Taught with a wrong reason

The kindergarten was a geometry class and a schoolteacher’s theorem are about how this subject is taught and who taught it. This is about what is taught alongside the results — the explanations, the rules of thumb, the sentences that begin because — and the finding is that several of them are wrong while their conclusions are right.

The examples below are not folklore collected from elsewhere. Two are in this site’s own text and one is in its own code, and the third put patterns on the printed shelf that nobody could fold.

The ring that reads as one letter #

What is said: a twist looks like a twist when the four creases round its central polygon carry the same letter. The polygon then lifts as one piece and rotates against the sheet, which is the visual signature of the family.

What is true: the conclusion is right — such letterings exist, thirty-two of them on a square twist, and every one satisfies every condition at every vertex. The explanation is right about appearance too.

What is wrong: not one of the thirty-two has a folded state. Each corner region between two consecutive pleats has to lie above one pleat and below the other, and going round the four pleats the requirement closes on itself — a loop of eight panels that no ordering can satisfy. The letterings that do fold, all eight of them, have a ring taking two letters in adjacent pairs.

So the sentence “a twist looks like a twist when the ring reads as one letter” is a correct observation about drawings and a false one about paper, and this collection used it as a criterion for years. Two printed patterns went out with no folded state, at true scale, with a note recommending one of them as the first to fold.

Four mountains and four valleys #

What is said: the preliminary base is the two diagonals and the two midlines of a square, and its assignment is what the symmetry suggests — four mountains and four valleys, alternating.

What is true: it is what the symmetry suggests, and it is what most people draw the first time.

What is wrong: Maekawa forbids it. At a flat-foldable vertex the mountains and valleys differ by exactly two, and four against four differs by nothing. The base’s actual assignment is three of one and five of the other, and the eight-fold symmetry of the drawing does not survive into any of its 112 admissible letterings.

This one is already in this site’s text as a note on the printed pattern, and it is the healthiest example of the three: the mistake is named, the reason it is made is given, and a reader who has made it is told what to do instead. The difference between this and the twist is that somebody wrote the correction down.

The unit is verified, so the tessellation is #

What is said: a twist tessellation is built by tiling a verified unit, and every vertex of the tiling is checked as the copies meet, so the tiling is as good as its unit.

What is true: every vertex of the tiling is checked, and passes. The copies meet correctly, the sector angles hold as an identity at every corner, and the local conditions are satisfied everywhere.

What is wrong: the forty-nine-panel patch this site drew had a loop of twenty-eight panels in the order its letters force, so no ordering of it exists and no paper could ever have taken that shape. The loop passes through panels from several different twist units. There is no unit to examine and no neighbourhood to check that would reveal it; the loop exists only once the pieces are joined into a ring.

The caveat that stood in the text was correct as far as it went — the unit is verified and the plane is not, because the general question is intractable — and it was read as nothing can be said. Something can: a loop is one pass over the crease list, and it says no on patterns whose orderings could never be enumerated.

What the three have in common #

Each one is a mechanism attached to a correct conclusion, and in each case the mechanism was never tested because nothing tests mechanisms.

A gate checks an output. the flat-folding assertion asks whether the vertices pass; the local checks ask whether labels fit and boxes are filled; the shared gates ask whether links resolve and pages are reachable. All of them are questions about the finished thing, and a finished thing can be correct for a reason nobody stated correctly.

The three examples are also all about the same gap, which is worth noticing because it means they are one failure rather than three. In each case the wrong reason is a statement about the pattern — the drawing, its symmetry, its vertices — offered as an explanation of the folded object. A drawing and a folded object are two things, a symmetry the drawing has need not survive into the letters, and every one of these mistakes is that substitution made in a different place.

Two more, from the same instrument #

The same shape appeared twice more in the same round of work, and both are worth recording because neither is anybody’s error — they are conventions that carry an implied reason.

