A construction assumes its sheet

A schoolteacher’s theorem re-runs Haga’s fold rather than quoting it. A corner is brought to the midpoint of the opposite edge; the crease is the perpendicular bisector of the corner and its target; and the numbers that come out are exact. The crease meets the vertical edges at three eighths and seven eighths, and the folded image of the top edge crosses the far edge at exactly two thirds.

Nothing is measured. A trisection falls out of one fold, which is why the construction is famous and why it is taught.

Read as a sequence of instructions it says: take a corner, take the midpoint of the far edge, fold one onto the other. Nothing in that sentence mentions a square.

Running it on a rectangle #

The construction generalises without any decisions having to be made. Take a sheet w wide and h tall, put the corner at the top left, put the target at the midpoint of the bottom edge, and take the perpendicular bisector of the two. Reflect the top edge across it and find where the image crosses the right-hand edge.

Every step is the same step. No parameter has been chosen, no interpretation has been made, and the arithmetic is the arithmetic of a reflection.

On an A-series sheet held tall the crease meets the left edge at seven sixteenths and the right edge at eleven sixteenths, and the folded edge crosses at two sevenths.

On a three-to-two sheet held tall, four ninths, two thirds and one quarter.

Every one of those is a clean small fraction. Seven sixteenths is not a number that looks like a mistake, and neither is two sevenths. A folder following the recipe on A4 gets a crease, gets a crossing, and gets a result that is exactly two sevenths of the way up — with nothing whatever to indicate that the construction was supposed to produce a third.

The other failure, which is the safe one #

Turn the same sheets through a right angle and something different happens.

On an A-series sheet held wide the perpendicular bisector meets the right-hand edge at one and a quarter times the sheet’s height, which is to say it does not meet it: the crease runs off the top of the paper. The fold cannot be made.

On a three-to-two sheet held wide it is worse — one and thirty-four hundredths — and on a six-to-five sheet held wide it is one and four hundredths, only just outside, which is the case a folder would probably try anyway and find impossible.

That is a failure with an alarm on it. The construction refuses itself, the folder cannot proceed, and nobody is misled about anything.

So the same rectangle fails in two different ways depending on which way round it is held, and the two ways are not similar. One returns a number and is silently wrong; the other returns nothing and is loudly right about being unable to help.

Which of the two is the dangerous class #

The distinction between the two failures is the whole of this rung, and it is worth naming why the quiet one is the serious one.

A construction that refuses itself is self-limiting. The folder discovers it immediately, at the fold, before any of the model is built. The cost is a wasted sheet.

A construction that returns a different exact number is not self-limiting in any way. Everything downstream of it is built on a value nobody checked, the value is a clean fraction rather than a suspicious decimal, and the error propagates into whatever the construction was a step of. In a long sequence it may not surface until the last fold does not meet the first.

And there is no way to tell from inside the construction which class a sheet is in. Both are the same alignment. The one that answers and the one that refuses differ only in whether a line happened to intersect a segment, which is a fact about the sheet’s aspect ratio and not about the geometry of the fold.

Which suggests the practical rule and it is a very small one. A published construction should state the proportion it was written for — which no notation this subject has invented has a place for, any more than it has a place for the sheet —, because the sequence itself does not carry it and the failure mode when it is wrong is silent. That is not a rule anybody would think to need until the arithmetic above is in front of them.

Why the numbers are still exact #

It is easy to read this as a construction becoming approximate or unreliable off its own sheet, and that is not what happens.

Every value computed here is exact, in the sense the exact-division ladder uses the word. Seven sixteenths is exactly seven sixteenths; two sevenths is exactly two sevenths. The construction has not degraded, lost precision or become approximate — it has done a different exact thing.

That is a sharper failure than an approximate one and a rarer one. An approximate result carries its own warning in its decimals; an exact result carries none, and a reader who checks the arithmetic finds it perfect.

The reason the numbers stay rational is worth a line. The construction is a reflection in a perpendicular bisector, which is a rational operation on the coordinates; the coordinates of a rectangle’s corners and edge midpoints are rational multiples of its sides; so on any rectangle with a rational aspect ratio every value the construction produces is rational. The recipe’s exactness is a property of reflections, not of the square — and so the square gets no credit for it, and no sheet loses it.

