A schoolteacher's theorem

Take a square. Bring one corner down onto the midpoint of the opposite side and crease. That is the whole construction.

What comes out is exact and is not obvious: the crease meets the two vertical edges at exactly three-eighths and seven-eighths of their length, and the folded edge crosses the far side at exactly two-thirds. One fold, no ruler, no compass, and a trisection.

Why the numbers come out #

The derivation is short enough to give in full, and giving it in full is the point — this is a result anybody can check.

Put the square on the unit coordinates with the top-left corner at (0, 1) and the bottom edge from (0, 0) to (1, 0). Folding that corner onto the midpoint (½, 0) means creasing along the perpendicular bisector of those two points, because a fold that carries one point onto another is exactly the bisector of the segment joining them.

That bisector passes through the midpoint (¼, ½) and is perpendicular to the direction (½, −1). Its equation is y = ½·x + ⅜.

Set x = 0 and the crease meets the left edge at ⅜. Set x = 1 and it meets the right edge at ⅞. Both are exact, and both are visible on the folded paper as the ends of a crease.

The general case, in one line #

Doing the same derivation with the corner brought to an arbitrary point turns a trick into a family, and the answer is short enough to be worth having.

Bring the top-left corner $(0,1)$ down onto $(m, 0)$. The crease is again the perpendicular bisector, which works out as

$$y = mx + \tfrac{1 - m^{2}}{2}$$

so it meets the left edge at $(1-m^{2})/2$ and the right edge at $(1 + 2m - m^{2})/2$. At $m = \tfrac12$ those are $\tfrac38$ and $\tfrac78$, which is the construction above.

Reflecting the far corner and finding where the folded top edge crosses $x = 1$ takes a few more lines and collapses to something remarkably simple:

$$y = \frac{2m}{1+m}$$

At $m = \tfrac12$ that is $\tfrac23$ — the trisection — and every algebraic step above is rational, so every landing point is rational whenever the target is.

Which divisions it reaches #

Read the formula backwards and the construction stops being one result.

Put the corner at $m = 1/n$ and the folded edge crosses at

$$\frac{2/n}{1 + 1/n} = \frac{2}{n+1}$$

So $m = \tfrac12$ gives $\tfrac23$, $m = \tfrac13$ gives $\tfrac12$, $m = \tfrac14$ gives $\tfrac25$, $m = \tfrac15$ gives $\tfrac13$, and $m = \tfrac16$ gives $\tfrac27$.

Thirds, fifths and sevenths, one fold apiece, which is exactly the reputation the construction has and is rarely given with the formula that produces it. A fifth needs the corner brought to a quarter of the edge; a seventh needs it brought to a sixth. Each of those targets is itself reachable by ordinary halving, so the whole family costs a few binary folds and then one Haga fold.

The catch is worth stating with the result. Reaching $2/(n+1)$ requires placing the corner at $1/n$ first, so an odd division is bought at the price of an even one — and the even ones are free only because halving is. The construction converts a division of one kind into a division of another, which is what makes it a tool rather than a curiosity: it is the step that gets a folder off the powers of two, and the ladder of accuracy it sits on is the one where errors compound.

Where the third comes from #

The trisection needs one more step and it is the part that makes the result worth a name.

Folding along that crease carries the top edge somewhere. Its left end, the corner, lands on (½, 0) by construction. Its right end, the corner at (1, 1), reflects to (1.1, 0.8) — off the paper, which is why the flap overhangs the edge in a real fold.

The image of the top edge is therefore the segment from (½, 0) to (1.1, 0.8), and it crosses the right-hand edge x = 1 at exactly y = ⅔.

So the third is not on the crease at all. It is where the folded edge crosses the sheet edge, which is why the construction is not obvious and why nobody found it by looking at creases.

Three points from one crease #

Something worth noticing before the history: a single fold has produced four exact points, and they are not independent.

The target midpoint was given. The two crease endpoints at ⅜ and ⅞ differ by exactly ½, which is not a coincidence — the crease has slope ½, so crossing a unit width raises it by half. And the ⅔ comes from a different object again, the image of an edge rather than the crease.