“A folded leaf is so many layers deep.” The number quoted is the lamina’s area over its packed footprint, which is the average depth. The deepest point of a corrugated leaf packet is 1.926 times that, at every patch size measured, and it is the deepest point that has to fit in a bud. The convention is not wrong; the sentence “so the bud has to be that deep” is.

“The Miura is used because it packs well.” It converts crease into packing at 0.67 sheet-widths per layer against the Yoshimura’s 0.23 — nearly three times worse than a pattern nobody deploys, and fourth of eight on this site’s printed shelf. What it buys is a one-parameter motion and a folded rectangle, and the packing is not why it flew.

Both are cases where an average or a headline figure carries an unstated mechanism, and the mechanism is what fails.

Why the shelf’s own curation did not catch it #

There is an obvious objection to the twist case: this site prints its patterns for readers to fold, so surely somebody would have noticed.

The curation rule was exactly that — a reader should be able to fold it — and it worked at the level it operates on. The eight printed patterns are the patterns of the subject: a base, two twists, a Miura, a Yoshimura, a waterbomb tessellation, a tapered corrugation, a fold-and-cut construction. Every one is a pattern that folds.

The failure was one level down. A pattern is a set of lines; a crease pattern is a set of lines with letters on them, and the letters were chosen afterwards by a routine. Nobody folded that particular lettering, because nobody chose it — the routine did, from among 256 that all pass every condition, on a criterion about how the drawing reads.

So curation by judgement selected correctly among patterns and had no view at all about letterings. That is not a lapse; it is a category the process did not have. The population it produced has the best record of the four this site keeps — nothing in it is refused now — and it still shipped two patterns that could not be folded.

The criterion was worse than a blind draw #

The twist case can be given a number, and the number is worse than “the criterion failed”. It is that the criterion was anti-correlated with the thing it was standing in for.

Two hundred and fifty-six letterings of the square twist satisfy every condition at every vertex. Eight of them have a folded state — one in thirty-two, or 3.1 per cent. The uniform-ring criterion selected thirty-two of the two hundred and fifty-six, and none of the thirty-two folds.

A blind draw of thirty-two from the admissible set would be expected to bring back one that folds. The criterion brought back none, and it did so every time it was applied, because the deficit is not sampling noise: a uniform ring is precisely the configuration that closes the corner loop. The reason did not merely fail to help. It selected against the property it was being used to find.

That is the strongest form of the failure this essay is about, and it is worth naming separately, because the weaker forms are much more common and much easier to forgive. A reason that is uninformative wastes the effort spent on it. A reason that is anti-correlated spends effort producing a result that is reliably wrong, and it looks like diligence while doing it.

What each reason costs once it is written as a rule #

The other two examples fall on the same scale, and putting all three in one column says what was actually available.

The symmetric preliminary base is uninformative rather than adverse: four-and-four is one lettering out of the two hundred and fifty-six the base admits, and it is the one Maekawa forbids. Selecting it is a single bad draw, not a bias, and the counting rule that refuses it takes one pass over eight creases.

The verified-unit argument is the middle case. It is right about every vertex and silent about the tiling, so it is not adverse — it simply has no view on the question it is being asked. What it costs is that its silence reads as reassurance. The loop test that does have a view is also one pass over the crease list, and on the forty-nine-panel patch it returns a refusal in the time it takes to read the letters once.

All three had a rule available that runs in a single sweep of the pattern, which is the part that stings. The reasons were not standing in for something expensive. They were standing in for something nobody had written down.

What separates a reason from a rule #

The distinction that comes out of the five is not subtle and it is worth stating as a test.

A rule is a statement with a decision procedure. Maekawa says the letters differ by two and there is a routine that counts them. Kawasaki says the alternating sums are equal and there is a routine that adds them. A rule can be run, and running it is what a gate does.