What the target does, which is a separate variable #

The corner may be brought to a point other than the midpoint, and it is worth separating that from the sheet, because the two are easy to conflate and only one of them is this rung’s subject.

Moving the target along the bottom edge is a parameter of the construction. It is a choice the folder makes, the result varies continuously with it, and the classical values are the values at the midpoint. Nobody is misled by that: a folder who brings the corner to a point two fifths along knows they have done something different, and the construction does not claim the same answer.

Changing the sheet is not a parameter of the construction. It is a change to the object the construction is performed on, it is invisible in the instructions, and the folder has no reason to think anything has changed.

A parameter is something the recipe knows about and a sheet is something it does not, and that is what makes the second dangerous and the first ordinary. The generator draws either, and it asserts the classical fractions only where they are the classical fractions — at the midpoint on a square — because a check applied at the wrong scope refuses correct figures, which this collection has done once already and recorded.

How far off square it has to be #

The two classes are separated by whether a line meets a segment, so there is a proportion at which the construction changes class, and it is worth finding because it says how fragile the recipe is.

The crease meets the right-hand edge at a height that rises as the sheet gets wider. On a square it is at seven eighths, comfortably inside. Widen the sheet and it climbs; at some width it reaches the top corner exactly and past that it is off the paper.

Solving for it: the crease reaches the far top corner exactly when the width is two over root three times the height, which is 1.1547 — a fifteen per cent departure from square, and a value pleasant enough to be worth having as a closed form.

So a sheet fifteen per cent wider than square refuses the fold and one fourteen per cent wider performs it. That is a wider tolerance than the loud failures on the page suggest — the A-series sheet held wide is forty-one per cent off square, nearly three times the threshold — and it is still narrow enough to catch a sheet somebody trimmed carelessly and called square.

And in the other direction there is no threshold at all. A sheet one per cent taller than square returns a value near two thirds and not equal to it, exactly, with the crease comfortably on the paper — so the silent class has no margin to be inside and the loud one has fifteen per cent of margin.

Which puts the construction’s tolerance in an awkward place. It fails loudly past fifteen per cent one way and quietly from nothing at all the other, so the direction with a signal is the one with a margin and the direction with a margin of zero has no signal.

What makes a construction portable #

The two failure classes invite a third question, which is which constructions have neither, and it turns out to have a clean answer.

A construction is portable across proportions exactly when every one of its alignments names objects that scale with the sheet. Halving names two opposite edges and gives a crease that is always in the same place relative to them, so it is portable — and halving repeatedly is what the A series is built on. Quartering is halving twice. Folding a diagonal names two corners and gives a crease that always joins them, so it is portable — though what it does is not the same on every sheet, since a diagonal makes a different angle.

Haga’s fold names a corner and a midpoint of the opposite edge, and both of those scale — so by that test it should be portable, and it is not. The difference is the third step: the reflection, whose result is compared against an edge that has not moved with it. The crease is where the construction says it is on every sheet; where the folded image lands relative to the far edge is a comparison between something that scaled one way and something that scaled another.

So the rule is about the comparisons, not the alignments. A construction is portable when the quantity it finally reads off is compared against something that scaled with everything else, and Haga’s is not, because the crossing point is read against an edge whose distance from the crease depends on the proportion.

That is a usable criterion and it explains the classification the ladder now owes. Constructions that end by marking an intersection of two things the construction itself produced are portable; constructions that end by marking where something the construction produced meets the sheet are not.

What the proportion is worth to the construction #

There is a reading of all this that runs the other way, and it is more encouraging than the rest of the rung.

If the same alignment gives two thirds on a square, two sevenths on an A-series sheet and a quarter on a three-to-two sheet, then the proportion is a parameter of a family of constructions rather than a source of error. A folder who wants two sevenths has a one-fold construction for it, and the construction is the classical one performed on the paper that comes out of the printer.

That is not a small thing. Two sevenths is a division a folder has no easy route to: seven is not a power of two, the exact division ladders reach it in several crossings, and here it arrives in a single fold from an ordinary sheet.

