Origami & Folding
what one sheet can do
Axioms and construction

One crossing, and then another

Folding a strip into thirds by Fujimoto's method halves the error at every fold and never reaches a third. There is a construction that arrives instead: cross the square's diagonal with a line through the mark you already have, and the crossing lands on the next fraction exactly — one fold per step, all the way down.

Assumes Folding a strip into thirds and Dividing without measuring.

18 min read 8 figures Folding beats the compassOne sheet, no cuts

This site has two essays about getting a fraction out of a square and both of them are about approaching one.

Folding a strip into thirds is Fujimoto’s method: guess a third, fold the rest over, fold again, and watch the error halve. It converges, it converges quickly, and the error is never zero. Dividing without measuring is Haga’s theorem, which is exact — and reaches one particular family of fractions, arriving at thirds and fifths as consequences of a specific construction rather than as members of a series.

Missing from both is the plainest thing anybody could want: a rule that reaches any fraction, exactly, without converging.

One crossing, and then another: 1/5 of the squareThe anti-diagonal is folded once. Crossing it with the line from the corner through the mark at one half gives one third; crossing it with the line through one third gives one quarter, and so on. Every crossing is exact, and the ladder costs one fold per rung after the first.1/25 folds to reach 1/5, and every mark on the way is exactthe solid line is the one fold used at every step; the dashed lines are the stepsnothing here converges — each crossing lands on its fraction and stops
Fig. 1 The ladder to a fifth. The solid line is the square’s anti-diagonal, folded once and used at every rung; each dashed line runs from a corner through the mark reached at the previous rung, and where it meets the anti-diagonal is the next one.

The crossing

Take the unit square. Fold the anti-diagonal from (0, 1) to (1, 0), which is one fold through two corners and therefore the cheapest fold there is.

Now suppose a mark at height 1/k is already available on the right-hand edge. Fold the line from the origin through that mark. The two lines cross at x = k/(k + 1), y = 1/(k + 1), so the crossing is at height 1/(k + 1).

Start with a half — the crossing of the two diagonals, or a single fold edge-to-edge — and the ladder runs 1/2, 1/3, 1/4, 1/5 and onward as far as anybody has patience for. Each rung is one fold, because the anti-diagonal is already there.

Exact at every rungEach rung of the crossing ladder, the fraction it lands on as an exact ratio of integers, the number of folds it took, and — for comparison — how far the halving method still is from the same fraction after ten folds. One arrives and the other approaches.partswhere the crossing landsfoldshalving, after 10 folds21/22off by 1.00e-131/33off by 6.51e-541/44off by 2.54e-651/55off by 1.91e-761/66off by 2.39e-871/77off by 4.25e-981/88off by 9.74e-1091/99off by 2.69e-10the fractions are computed as pairs of integers, so “exact” is settled by a comparison and not by a tolerancethe same construction in doubles agrees to about three parts in 10¹⁷, so this is not a claim about roundingit is a claim about arriving
Fig. 2 Each rung, the fraction it lands on as an exact ratio of integers, the folds it took, and how far Fujimoto’s method still is from the same fraction after ten. One arrives and the other approaches.

What “exact” is being claimed

There are two things that phrase can mean and only the strong one is true here, so the check was built to distinguish them.

Every coordinate on the ladder is computed as a pair of arbitrary-precision integers. The intersection of two lines through rational points is rational, so the whole construction stays inside the integers, and the claim that the eleventh rung is one eleventh is settled by comparing two integers rather than by asking whether a number is within a tolerance of another. Checked that way for every n from 2 to 24, the ladder lands on 1/n — and the fold count comes out at n, one for the anti-diagonal and one per rung.

The same construction was then run a second time in doubles, and the result was not the expected one. The two agree to about three parts in 10¹⁷ at every rung, which is to say the floating-point route is fine. That is worth reporting rather than hiding, because it locates the claim: exactness here is not a statement about rounding — an ordinary double would do — it is a statement about the construction, which lands on the fraction rather than approaching it.

