Dividing a loop into n

Folding a strip into thirds is the standing example of exact division by folding, and the method — Fujimoto’s — is one of the prettiest procedures in the subject.

Guess a third. Fold there. The leftover is two thirds; fold it in half and the new mark is nearer the true third than the guess was, by a factor of two. Repeat, and the error halves every time.

The method works because the strip has two ends, and the leftover is always measured against one of them.

Where the edges enter #

The iteration is: mark an estimate, fold the material between the mark and the far edge in half, and take the new crease as the next estimate.

The far edge is the operative phrase. It is the reference the leftover is folded against, it is what makes the fold exact rather than another guess, and it is a feature of the sheet rather than of the method.

So the method is not halve the error in the abstract. It is fold the leftover against the edge, which halves the error as a consequence.

A loop has no far edge #

Join the strip’s ends into a band. Now mark an estimate of a third of the way round.

The leftover is the two thirds between the mark and — where? There is no far end. Going one way from the mark, the material runs all the way round and comes back to the mark from the other side.

So fold the leftover in half has no meaning: the leftover is the whole band minus a point, and folding it in half against nothing is not an operation.

What replaces the edge #

The closure does, and it is a different kind of reference.

A band divided into $n$ parts has $n$ marks, and the condition on them is not that each is a certain distance from an edge but that the $n$ gaps between them are equal and sum to the whole loop. That is a condition on the set of marks rather than on each mark separately.

The practical procedure that results is different in shape. Mark an origin arbitrarily — a cornerless sheet needs one — and then the origin plays the part the edge played, and Fujimoto’s iteration runs exactly as before.

So the method survives, at the cost of one arbitrary act, and the loop’s closure is what the final check is against: after $n$ steps the marks should come back to the origin, and the amount by which they miss is the error.

The iteration, written out #

The arithmetic is short and worth having, because the essay’s claim is about what the arithmetic is measured against.

To divide a strip of length one into $n$ parts, guess a position $x$ near $1/n$. The leftover is $1 - x$, which should be $(n-1)/n$.

Fold the leftover into $n-1$ parts by repeated halving where $n-1$ is a power of two, or by the same procedure recursively otherwise. For thirds, $n - 1 = 2$: fold the leftover in half, and the new estimate is $x' = 1 - (1-x)/2 \cdot 2 / 2$ — more simply, the new mark sits at $(1+x)/2$ measured from the far end, and the error in it is half the error in $x$, with the sign reversed.

Iterate. After $k$ steps the error is the original divided by $2^k$, so ten steps take a first guess that is a centimetre out on a thirty-centimetre strip down to ten micrometres, which is well below the width of a crease.

That is why the method is exact in practice and not in principle: the limit is exact and every finite stage is not, and the finite stages get below the material’s own resolution after about ten folds.

Every step of that involves the far end. Remove it and the recursion has no base.

Two ways to divide a band #

Given the analysis, there are two procedures for a band and they are worth comparing.

Fix an origin and run Fujimoto. Mark a point arbitrarily, treat it as an edge, and proceed exactly as on a strip. The result is $n$ marks whose gaps are equal, positioned relative to the origin, and the closure at the end is a check.

Fold the band flat into $n$ layers. Collapse the band directly into a stack of $n$, which for even $n$ is what pressing it flat does. The creases are then at the division points automatically, and no iteration is involved.

The second is much better where it works and it works only for the $n$ a band can be collapsed into, which is an even number for a plain cylinder. For odd $n$ the collapse is impossible and the iteration is the only route.

That is a pleasing connection: the parity condition that decides whether a band folds flat also decides which divisions of it are available by collapsing rather than by iterating.

The closure check, used properly #

The amplified error is the most useful thing a band gives a folder and it is worth describing as a procedure.

Divide the band into $n$ by whatever means. Now walk round applying the constructed gap $n$ times from the origin. If the marks were exact, the last one lands on the origin; if not, it lands short or long by $n$ times the individual error.

