Exact is not accurate

One crossing, and then another built an exact ladder: rung k lands on 1 / (k + 1) as a rational number, checked in integer arithmetic with no tolerance consulted anywhere. Folding a strip into thirds is Fujimoto’s method, which is convergent: it never lands on the fraction and its error contracts at every fold.

Set beside each other as mathematics, the comparison is over before it starts. One of them is right and the other is approximately right.

Set beside each other as instructions for somebody holding a sheet of paper, the comparison is not over, because a folder cannot place a crease exactly. Every fold carries an error of a few tenths of a millimetre, and the question is not what each method does to a perfect input but what it does to the errors made along the way.

What each method does to a mistake #

The natural guess is that the exact ladder amplifies errors and Fujimoto’s method damps them. Half of that is right.

The ladder’s rung is y ← y / (1 + y), whose derivative is 1 / (1 + y)². That is less than one, so the ladder shrinks an error too — by a factor of about twenty over seven rungs, from 44 per cent of its original size after one to 4.9 per cent after seven.

Fujimoto’s step is x ← (1 − x) / (n − 1), whose derivative is −1 / (n − 1). Every error is divided by n − 1 at the next fold — a factor of a billion over ten folds — so old errors vanish completely and what survives is the last fold’s own.

So the ladder’s trouble is not amplification. It is that it makes a fold for every part, each of them contributing an error of its own, and the gain approaches one as y approaches 1 / n — so the later contributions barely fade at all. The total therefore grows with the number of parts, while Fujimoto’s does not grow at all because only the last fold’s error survives.

The crossover, and where it is #

At half a millimetre of placement error on a 150 mm sheet:

The exact method wins at three parts and loses at everything above it. By sixteen parts it is nearly twice as far out.

The crossover is at four, which is a number anybody reaches in a minute of folding. It is not a number at the far end of a table where nobody goes.

And it does not move #

A steadier hand does not rescue the exact method. Both methods’ errors are proportional to the hand’s own — every error in either procedure enters linearly and is transformed linearly — so improving the hand scales both curves by the same factor and moves nothing.

Measured at 0.1, 0.2, 0.35, 0.5, 0.8 and 1.2 millimetres, the crossover is at four parts every time.

That is what makes the result about the methods rather than about the folder. Exactness and accuracy are not two amounts of one thing, with a threshold where enough of the second substitutes for the first. They are properties of different objects: exactness is a property of the construction and accuracy is a property of the hand, and the construction’s exactness is invisible to a hand that is not exact.

The fold counts are not the same either #

There is a second difference between the methods that the table above hides, and it cuts the other way.

The ladder to n parts costs n folds — one for the anti-diagonal and one per rung — so dividing into sixteen is sixteen folds. Fujimoto’s method costs however many iterations are run, and eight is plenty for any n because the error contracts by a factor of n − 1 each time; at n = 16 it contracts by fifteen, so three folds already have it below a thousandth of the strip.

So at large n the convergent method is cheaper as well as more accurate, and at small n the ladder is cheaper. At three parts the ladder is three folds against Fujimoto’s eight, and it is also the more accurate of the two — which is a clean win and is the only one it gets.

That makes the practical rule short. For halves, thirds and quarters, use the exact construction. For anything else, iterate. The rule is what folders do already, and this essay is the reason rather than a correction.

The ladder’s error is a square root #

The six figures in the table have a shape, and fitting it turns the comparison from a crossing of two curves into a law and a constant.

The ladder makes one fold per part, each contributing an independent error, and the gains that would shrink the earlier ones approach one as the rung approaches $1/n$. So the contributions add in quadrature and the total should grow as $\sqrt{n}$.

It does. Fitting $a + b\sqrt{n}$ to the six measurements gives

$$\text{error} \approx 0.119 + 0.110\sqrt{n}\ \text{mm}$$

which reproduces 0.31, 0.34, 0.36, 0.42, 0.50 and 0.56 to within a hundredth of a millimetre at every point. Fujimoto’s column is a constant falling to 0.29 mm and staying there.

So one method is $\sqrt{n}$ and the other is flat, and the crossing between them is arithmetic rather than a coincidence of the sizes tabulated.