Folding a strip into thirdsA guess, and then halving. Each fold moves the mark to the midpoint of one of the two pieces, and each fold halves the distance to the exact division — so the error falls geometrically from whatever the first guess was. The sequence of halvings is read off the binary expansion of the fraction rather than chosen, and the halving of the error is asserted rather than observed.guessoff by 0.16667fold 1off by 0.08333fold 2off by 0.04167fold 3off by 0.02083fold 4off by 0.01042fold 5off by 0.00521fold 6off by 0.00260solid mark — the fold is aiming at 1/3; dashed — at where 1/3 has gonehalving word R L — period 2, from 2^2 − 1 = 3 × 1every fold halves the error exactly, so 6 folds divide it by 64
Fig. 3 Fujimoto’s method for comparison, from the rung below. The error falls by exactly one half at every fold, which is a splendid rate and is not zero at any number of folds; recomputed here rather than quoted, so the two methods are measured by the same file.

The difference that makes

A method that converges and a method that arrives are not distinguishable by their outputs at any practical precision. Fujimoto’s third after ten folds is 0.333333, which is a third to more decimal places than a sheet of paper has. If the only question were how accurate, there would be nothing to choose between them.

The difference is what happens next. A mark produced by convergence is an approximation and everything built on it inherits that; a mark produced by a crossing is a reference point, exactly where it says, and can be used as the input to the next construction without a second thought. That is the whole reason the ladder can be a ladder: each rung uses the previous rung’s mark as an exact input, and a converging method has no exact input to give.

One crossing, and then another: 1/7 of the squareThe anti-diagonal is folded once. Crossing it with the line from the corner through the mark at one half gives one third; crossing it with the line through one third gives one quarter, and so on. Every crossing is exact, and the ladder costs one fold per rung after the first.1/27 folds to reach 1/7, and every mark on the way is exactthe solid line is the one fold used at every step; the dashed lines are the stepsnothing here converges — each crossing lands on its fraction and stops
Fig. 4 The difference that makes: sevenths, reached by the ladder rather than by consulting a set of reference points. Each rung is one crossing of two folds, and the rung after it uses what the last one made.

Convergence is a property of a process and exactness is a property of a point. Fujimoto’s method has the first and never the second; the crossing has the second and needs no first.

Reading the ladder as a continued fraction

There is a second way to see why the crossing does what it does, and it is worth having because it explains the shape of the recursion rather than merely verifying it.

Write the crossing’s rule as a map on the height: the mark y goes to y/(1 + y). Take reciprocals and the map becomes 1/y ↦ 1/y + 1 — adding one. So the ladder is the sequence 2, 3, 4, 5 seen through a reciprocal, and the reason it hits every whole number is that adding one to a whole number hits every whole number.

That is also why nothing more interesting comes out of it. The map is as simple as a map can be, its orbit is an arithmetic progression, and the fractions it reaches are exactly the reciprocals of the integers. Starting the ladder somewhere else — from 2/5 rather than from 1/2 — walks a different orbit and lands on 2/7, 2/9, 2/11, which is the same recursion applied to a different seed and reaches a different family.

One crossing, and then another: 1/9 of the squareThe anti-diagonal is folded once. Crossing it with the line from the corner through the mark at one half gives one third; crossing it with the line through one third gives one quarter, and so on. Every crossing is exact, and the ladder costs one fold per rung after the first.1/29 folds to reach 1/9, and every mark on the way is exactthe solid line is the one fold used at every step; the dashed lines are the stepsnothing here converges — each crossing lands on its fraction and stops
Fig. 5 The ladder run to a ninth. Nine marks, nine folds after the first, and the spacing visibly compressing toward the corner — which is the reciprocal map doing its work, since equal steps in 1/y are unequal steps in y.

The compression is a practical limit rather than a mathematical one. Twelve rungs down, consecutive marks are a hundredth of the sheet apart, and a fold aimed at one of them by hand is aiming at a target narrower than the crease it will make.