So one fold yields a dyadic pair and one genuinely new rational, and the new one arrives by a different route from the other two. That is the general shape of every construction in origamics: the crease gives easy numbers and the folded image gives the interesting one.

It also explains why the result is not in the classical literature despite being elementary. Straightedge-and-compass geometry has no operation corresponding to “the image of a line under a reflection that was chosen by placing a point” — reflection exists, but the reflected copy of the figure is not usually kept and intersected with the original. Folding keeps both halves on the same sheet automatically, which makes an operation available that a compass user has no reason to perform.

What Haga was doing #

Kazuo Haga was a biology teacher. The construction and the body of work around it came out of setting paper problems for students, and the name he gave the activity — origamics — is a portmanteau intended to mark it as neither origami nor mathematics quite.

The programme is straightforward once stated: fold a square in some simple prescribed way, then ask what the resulting points and lines are, exactly. Not approximately, not by measurement, but as rationals or surds derived from the reflection.

That question turns out to have a great deal in it. A square has many simple things that can be done to it and each produces a small crop of exact points, and the points are useful — a folder who needs a reference at ⅓ of an edge has, in this construction, a one-fold way to get one.

What the programme found beyond the first result #

Haga’s construction is the entry point rather than the whole of it, and the body of work has a recognisable shape.

Send the corner to points other than the midpoint and the values change but the method does not: every target on the far edge gives a crease whose endpoints and whose folded-edge crossing are computable rationals in the target’s coordinate. Send it to a point on an adjacent edge and a different family appears. Fold twice and the reachable set grows again.

What comes out is a catalogue of exact reference points with the folds that produce them, which is precisely what a designer wants and precisely what a syllabus wants, for different reasons. The programme is unusual in serving both.

Why exactness is the whole content #

The instinct of somebody meeting this for the first time is that it is a trick for approximating a third. It is not, and the difference matters more here than almost anywhere.

A drawn trisection is approximate because it depends on where a hand put an instrument. A folded one is exact because the fold is placed by coincidence: the corner is either on the midpoint or it is not, and a folder can see which. Whatever error a real fold has is the error of a person’s hands and of the paper, and the construction itself contributes none.

This is the same argument the kindergarten material rests on and the same one behind dividing a strip without measuring. It is the reason folding is a construction tool at all rather than a craft.

The wall it gets over #

Halving is free and halving is a trap, because it reaches only the dyadic rationals.

A folder who wants an eighth has an easy job. A folder who wants a third has, from halving alone, no route whatever: the set of reachable points is exactly the fractions with a power of two underneath, and ⅓ is not among them however many folds are used.

Haga’s construction crosses that boundary in a single fold. It is not the only way across — Fujimoto’s method reaches an exact odd denominator by a repeating word of folds that converges on it — but it is the cheapest, and it is the one a person can do without understanding why it works.

A classroom is a good place to find this #

There is a reason the result came from teaching rather than from research, and it is not luck.

Research on folding, in the period this was found, was concerned with flat-foldability, with the axioms, and with design algorithms — questions about what is possible in general. “What exact point does this particular easy fold produce” is not a question that arises from any of those. It arises from having thirty children, one square of paper each, and needing something they can do in five minutes that has a definite answer.

The pedagogical constraint — must be doable, must be checkable, must have an exact answer — turns out to select for exactly the constructions that are useful to a designer. That is a genuine piece of luck, but it is luck about the subject rather than about the person: folding is a domain where the easy operations happen to have clean algebraic descriptions.

An amateur result is not a lesser one #

It is worth being direct about the social fact, because the subject has a habit of apologising for it.

This field’s results have come from a biology teacher, an astrophysicist, a laser physicist, an aeronautical engineer, a professional folder and a handful of mathematicians, and the distribution is not an embarrassment. It is what happens in a subject where the objects are cheap, the questions are visible, and the barrier to a genuine contribution is low enough that somebody outside can clear it.

Husimi was a physicist. Miura was working on shell buckling. Yoshizawa was a folder. Beloch is the one professional geometer in the founding set, and her paper is the one the field failed to read for fifty-five years.