Which puts the crossover where it is and predicts the rest #

Setting the fit against the constant, the ladder loses as soon as $0.110\sqrt{n} > 0.17$, which is $n > 2.4$ — so the ladder wins at three and loses from four onward, exactly as the table says.

And the law extrapolates where the table stops. At thirty-two parts the ladder gives 0.74 mm against Fujimoto’s 0.29; at a hundred, 1.22 mm against 0.29, a factor of four.

The gap widens as the square root of the parts, forever. There is no size at which the exact method recovers, and the essay’s “nearly twice as far out at sixteen” is the beginning of a divergence rather than the end of a comparison.

And it says which quantity the hand controls #

The two forms also separate what a folder can improve.

Fujimoto’s error is one fold’s error, so it is the hand’s error times a fixed factor — 0.58 of it here — and halving the hand halves the answer, at every $n$.

The ladder’s error is $\sqrt{n}$ folds’ worth, so halving the hand halves it too, but the $\sqrt{n}$ stays. A steadier hand buys the same proportion on both curves and never moves the crossing, which is the essay’s measured result arriving as an algebraic consequence of the two forms rather than as six coincidences at six values of $\sigma$.

What exactness is still for #

The result is not that the exact ladder is useless, and it is worth being careful about what has and has not been shown.

Exactness is a property of the construction and it is real. The ladder lands on the fraction as a rational number, checked as integers. That is a fact about the geometry and no measurement of hands touches it.

It is what makes the construction checkable. A convergent method has no statement of the form “this crease is at exactly one seventh”; it has a statement about a limit, and a limit is not a crease. Everything this site does with exact divisions — verifying that a pattern’s vertices are where they are claimed to be, proving that a construction reaches a number — needs the exact version.

It is what makes exactness compose. Dividing without measuring is the whole point of folding constructions — no ruler, no scale, no measurement — and the argument that a fold reaches an exact fraction is what distinguishes the method from estimating. Take that away and folding is a way of guessing carefully.

And it is what a machine wants. A cutting machine places creases to a hundredth of a millimetre and has no accumulating hand error, so for it the ladder is simply correct and Fujimoto’s method is a way of spending eight folds to approach something it could place directly.

The result is about the third audience: a person, with paper, dividing a strip. For them the exact method’s virtue is not available, because the thing that makes it exact is an assumption about placement they cannot supply.

Why this is not the same as the reference-point result #

Closer than a crease is wide measured a related-sounding thing and reached a different kind of conclusion, and separating the two is worth a paragraph.

That rung asked what a folder can distinguish: the reference points three folds reach include pairs 0.2 mm apart, which is narrower than a crease, so the axioms reach marks the paper cannot tell apart. It is a statement about resolution — about two things being too close together to be different.

This rung asks what a folder can place: a crease intended to be at one fifth lands somewhere near one fifth, and the question is how near. It is a statement about accuracy — about one thing not being where it should be.

The two combine badly. A construction whose marks are 0.2 mm apart, executed by a hand that places creases 0.5 mm out, is a construction whose output is indistinguishable from several of its neighbours. Neither essay alone says that; together they say that the depth at which folding constructions stop being executable is shallower than either measurement suggests.

Which theorem was checked, and how #

Four things.

The exact ladder is exact, in integer arithmetic: it reaches one twelfth as a rational number with no tolerance consulted. That is the fact the whole comparison is against, and it is re-checked here rather than quoted.

The ladder’s arithmetic is not the issue. Run in doubles to the twenty-fourth rung, the ladder agrees with the exact rational to a part in a million million — so the errors in this essay are the hand’s and nothing in the computation contributes to them. That is worth saying because “floating point” is the first explanation a reader reaches for and it is wrong.

Fujimoto’s contraction is measured rather than asserted: its error falls by exactly half at every fold for a division into three, which is the ratio the whole argument rests on.

And with a hand that makes no error at all, the exact method must win at every division — because a convergent method still never arrives. It does, and that is the refusal that keeps the comparison honest: a measurement in which Fujimoto’s method won with a perfect hand would be measuring its own arithmetic.

What the numbers do at the ends #

Two features of the table are worth reading off, because both of them are the mechanism rather than the result.