A reason is a statement about why a rule holds, or about what a pattern is like. “The polygon lifts as one piece”, “the symmetry suggests”, “the unit is verified so the tiling is” — none of these has a routine, none can be run, and each is therefore untested however many times its conclusion is confirmed.

The practical consequence is not “stop giving reasons”. It is that a reason should be written so that it could be a rule, and where it cannot, it should be marked as description. The three failures above are all reasons that were being used as rules — as criteria for choosing a lettering, as a prediction of an assignment, as a warrant for a tessellation.

Not the same as an attribution error #

This site keeps a separate record of a different kind of mistake, and the two should not be run together.

Nothing here is as old as it sounds is about dates: practices credited to centuries they postdate, results credited to the wrong person, a median overrun of two hundred years across the documentary record. Those are errors about provenance — who and when — and they are checked against sources.

The errors in this essay are about mechanism — why — and there is no source to check them against. A reason is not something a citation settles; it is either recomputable or it is not, and the three above were not.

The two kinds also fail differently. A wrong attribution misleads a reader about history and leaves the geometry intact. A wrong reason used as a criterion puts an unfoldable pattern on a printed page. The second is the one that reaches somebody’s hands, which is why this site’s copy of it is worth an essay and somebody else’s dates are worth a table.

How the corrections were found #

All five came from the same source and it is worth saying what it was, because it was not care.

A new instrument was built for a different question. The subject in hand was layer order on a sheet, and the machinery was written to answer a question about how many folded states a pattern has. Pointed at the shelf, it refused two printed patterns; pointed at a tessellation, it refused a patch; pointed at a leaf, it separated an average from a maximum.

None of the three explanations was under suspicion. Nobody re-read the twist code looking for a bad criterion; the criterion was found because something new asked the pattern a question the criterion had an implied answer to.

That is the argument for building instruments rather than for checking more carefully. Care re-runs the tests that exist. An instrument asks something the tests do not, and what it finds is whatever was resting on an untested reason — which, on the evidence of one round of work, is a fair amount.

What was changed, and what was not #

The corrections are all in the machinery rather than in the prose, which is deliberate.

The pattern builders now choose a lettering by whether a folded state exists, and break the remaining tie on appearance rather than the other way round. The square, triangle, pentagon and hexagon twists come out with two letters on their rings; the fold-and-cut triangle is chosen from the eighteen letterings that fold rather than from the thirty that pass.

The tessellation builder redraws. Where the propagated letters demand a loop, it draws again with the branch order randomised until they do not — seven of forty draws are clean on the square tiling — and where no draw is clean it keeps what it had and reports that it could not clear it.

And the figures say which verdict they got. A twist’s readings now carry a line saying whether its panels can be ordered, and undecided is one of the three things that line can say.

No essay’s argument changed. The counts over admissible letterings are still counts over the right set; the symmetry measurements, the piece counts, the cut census are all measurements of the first sieve and the first sieve is unaltered. What changed is which single member of that set got printed, and what a caption is allowed to imply about it.

Three things this does not say #

It does not say the literature is unreliable. Every conclusion in the five is correct. What failed is a layer of explanation that no publication is expected to prove and that readers reasonably take on trust.

It does not say reasons are optional. A subject taught as a list of routines to run is worse, not better, and this site’s own habit is to give the mechanism every time. The point is that a mechanism offered as a criterion has become a rule and should be tested like one.

And it does not claim the five are all of them. They are what one instrument found in one round of work. There is no reason to think the rate falls.

What a folder should take from it #

When a pattern is drawn a particular way, ask whether the reason is a rule. “It looks like a twist” is not a criterion, and using it as one produced a sheet nobody can fold.

A correction written down is worth more than a correction known. The preliminary base’s forbidden assignment is the one of these that never caused trouble, and the difference is that somebody put it in the note beside the pattern.

And the conclusions are still right. Twists rotate, the base’s assignment is not symmetric, tessellations tile, leaves pack and the Miura flies. Every one of those survives; only the sentences beginning because did not.