For $n = 12$ that is a twelvefold magnification, which turns a tenth of a millimetre into more than a millimetre — visible, measurable, and correctable by adjusting the gap by a twelfth of the discrepancy.

That procedure is standard in metalwork and in clockmaking, where dividing a circle is a routine problem and the closure is the check that has always been used. Its arrival in folding is a consequence of the sheet being closed, and it is the one respect in which a band is a better object to construct on than a strip.

The error, which now accumulates visibly #

That last point is the interesting difference, and it improves the method rather than damaging it.

On a strip, the error after $k$ iterations is a small distance from the true position and there is no way to see it: the mark is where it is, and only a measurement would reveal that it is a hair off.

On a loop, the $n$ marks have to close. Walk round applying the same gap $n$ times and the accumulated error is $n$ times the individual one, and it appears as a visible gap at the origin.

So a loop makes the error legible at $n$ times its size, which is a genuine advantage: a division that looks perfect on a strip shows a visible mismatch on a band, and a folder can correct it.

That is the same amplification a machinist gets from measuring a circle’s circumference rather than a length, and it is used for the same reason.

Doing it with a band #

The whole essay fits on one loop of paper and takes about five minutes.

Cut a strip thirty centimetres long and three wide, and tape the ends into a plain band. Mark a point on it arbitrarily — that is the origin, and the arbitrariness is the point.

Now divide the band into thirds. Guess a third of the way round from the origin and pinch. Then fold the material between that pinch and the origin the other way round in half, and pinch again; the second pinch is nearer the true third.

Three or four iterations and the pinch stops moving perceptibly. Now put the third mark at the same gap again, and the fourth should land on the origin.

It will not, quite. The gap between where it lands and the origin is three times the error in the individual division, and it is visible where the individual error is not.

Adjust the gap by a third of the discrepancy, walk round again, and it closes. That is the whole procedure, and it converges faster than the halving does because the closure gives feedback the strip cannot.

Why the strip cannot give feedback #

The asymmetry is worth stating, since it is the essay’s one positive finding.

On a strip, a mark is at some position and there is no way to know whether it is the right position without measuring. The construction is exact by argument — the halving converges — and the folder has no independent confirmation.

On a band, the construction has a consistency condition: the parts have to fill the loop. That condition is not used in the construction and it is checkable afterwards, and it fails visibly when the construction is imperfect.

So a closed sheet supplies something a flat one does not: a place where an accumulated error becomes visible. That is the same phenomenon as the thickness offsets having to close round a loop, and in both cases the closure turns an accumulation into something checkable.

Which is a mild consolation for everything else a closed sheet takes away.

Which n a band admits by collapsing #

The alternative to iterating is collapsing, and it is worth being precise about when it is available, since it is much the better method where it is.

Pressing a plain band flat produces two creases — the two edges of the flattened strip — so it divides the band into two. Pressing the flattened strip flat again gives four, then eight, and so on: every power of two, by repeated collapsing, with no iteration and no error at all.

Any other $n$ needs the material to be gathered rather than folded in half, and that is the iteration.

The parity condition enters because a band collapsing into $n$ layers is a band with $n$ creases running round it, and an odd $n$ has no flat state. So a band cannot be collapsed into three, and thirds have to be iterated.

That is a satisfying alignment of two results that come from different places: the parity says which flat states exist, and the flat states are exactly the divisions available for free.

Powers of two by collapsing, everything else by iteration, and the odd numbers refused by a condition that has nothing to do with division.

The band that has to be divided #

The task is not hypothetical, which is worth saying since the essay is otherwise about a procedure nobody has needed.

A folded tube has facets round it, and how many is a design decision with a parity condition attached. Making one means dividing a band into that many equal parts — physically, on the material, before the creases go in.

For a manufactured tube the division happens on the flat sheet before joining, which is why the problem does not usually arise: the flat sheet has ends, Fujimoto works, and the tube inherits an exact division.

It arises when the tube exists first. Retro-fitting creases to an existing cylinder — a tube, a can, a rolled sheet already joined — means dividing a band, and then the procedure above is what is available.