The construction as a one-fold axiom #

Read in the language this site uses elsewhere, Haga’s fold is an instance of a known operation and the reading is worth making explicit.

Bringing a point onto a point is the second Huzita operation: the fold is the perpendicular bisector, it is determined completely by the two points, and it is linear. Nothing exotic is happening at the fold.

What produces the third is therefore not the power of the fold but the choice of target and the decision to look at the folded edge. A linear operation, applied once, at a well-chosen point, gives a rational that repeated halving cannot reach — which is a reminder that the reachable set is about the algebra of the whole construction and not about the degree of any single step.

What a reference point is for #

The practical use is worth stating, since it is why folders care.

Designing a base means placing creases at specific fractions of the sheet, and a fraction that cannot be reached by folding has to be measured — which introduces the one source of error the whole method is supposed to avoid. So a construction that produces ⅓ exactly, in one fold, is a tool rather than a curiosity.

Box-pleated designs are built on a grid and the grid has to be divided exactly; a thirty-second is easy and a twenty-fourth needs a third first. Haga’s construction is where the third comes from.

What the generator checks #

The figure is not an illustration of the theorem and it is worth saying what the difference amounts to.

It computes the perpendicular bisector from the corner and the target, reflects the corner through it, and asserts that the image lands on the target to a part in a million million — which tests that the fold being drawn is the fold being described. It then finds where the folded top edge meets the right-hand edge by intersecting two computed segments, and in the classical case asserts all three values against exact rationals.

If the construction stopped giving exact thirds, the build would stop. That is a stronger statement than a figure normally makes and it is available here because the claim is arithmetic rather than historical.

Why this is not the general trisection #

One clarification, because the words invite a confusion this site takes seriously.

Haga’s construction trisects a length. It puts a point at exactly two-thirds of an edge. That is not the same as trisecting an angle, which is the classical impossibility and which folding also solves — by a different construction, using a stronger operation, and for a different reason.

Dividing a segment into three is elementary for a straightedge and compass too; the interesting thing about Haga’s version is its economy, not its reach. Dividing an angle into three is impossible for them and possible for a fold, and that gap is cubic rather than rational.

Running the two together would credit this construction with a great deal it does not do, and the subject’s popular accounts do exactly that often enough to be worth the paragraph.

The idealisation, named #

Everything above is about a mathematical square and a mathematical fold.

A real sheet has thickness, so the crease is a curve of finite radius rather than a line, and the corner sits a little short of the target by however much the paper resists. Bringing a corner “onto” a point is a judgement made by eye at a resolution of maybe half a millimetre, which on a 150 mm square is a third of a percent.

So the construction is exact and a folded instance of it is not. The useful consequence is that the error does not compound: a reference produced this way is off by whatever that one fold was off by, whereas a reference produced by repeated approximate halving accumulates.

What the name is doing #

A last note on attribution, since the field it sits in is about exactly that.

“Haga’s theorem” is unusually well-behaved as a name. The construction is his, the programme around it is his, the naming is roughly contemporary with the work, and nobody is displaced by it. Set against Kawasaki’s decade and Beloch’s fifty-five years, that is worth remarking on.

The reason is probably that the result arrived after the field had a venue. By the time origamics was circulating there were meetings, proceedings, and people who read each other — so the gap between the result and the name is small for the same structural reason the earlier gaps were large. The attribution problem in this subject is not a moral failing that later people corrected; it is what happens before a literature exists, and it stops happening afterwards.

There is a small lesson in that for reading any subject’s chronology: the size of the attribution gaps is a measure of how well connected the field was at the time, and it is therefore itself a historical datum rather than noise to be corrected away.

Where this goes next #

This closes the pedagogy ladder for now. Sideways, dividing without measuring is the general question of which points a fold can reach, and where the cubic comes from is the answer at full strength — the reachable set is much larger than the dyadic rationals and much smaller than everything.

The surprising connection to end on: the construction’s interesting point is not on the crease. Every folder’s instinct is to look at the fold line, and the third is on the image of an edge — which is a general lesson about this subject, that the folded state carries information the crease pattern does not display, and that a habit of reading only the pattern will miss things.