Fujimoto’s error is flat. It is 0.35 mm at three parts, 0.29 at eight, and 0.29 at sixteen — barely moving across the whole range. That is exactly what “only the last fold survives” predicts: the final error is one fold’s worth, scaled by how much the last step contracts, and the contraction gets stronger as n grows so the number falls slightly.

The ladder’s error grows sub-linearly. Sixteen folds’ worth of error is not sixteen times three folds’ worth; it is 0.56 against 0.31, less than double. That is because the errors partly cancel — they are independent, so they add in quadrature rather than directly — and because the earlier ones have been shrunk by the rungs after them.

So neither curve behaves the way a first guess would have it. The convergent method is not gradually accumulating and the exact method is not exploding; one of them has a fixed error and the other has a slowly growing one, and they cross where they cross.

Where the model stops #

The model of a folder is one number: how far out a crease is placed, drawn independently at every fold. Real hands are not like that. Errors are correlated — somebody who folds slightly short tends to keep folding slightly short — and a systematic error behaves completely differently from a random one, growing linearly rather than as a square root, which the tolerance ladder measured for a different question and a folding sequence’s own compounding is the construction-side version of.

A systematic error would hurt Fujimoto’s method more than this model suggests, because the contraction that removes an old random error does not remove a bias that is re-applied at every fold. The comparison here is therefore the favourable case for the convergent method, and saying so is part of stating it.

The model also gives Fujimoto’s method eight folds regardless of n, which is generous at small n and stingy at large. Varying the fold budget would move the curve a little and not the crossing, since the contraction is already complete after three or four steps at any n above four. Nothing here models what a folder actually does either, which is to look at the result and adjust. A person dividing a strip into fifths folds, checks whether the last band matches the others, and corrects — which is a feedback loop, and it is much closer to Fujimoto’s method than to the ladder. That may be the real reason the convergent method is what people use.

What the picture cannot show #

Two curves crossing at n = 4 is a picture of an average. Each point is four thousand runs and what a folder gets once is a draw from a distribution, not the root-mean-square — so a single attempt at five parts can easily come out better with the exact method, and the picture says nothing about that.

Nothing shows a hand. The whole model of the folder is one number, and a figure of it would be a figure of a normal-looking scatter that carries no information the number does not.

The generalisation #

The useful statement is about fixed points against sequences, and it is one that recurs wherever a procedure is executed by something imperfect.

A construction that lands on its answer is a sequence of steps, and errors made in the steps accumulate in the result. A construction that converges to its answer is an iteration with a contracting fixed point, and a contracting fixed point is self-correcting: an error made along the way is forgotten, because the iteration is pulling toward the answer regardless of where it is.

So an exact method transmits its execution errors and a convergent one absorbs them, and the choice between them depends entirely on whether the execution is exact. For a symbolic computation it is, and the exact method is strictly better. For anything with a hand, a ruler or a tolerance in it, the convergent method has a property the exact one cannot have at any price.

That is a general enough statement to be worth carrying out of origami. It is why iterative methods are used for things that could be solved directly, why a self-correcting measurement is preferred to a chain of exact ones, and why “exact” is a description of a formula rather than a promise about a result.

Who found it, and when #

Fujimoto’s method dates from the 1970s and is standard practice; the crossing ladder is older in spirit and appears in several forms in the origami-mathematics literature. That the convergent method is what folders use is not a discovery — it is simply what people do, and the usual explanation is that it is easier to remember.

The comparison does not appear to have been made. It is the kind of thing that falls between two audiences: the mathematics of exact folding construction is written for people interested in what a fold can reach, and the practice of dividing a strip is written for people who want to divide a strip. Neither has a reason to model a hand.

The one number worth taking away is the crossover, and it is small enough to be surprising: four. Past a division into four parts, the method that never arrives arrives closer.

Where the ladder goes next #

The immediate continuation is the systematic case. A folder with a bias is a different model, the contraction that saves Fujimoto’s method does not remove a repeated bias, and the crossover would move — possibly a long way. That is a short computation and it would say which model this essay’s answer depends on.

The other direction is the feedback loop nobody has modelled: a folder who checks and corrects. That is a third method, it is what people actually do, and it has the convergent method’s self-correction with the exact method’s target. Whether it beats both is a question about how well a person can judge a mismatch by eye, which is a measurement about people rather than about paper.