One more fold reaches everything

The reciprocal reading says the ladder’s orbit is an arithmetic progression, and that is a limitation as well as an explanation. Seeded at 1/2 it reaches the reciprocals of the integers; seeded at 2/5 it reaches 2/7, 2/9, 2/11 — the denominators rising by two each time. In general a seed of p/q reaches p divided by q, q + p, q + 2p, and no other fraction at all.

So the ladder on its own does not reach every fraction. It reaches one arithmetic progression per seed, and 3/5 is not on any progression the square hands over for nothing.

One more fold fixes it, and it is a fold the ladder is already making. Folding the edge to a mark at height y leaves a mark at 1 − y, which costs one fold and is the same operation the ladder uses to turn its marks into a division. In the reciprocal coordinate the crossing is add one and the complement is divide by one less than itself, and those two together do not stay inside any progression.

Run them on 3/5, which the ladder alone cannot reach. Start at a half. Cross: a third. Complement: two thirds. Cross: two fifths. Complement: three fifths. Four operations, from the two the square already provides.

The continued fraction is what is being walked

That is not an accident of the example, and the structure underneath it is the one this recursion has always belonged to.

Adding one and taking the complement are exactly the two moves a continued-fraction expansion is built from, so the reachable set from a half is every rational strictly between nought and one — and the number of moves a target takes is governed by the length of its expansion rather than by the size of its denominator.

That explains the ladder’s cost, which otherwise looks like an unfortunate accident. One over n has the shortest possible expansion, a single term equal to n, and a single term of size n is walked by adding one n times. So the reciprocals are simultaneously the easiest fractions to state and the most expensive ones to reach: their expansion is short and its one term is large.

A fraction with several small terms goes much faster. Three fifths took four moves against a fifth’s five, and a denominator in the hundreds with a well-behaved expansion can be cheaper than a thirteenth. The fold count tracks the continued fraction and not the denominator, which is the opposite of what the ladder’s own table suggests, because the table is a table of exactly the worst family.

That also disposes of the essay’s remark about composite denominators taking a shorter route through their factors. It is true and it is a special case: factoring is one way to find a shorter walk, and the continued fraction finds the shortest one whether the denominator factors or not.

Which folds these are

Each rung uses two folds and neither is exotic.

The anti-diagonal is axiom one: the fold through two given points, here two corners of the square. The line from the origin through the mark is axiom one again. So the entire ladder is built from the simplest of the seven, used repeatedly, and it never needs the bisector, the perpendicular or anything cubic.

The seven ways to specify a foldThe Huzita–Hatori axioms: every fold that can be specified by bringing existing points and lines into coincidence. The list is complete — six were catalogued in 1991, the seventh in 2001, and no eighth exists. The sixth is the one a compass cannot reach.axiom 1through two pointslinearaxiom 2point onto pointlinearaxiom 3line onto linelinearaxiom 4through a point, square to a linelinearthe degree each axiom can solve — one of them is why paper beats the compass
Fig. 6 The first four axioms. The ladder uses only the first of them, twice per rung — which is why it stays inside the rationals, and why the reference closure that includes the bisector reaches a different and larger set.

That is not an accident, and it is the same fact the reference closure found from the other side: the linear axioms produce rational coordinates and nothing else, because intersecting two lines through rational points is a rational operation. The ladder is a directed walk through that rational set, aimed at 1/n.

Where the marks actually land

There is a second thing on the ladder worth noticing, and it is the x-coordinate.

The crossing that gives 1/(k + 1) in height gives k/(k + 1) across, so the same fold that produces a fifth on one edge produces four-fifths on the other. Every rung therefore hands over two marks, and running the ladder to n leaves the marks 1/2, 1/3, …, 1/n on one edge and their complements on the other.

A mark is not a division: a square divided into five equal strips needs the marks at 1/5, 2/5, 3/5 and 4/5, and the ladder gives 1/5 and 4/5. The others follow by folding edge-to-mark, which is more axiom one — so a full division into n parts costs the ladder plus about n folds, and the ladder is the part that had to be thought about.