So the same arithmetic reads as a warning and as a technique, depending on whether the proportion was chosen. A construction performed on the wrong sheet is a bug; a construction performed on a chosen sheet is a parameterisation, and the two are the same computation.

Which is worth saying because it changes what the classification would be for. A table of what each construction does on each proportion is not only a list of ways to go wrong; it is a table of one-fold divisions, indexed by the paper, most of which nobody has looked up.

Which theorem was checked and how #

The square’s three classical values are asserted exactly. Three eighths, seven eighths and two thirds, to within a part in a trillion, computed from the reflection rather than quoted — so a slip in the generalised geometry breaks the case it was generalised from.

Both classes must be present. The figure refuses a comparison in which every non-square sheet fails the same way, because the essay’s finding is that there are two classes and a picture showing one of them would argue nothing.

The quiet failures must not be near misses. The largest silent error is required to exceed a tenth, so a sheet returning something close to two thirds could not be presented as returning a different number.

And the orientation effect is asserted rather than left to the reader. Among the sheets drawn there must be a proportion whose two orientations fall into different classes, or the claim that a right angle changes the class has no instance on the page.

Where the model stops #

One construction is analysed. Haga’s fold is the example because it is the one this collection has already computed exactly, and the argument about two failure classes is a general one that this rung demonstrates rather than proves.

Only rectangles are compared, and only at rational and one irrational proportion. A sheet with a curved edge, or a hole, changes which alignments exist at all.

The construction is taken as its alignments. A published sequence often carries an instruction like “fold the sheet in half first”, which is itself a construction and changes the effective proportion; nothing here follows a real published sequence through.

And the two classes are the two this construction has. A longer construction has more ways to fail — an intermediate off the paper, a crease that cannot be reached, a reference that has become crowded — and the two here are the simplest members of a larger family.

What the picture cannot show #

The table gives what each sheet returns and cannot show what a folder would do next. The silent failure is only dangerous inside a longer sequence, and the sequence is not drawn — so the picture states the error and not its consequence.

Nor can it show how a construction would be written to be proportion-independent. Some are: a construction that only halves and quarters works on any rectangle, because halving is defined by the edges rather than by a length. Separating the constructions that survive from the ones that do not is a classification this rung suggests and does not make.

The clearest absence is a warning that could be automatic. A construction expressed as a sequence of axiom calls with named arguments could be checked against a proportion mechanically — every intermediate tested for lying on the sheet, every value compared against the one the square gives — and nothing in the field’s file formats carries enough to do it, which is what the notation ladder is about arriving from another direction.

The idealisation, named #

The sheet is a rectangle with exact corners, the fold is exact, and the crease is a line. The first of those is the one doing work.

A real sheet of A4 is not exactly one to root two; it is 210 by 297 millimetres, which is 1.4143 rather than 1.4142. The construction on the real sheet gives a value near two sevenths and not exactly it, so the silent failure is silent in a second way — the number is not even the clean fraction the model says, and a folder checking the arithmetic against the model would find a small discrepancy and attribute it to their hands.

That compounds the problem rather than relieving it. An exact wrong answer is hard to catch and an almost-exact wrong answer is harder, because it has the texture of a measurement error, and measurement error is the thing a folder is trained to expect and forgive.

Where the ladder goes next #

The classification is the rung this anchor owes and it is a definite piece of work. Which of this collection’s own constructions survive a change of sheet? Halving and quartering survive by construction. The trisection ladders divide a length, so they survive on any rectangle applied along the right edge and not the other. Haga’s fold does not survive. Sorting the site’s own constructions into those three classes is an afternoon and it would produce a table nobody appears to have.

Sideways, the finding belongs beside what the sheet does to the reachable set, which measures the same dependence from the other end: the square reaches the fewest distinct references of any proportion, and the constructions written for it are the ones most likely to have coincidences baked into them. A construction written on the sheet with the most coincidences is the one most likely to be relying on one, which is a reason to expect the silent-failure class to be large.

The habit worth carrying is a question to ask of any recipe. What does it assume that it does not mention? Here it is a proportion, it is invisible in every step, and the failure when it is wrong looks exactly like a success.