That is a niche and it is a real one, and it is worth knowing that the answer is an arbitrary origin, then the usual iteration, with a closure check that makes the error visible.

What the strip’s ends were doing #

A last look at the mechanism, since the essay’s claim is that a familiar procedure has an unfamiliar ingredient.

Fujimoto’s method is normally described as halve the error, and that description is a summary of the outcome rather than of the operation. What is actually done at each step is to take the material between the current mark and the sheet’s far end, and fold it in half.

Folding something in half is exact because the two ends of the thing being halved are both available: one is the mark and the other is the sheet’s edge. Bring them together and the crease is exactly at the midpoint.

Remove the edge and there is nothing to bring the mark onto. The material between the mark and — nowhere — is not a thing with two ends.

That is why an origin has to be supplied. It is not a technicality about coordinates; it is that the operation the method performs needs two endpoints and a loop supplies one of them and not the other.

Exact, and what that word is doing #

A last note on vocabulary, since exact division is the anchor this essay sits under and the word carries a specific meaning.

A construction is exact when its limit is the true value and every step of it is a fold that could in principle be made perfectly. Fujimoto’s iteration is exact in that sense: the sequence converges to $1/n$ and each fold is a well-defined alignment.

It is not exact in the sense of terminating. There is no finite number of folds after which the mark is exactly at a third; there is a sequence of marks approaching it, and the folder stops when the error is below what the paper can hold.

Exact is not accurate is the collection’s phrase for the distinction, and it survives being moved onto a band unchanged. What the band adds is that the accumulated inaccuracy becomes visible at $n$ times its size, which does not make the construction more exact and does make it easier to get right.

Those are different virtues and it is worth having both words.

The general shape, once more #

Every classical construction in this subject measures against the sheet’s boundary, and the boundary has been so reliably present that the measuring has never been named as a step.

Fujimoto folds the leftover against the far edge. Haga’s construction brings a corner onto a point on an edge. Binary division starts by folding one edge onto the opposite one. Each is an operation on marked features of the sheet and each is described as though the sheet were incidental.

Remove the boundary and each of them needs an origin supplied by hand, after which they work exactly as before. So the correction is uniform across the whole of construction: one arbitrary act, and then everything.

That is a smaller correction than the ones this phase has made to flat-foldability, where a closed sheet acquires conditions that no amount of arbitrary choice removes. Construction loses a starting point; folding gains a constraint.

What is not available #

Two things, for completeness.

A canonical division. The origin is arbitrary, so the thirds of this band is not a well-defined set of points. Two folders dividing identical bands produce divisions that differ by a rotation, and neither is more correct.

Anything using both ends. Several classical divisions use both edges of a strip at once — folding one end onto a mark near the other, for instance — and a band has neither end.

Neither is fatal and both are the ordinary cost of a sheet with no distinguished points. The first is what a physical mark or a seam supplies in practice; the second means the procedure has to be chosen from the ones that work with a single reference.

What converges and what does not #

Worth being precise, since the method’s convergence is its whole claim.

The individual step still halves the error. Folding a known interval in half is exact, and the arithmetic that makes Fujimoto’s iteration contract is arithmetic about intervals rather than about sheets.

The origin is arbitrary and never converges. It is not an error; it is a choice, and the result is exact relative to it.

The closure is a check rather than a step. It does not participate in the convergence; it reports the accumulated error at the end.

So the honest description is: the method transfers, it needs an arbitrary origin, and it gains a closure check the strip version does not have.

Why this is not a limitation #

Dividing a band into $n$ parts is a real task — a folded tube with $n$ facets round it needs exactly that, and an even $n$ if it is to flatten — and the procedure above does it.

What the essay is about is that the method’s mechanism is not what it appears. Halve the error is the description everybody gives and it is a consequence; fold the leftover against the sheet’s edge is the operation, and the sheet is doing a job that its ubiquity hides.

Which is the same observation this phase has made in every other corner of the subject, arriving here in a procedure that is two hundred years old and works perfectly.