The other seed, and what it costs

The reciprocal reading suggests an obvious economy and it is worth checking whether it pays.

Reaching 1/12 by the ladder is twelve folds. Reaching it as 1/4 of 1/3 — three folds to a third, then two halvings of that — is fewer, and reaching it as the sixth rung of a ladder seeded at 1/2 is the same thing said differently. So a composite denominator has a shorter route through its factors, and the ladder’s uniformity is paid for at exactly the numbers where uniformity is not needed.

The primes are where it earns its keep. A seventh, an eleventh, a thirteenth have no factorisation to exploit, so a construction that reaches them at all is worth having even at seven, eleven or thirteen folds. That is the same asymmetry the constructible polygons show from the other direction, where the composite cases fall out of the prime ones and the prime ones are the whole difficulty.

Where the model stops

Nothing here says this is the shortest route. The ladder reaches 1/7 in seven folds, and the reference closure reaches 1/5 in two — so for some fractions there are much shorter constructions, found by searching rather than by following a rule. What the ladder guarantees is that a route exists for every n and that it is uniform; it does not claim to be efficient, and the next rung is about exactly that gap.

A fold has a width and a hand has an aim. The construction is exact and the folding of it is not: every physical fold lands within a fraction of a degree of where it was meant to, and that error compounds down a sequence. A seven-rung ladder folded by hand is seven opportunities to be slightly wrong, and its final mark is worse than Fujimoto’s third after the same effort. The exactness is a property of the geometry, and a folder who wants a good third should still use Fujimoto’s method.

Exact against accurateHow far each method's final crease lands from the true fraction, for a folder who places every crease 0.5 millimetres out on a 150 millimetre sheet. The exact method's error grows with the number of parts because it makes a fold for each of them; the convergent one's does not.4681012141600.10.20.30.40.50.60.7parts the strip is divided intohow far the crease lands out, mmthe exact ladderFujimotothey cross at 4
Fig. 7 Where the model stops: the same construction with a hand’s half-millimetre added at every crease. Exact on paper and long in practice are different properties, and this is the second one measured.

The ladder is one construction and there are others. Any pair of lines through rational points crosses at a rational point, so the space of exact constructions is enormous. What makes this one worth writing down is that it is a rule — one sentence, applicable at every n — rather than a family of special cases.

Nothing here reaches an irrational. Every mark on the ladder is a fraction, which is a consequence of using only the linear axioms. Reaching √2 or a cube root needs a different axiom and produces a different kind of number, and what a fold reaches is a subject with its own boundary.

The rung below, revisited

It is worth being precise about what this does and does not do to Fujimoto’s method, because the rung below is not being overturned.

Fujimoto’s method is a self-correcting procedure, and that is its real virtue. Start with a bad guess at a third and it converges to a third; start with a bad guess and make a bad fold and it still converges, because each step folds the current error in half regardless of where the error came from. A construction that is exact in principle has no such property: one bad fold on the ladder and every later rung is wrong, with nothing to bring it back.

So the two methods differ in which kind of error they are robust to. Fujimoto tolerates execution error and cannot be exact. The ladder is exact and tolerates nothing. That is a genuine trade and it is the reason both belong in the subject, rather than one superseding the other.

Who worked this out

The crossing construction is old and its history is diffuse in the way elementary constructions usually are. It appears in Japanese folding literature under several names, in the Western recreational-mathematics tradition as a puzzle answer, and in more than one school textbook as an exercise in similar triangles — which is what the derivation is, once the two lines are written down.

There is a reason it has no famous name attached. It is not surprising: anybody who writes the two line equations and solves them gets the answer in two lines of algebra, and the interesting content — that iterating it reaches every fraction — is a one-line observation once the crossing is in hand. Constructions like that are rediscovered continuously and attributed rarely, which is a pattern this site has documented at a vertex rather than at a fraction.

