all, and how such a mechanism’s freedoms are counted, is a subject with its own owner and its own vocabulary, and nothing above needs any of it. The drift computed here would be identical on a sheet of paper folded by hand, which is the test of whether an argument is about mechanisms or about geometry.

What the model leaves out, and what happens then #

Three idealisations, and each makes the real answer worse rather than better.

The paper has no thickness. A real stack of thirty-two layers is millimetres deep, and the outer layers have further to travel round every fold than the inner ones. That is a systematic error, in the sense that matters here: it goes the same way every time.

The crease’s radius is the clearest case of that. It costs a fraction of a millimetre per fold, which is nothing once and several millimetres across a large tessellation — and it always costs in the same direction, so it accumulates linearly, exactly as a biased angular error does. An ambitious tessellation comes out short for the same reason a miscalibrated strip comes out fanned.

The creases are lines. A scored crease is a band, and where inside the band the paper actually breaks is another unbiased error on top of the placement.

The panels are rigid. They are not, and this is the one that helps: paper bends, and a stack that is fanning slightly can be persuaded back into line by the folder’s hands, distributing the error into small curvatures rather than a visible drift. That escape route closes completely for sheet metal and for panelled hardware, which is why the tolerance question is much more serious there than on paper.

The folklore limit on halving a sheet is the same shape of argument arriving from the material side.

What a folder and a factory each take from it #

For a person folding by hand, the useful reading is about when to check rather than how carefully to fold. Errors of the random kind grow slowly and forgivingly; errors of the biased kind grow without limit. So the check worth making is not “is this fold accurate” but “are my folds all wrong in the same direction”, and that is visible in the stack long before it is visible in any single crease.

For a process that makes the same fold many times, the reading is the reverse of the intuitive one. Improving the precision of a folding machine buys √n — a fourfold improvement in scatter buys a factor of two in the drift after any number of creases. Improving its calibration buys n. On a pattern with fifty creases, removing a systematic half-degree bias is worth more than making every fold four times more repeatable.

That is the practical content of the two exponents and it is why they are worth separating. The same specification — “creases accurate to half a degree” — describes two processes with drift figures differing by a factor of six.

There is a design response as well as a process one, and it follows from the lever arm rather than from the exponents. The drift at a point is the accumulated angular error times the distance still to go, so a pattern folded from the middle outward halves its worst lever arm compared with the same pattern folded from one end. That is why an experienced folder establishes the centre lines of a tessellation first and works outward in both directions, and why a machine that indexes from a datum in the middle of the sheet beats one that indexes from an edge.

It also argues against a particular kind of convenience. Registering every crease against the previous crease is easy — the paper is already there to align to — and it is the worst possible arrangement, because it makes every error part of the reference for every fold after it. Registering each crease against a mark on the original sheet keeps the errors independent, which converts the linear law into the square-root one. The same folds, the same accuracy, and a difference of a factor of six at thirty-two creases, decided entirely by what each fold is lined up against.

The other systematic cost does not respond to any of that, because it is not an alignment error at all: the paper genuinely is consumed, and no reference scheme puts it back.

The consequences are largest where the folding is done by machine and the sheet is long.

Who worked this out, and when #

Error propagation through a chain of operations is old and general and belongs to no field in particular; the specific form here, where the operation is a reflection and the chain is a fold sequence, is a consequence of writing folded states as compositions of isometries, which is how they have been treated since the field acquired a computational vocabulary.

The engineering side is much older and arrived from measurement rather than from geometry. Anybody making a folded metal component has known that repeated bends accumulate, and the working rule — that the last bend in a long sequence is the one that is out — is the linear law being observed rather than derived. What the geometry adds is the distinction between the two growth rates and therefore between the two remedies.

One thing the strip deliberately avoids is worth naming, because a real crease pattern does not avoid it.

Where the ladder goes next #

Two directions, and both are about what the idealisations were hiding.

The material direction is the crease that has a radius and everything else that is not true of paper. Every one of those is an error source, and the argument above sorts them by whether they are biased.

The geometric direction is back toward how rare a foldable pattern is. A pattern that satisfies the conditions exactly is a measure-zero object, and a folded sheet never satisfies them exactly. That two facts so uncomfortable together produce working models at all is a statement about how much a bending material forgives — and the size of that forgiveness is the real tolerance of the subject.