Cheap where it reaches

Two essays on this site answer the same question in incompatible ways.

A fold needs something to align computes what a folder can refer to: start with a bare square, apply every axiom to every pair of things present, and see what marks appear. It is a breadth-first search over folding itself, and it answers “what is reachable in k folds”.

One crossing, and then another gives a rule that reaches 1/n for every n, exactly, by a construction anybody can follow without searching for anything. It answers the question a folder actually has, which is how to reach one particular fraction.

Put the two side by side and they disagree about everything.

What two folds reach #

One fold from a bare square leaves nine references, and every coordinate among them is nought, a half or one. The only fraction available is the obvious one.

Two folds change that completely. Using only the linear axioms — the fold through two points, the fold placing a point on a point, the fold placing an edge on an edge — the second round leaves 133 references, and the fractions that appear among their coordinates are 1/2, 1/3, 1/4, 1/5, 1/6, 1/8 and 1/12.

A third in two folds. A fifth in two folds. A twelfth in two folds.

The ladder takes three folds for a third, five for a fifth and twelve for a twelfth. So wherever the closure arrives, it arrives sooner — usually much sooner — and the gap grows with the denominator.

And what it does not reach #

The list above is complete for two folds, and what is missing from it is the interesting half.

There is no seventh. No ninth, no tenth, no eleventh. The closure at that depth simply does not contain them, and the search’s output is not “not yet” — it is a list, and they are not on it.

So the two methods fail in opposite ways. The ladder is uniform and slow: it reaches every fraction and takes its time about it. The search is fast and patchy: it reaches some fractions almost immediately and is silent about the rest, in the strong sense that a search which has enumerated a depth completely has nothing more to say about that depth.

The bisector buys nothing at all here #

The closure can be run with the angle bisector included, and doing so quadruples the references: 565 instead of 133 at the second round, with 432 of them carrying a coordinate that is not a fraction.

It adds not one new unit fraction. The list is 1/2, 1/3, 1/4, 1/5, 1/6, 1/8, 1/12 with the bisector and without it.

That is a sharper version of a finding the rung below made about the wall between the rationals and everything else. The bisector is the quadratic operation in the axiom list; its neighbours are linear; and what it produces is a whole new population of irrational references — 22.5° angles and the two halves of the square’s bisected diagonal — while leaving the rational part of the closure exactly where it was. The two halves of the axiom set reach into different number systems and they do not help each other.

The two lists, side by side #

It is worth writing the comparison out, because the shape of the disagreement is more interesting than either column.

The closure’s column is almost constant and the ladder’s is linear, so the ratio between them grows without limit over the fractions the closure reaches — and at the fractions it does not, the ratio is undefined rather than large.

There is no denominator at which the ladder wins on folds. What it wins on is coverage, and coverage is a different axis entirely.

Why the ladder is slow #

The ladder’s inefficiency is not an accident of a badly chosen construction, and the reason is worth having.

Each rung uses exactly two lines: an anti-diagonal that is already there, and a line through one previously found mark. So each rung consults one earlier result. The closure at each round consults every pair of everything — nine references and a handful of lines at the first round gives hundreds of candidate folds at the second — so it is doing an enormous amount of work per round and getting an enormous amount of reach for it.

That is the ordinary trade between a rule and a search, stated in folds. A rule is a path; a search is a frontier. A path reaches a chosen destination and takes as many steps as the path is long; a frontier reaches everything nearby at once and stops.

Where the model stops #

The third round is not computed. The closure is enumerated to depth two and refuses to attempt depth three, because the candidate count grows faster than anything worth running in a build. So “no seventh in two folds” is exactly what is claimed; whether a seventh appears at three folds is unknown here, and it very likely does.

That limitation is the whole reason the ladder matters. A search that goes silent at depth two has said nothing about depth three, and a rule that reaches 1/7 in seven folds has said something — a route exists, here it is, and it will not need a search.

Nothing here says the closure’s routes are the shortest. The closure establishes that a fifth is reachable in two folds and it does not establish that two is the minimum, which needs the first round to have been checked as well — and it has: one fold reaches only halves. So for the fractions in the table the closure’s counts are genuinely minimal, and that is a property of the enumeration being complete at both depths rather than of the search being clever.

Fold counts are not effort. A fold in the closure is a fold anybody can make and a fold in the ladder is too, but the finding is not free: a folder using the closure’s answer has to know which of hundreds of candidate folds to make, in which order, and that knowledge is the output of a computation nobody performs at the table. The ladder needs no such knowledge. Comparing the two by fold count flatters the search.

Unit fractions are not everything. The closure contains many rationals that are not of the form 1/n, and the comparison above is restricted to unit fractions because those are what the ladder produces. A fuller comparison would ask which whole rationals each method reaches, and the answer would favour the search still further.

Neither method knows about the other. The ladder does not consult the closure and the closure does not know the ladder exists. That is deliberate — they are independent constructions, and their disagreement is evidence rather than a bug — but it also means neither is optimised against the other, and a hybrid that used the closure where it reaches and the ladder where it does not would beat both.

What “reachable in two folds” means for a hand #

There is a gap between the closure’s arithmetic and a folder’s afternoon, and it deserves stating rather than being left as an asterisk.

The closure’s second round is 133 references, which is the set reachable in two folds. It is not a set anybody carries around. A folder wanting a fifth has to know which two folds, in which order, out of some hundreds of candidate pairs — and knowing that is the output of the enumeration, not something the enumeration makes obvious.