What is worth attaching a date to is the alternative. Fujimoto’s method was published in the 1970s and is genuinely his; Haga’s theorem was published in 1979 by a schoolteacher and is genuinely his. Both are named because both are surprising. The crossing is anonymous because it is not.

What this is worth to a folder

Very little directly, and that is the honest summary. Anybody who wants a sheet in fifths will use Fujimoto’s method or a printed guide, and the ladder’s seven folds for a seventh are not competitive with either.

Its value is to the machinery. A construction that produces exact references is what every later construction takes as input, and this site’s own reference closure, its axiom work and its constructible-polygon results all depend on knowing that certain marks are exactly where they say. A converging method cannot be composed; an exact one can, and the ladder is a supply of exact marks at every denominator, produced by one rule.

What each method does to one mistakeA single error injected at the first fold, with nothing else going wrong, and what is left of it at each fold after. The exact ladder shrinks it too — its trouble is that it makes a fold for every part and each of them adds one of its own.12345678910-8-6-4-20folds since the mistakewhat is left of it, powers of tenthe exact ladderFujimoto
Fig. 8 What this is worth to a folder, at a division nothing else reaches easily: how the ladder’s error behaves at ninths. It does not decay, because there is none — the marks are where the arithmetic says and the rungs do not accumulate.

That is the ordinary division of labour between a practice and its foundations, and it is worth naming because the two look alike on the page. A construction can be the right one to use and the wrong one to build on, and the two verdicts here point in opposite directions.

Where the ladder goes next

The immediate question is the one this construction raises against itself. The ladder is uniform and it is not short — 1/7 in seven folds, when a two-fold search reaches 1/5 and 1/6 — so the natural next measurement is where the two disagree, and by how much. The closure a folder can search is cheap where it reaches and silent where it does not, and the two together say something neither says alone.

The other direction is the rectangle. Everything above is on the unit square, whose anti-diagonal happens to be a fold through two corners. On a rectangle the same construction works with a different first fold and lands on the same fractions, which suggests the ladder is about the crossing rather than about the square — and the rectangles that divide into themselves are a family the construction has not been asked about.

Where this goes next

guessoff by 0.16667fold 1off by 0.08333fold 2off by 0.04167fold 3off by 0.02083fold 4off by 0.01042fold 5off by 0.00521fold 6off by 0.00260solid mark — the fold is aiming at 1/3; dashed — at where 1/3 has gonehalving word R L — period 2, from 2^2 − 1 = 3 × 1every fold halves the error exactly, so 6 folds divide it by 64 Folding a strip into thirds A third cannot be constructed by the axioms, so it is not constructed. It is guessed, and then halved into place — an algorithm rather than a construction, with an error that falls by exactly half at every fold. 3 equal partsestimated by eyethe left-hand divisions are exact — a consequence of the fold, not of carethe right-hand ones are a guess, and the error compoundsvalleymountain Dividing without measuring A square can be divided into any whole number of equal parts by folding alone — exactly, with no ruler, and with no error to accumulate. The construction is one fold and a theorem nobody expected. the bare sheet4 references · 4 linesnothing has been foldedafter 1 fold9 references · 12 lineshalves, and nothing elseafter 2 folds565 references · 92 lineshalves, thirds, fifths — and worsea fold is an alignment, and an alignment needs something already on the paper to align565 references after 2 folds, and the count is finite however many folds are allowed A fold needs something to align Every axiom names things that must already be on the paper — a point to fold onto a point, a line to bring to a line. So what a folder can build is bounded by what they can refer to, and that set is finite at every depth: nine references after one fold, several hundred after two, and every one of them computable in advance.

What this makes readable

Essays that name this one as a prerequisite.

Named alongside this one

Essays reaching for the same objects. Nobody chose these; they are what the concept index makes visible.

What links here

The 8 essays that link to this one and share the most of its objects, of 12 that link here.

The objects this essay names

Each one links to every other essay that touches it.

ConstructionConvergenceExact divisionRational divisionReference pointThe axioms