The tradition solved this by memorising the useful cases. Haga’s theorem is one of them written down and named; the standard ways to find a third, a fifth and a sixth are others, passed around as recipes. Those recipes are the closure’s short routes, discovered one at a time over decades by people folding paper, and the enumeration’s contribution is to say that the list is complete rather than merely long.

What a silent search is worth #

There is a general point here about search results and it is worth separating from the folding.

A completed enumeration to depth two produces two different kinds of statement. A third is reachable in two folds is a positive result with a witness — here are the two folds — and anybody can check it by making them. A seventh is not reachable in two folds is a negative result, and it rests entirely on the enumeration being complete.

This site has written about that asymmetry directly, and the closure is a good example of the honest version: the enumeration is exhaustive by construction, it refuses depths it cannot complete rather than sampling them, and the negative it produces is scoped to a depth rather than stated absolutely. “No seventh in two folds” is a claim that can be checked by re-running the same enumeration; “no seventh” would be a claim about all depths and is not made.

What the two methods bracket #

Neither method computes the true cost of a fraction. Together they bound it, and a bracket is a stronger object than either column on its own.

The closure is a closure: each round contains the one before it, because the references a fold produces are added to the marks already on the paper rather than replacing them. So a fraction reachable in two folds is reachable in three, and the closure’s counts are upper bounds which happen to be tight at the depths it has finished. The ladder is an upper bound of a different kind — it exhibits a route of length n for 1/n and says nothing at all about whether a shorter one exists.

Put together they give every unit fraction an interval. A half costs exactly one fold, because the first round is a complete enumeration and it reaches halves. A third, a quarter, a fifth, a sixth, an eighth and a twelfth cost exactly two: the second round finds them and the first round, enumerated completely, contains none of them, so two is minimal rather than merely sufficient. A seventh costs at least three and at most seven. A ninth costs at least three and at most nine, a tenth at least three and at most ten, an eleventh at least three and at most eleven.

Seven of the first eleven unit fractions therefore have an exact answer, and four have a range spanning most of an order of magnitude. That is the state of the knowledge, and the shape of it — exact where the search finished, wide where it stopped — is the entire relationship between the two methods, stated as a table of intervals rather than as an argument about which is better.

The gaps are not where composition would put them #

There is a reason to expect a tenth to be cheap which turns out to be wrong, and the error is worth making explicitly because it is the natural one.

A tenth is half of a fifth. A half costs one fold and a fifth costs two, so a tenth ought to cost three: construct the fifth, then halve it. But the reachable set at a fixed depth is not closed under that kind of composition, and there is no depth at which it is. Composition costs folds, and it costs them because a construction that halves an existing reference is not free merely because halving was cheap the first time — the second halving starts from a different sheet, with different marks on it, and the axioms have to find the new point from the ones now present.

What is closed is the number system. Every reference the three linear axioms produce has rational coordinates, and the rationals are closed under products and quotients alike; a tenth is certainly reachable and is certainly not far away. The closure’s silence about it at the second round is a statement about a budget, not about the arithmetic, and reading the one as the other is the standard way to misread a reachability result.

So the four missing denominators are missing for two different reasons that the enumeration cannot tell apart. A tenth is absent because the third round was not looked at. A seventh may be absent for that reason too, or because seven is genuinely awkward in the way it is awkward for the compass and for the ladder alike. At depth two the search says only that neither has arrived, and it is exactly the same silence in both cases — which is why the next round is the computation worth doing rather than the argument worth having.

Which one a folder should use #

Both, and the division is straightforward once the two are laid side by side.

For a fraction the closure reaches — halves, thirds, quarters, fifths, sixths, eighths, twelfths — the closure’s route is shorter and the folds are ordinary. That covers essentially every division anybody does in practice — the exact-division constructions the tradition carries are all in that range, and so are the grids box pleating is laid out on — which is why nobody has ever needed a general rule: the fractions people want are the ones two folds happen to reach.

For anything else the ladder is what there is. A seventh, a ninth, an eleventh: the rule gives a route, the route is exact, and the route is long. Those are also the fractions that come up when a construction demands them rather than when a person wants them — dividing a strip for a tessellation, placing a reference for a design — and in that setting the fold count matters less than the existence of a route.

Seven is the first number that costs anything, in both directions. The closure misses it; the ladder needs seven folds for it; the compass cannot reach the heptagon and a fold can. It is a good number to keep an eye on.

The shape of the disagreement, generalised #

Two methods that answer the same question and disagree about cost are common enough. What makes this pair worth an essay is that they disagree about which questions have answers, and that is rarer.

The ladder has an answer for every n and a bad one. The closure has an excellent answer for some n and none at all for the rest. Neither is a refinement of the other, neither dominates, and the union of the two is strictly better than either — which is unusual, because most pairs of methods for one problem are related by one being a special case of the other.

The reason they are independent is that they are searching different things. The ladder searches nothing; it is a path chosen in advance, and the fraction is the input. The closure searches folds and reports fractions as a by-product, so the fraction is an output and nobody asked for the one they wanted. A method whose answers are outputs will always have gaps where a method whose answers are inputs does not.

Where the ladder goes next #

The obvious continuation is the third round, and it is a computation rather than an idea: enumerate the closure one fold deeper, find where the seventh first appears, and put a number on how much the search beats the rule at the denominators it currently misses. The obstacle is size rather than method, which makes it the kind of thing that gets done when somebody wants it enough.

The other direction is the hybrid. A construction that ran the closure to whatever depth is affordable and fell back to the ladder only where the closure is silent would reach every fraction with the shortest known route to each — and the interesting output would not be the constructions but the table: the true fold cost of every unit fraction, which nobody has and which is not obviously anything simple.