A length needs a scale

How much line is on the paper opens these essays by measuring a quantity nobody had measured: the total length of crease in a pattern, which is what a folder actually spends. It reports the Miura fold as thirty-eight creases and six point seven metres of folding on a sheet seventeen centimetres across.

It is thirty-eight creases and one point zero five metres.

The error is a missing divisor and it is worth the whole of an essay, because the reason it survived four essays is not carelessness. It is that a length has no meaning without a scale, the scale was implicit, and the implicit scale was right on six of the eight patterns these essays measure.

What the measurement is #

A crease pattern is a list of vertices with coordinates and a list of edges between them. The crease length is the sum of the distances between the ends of every edge not marked as the sheet’s boundary — a plain sum, and the only thing in it that is not obvious is which edges to leave out.

That sum is in the pattern’s coordinates. To turn it into millimetres it has to be multiplied by however many millimetres one coordinate unit is, and that scale is the printed size of the sheet divided by how wide the pattern is in its own coordinates.

The shelf multiplied by the printed size and stopped there, which is correct exactly when the pattern is one coordinate unit wide.

Six of the eight are. The preliminary base, the square twist, the hexagon twist, the Yoshimura, the waterbomb and the fold-and-cut triangle are all built on a unit square, because their builders take a sheet and put creases in it. Their coordinates are sheet widths, the missing divisor is one, and the two readings are the same number.

Two are not. The Miura’s builder takes a cell width, a cell height and a count, and lays out six cells of unit width — so the pattern spans 6.365 units, not one. The tapered corrugation spans 1.176 for a similar reason.

So the divisor was missing on every pattern and visible on two.

What moved #

The Miura was reported at six point four times its length. The tapered corrugation at 1.18 times.

That changes three things in these essays and one outside it.

The length ordering. The Miura was the longest pattern on the shelf by a factor of nearly three and is now fifth of eight, between the hexagon twist and the tapered corrugation. The shelf’s total falls from 15,195 millimetres to 9,379.

The mean crease length. The first of these essays reports the Miura’s average crease as 1.03 sheet-widths — longer than the paper is wide — and builds an argument on it: the zigzags run corner to corner, so a crease can outrun the side of the square it lives on. The mean is 0.162 sheet-widths, which is the second shortest on the shelf. The geometric observation about the zigzags is true and the number it was attached to was not.

The density. The first of these essays’ closing figure runs from 1.72 sheet-widths of crease per unit area to 39.3, “a factor of twenty-three between the emptiest pattern and the busiest”. The busiest is the waterbomb at 14.3, and the factor is eight.

And the one outside the essays here #

The worst rate on the shelf takes the same measurement into the deployables field and divides it by compaction: how much folding buys one layer of packing. It reports the Miura at 4.26 sheet-widths per layer against the Yoshimura’s 0.23 — a factor of eighteen — and concludes that the pattern which actually gets built is the worst converter on the whole printed shelf.

Corrected, the Miura is 0.67 sheet-widths per layer. It is fourth of eight. The worst is the hexagon twist at 1.88, and the factor between the Miura and the Yoshimura is 2.9.

That essay’s title is now false and its argument needs redoing rather than patching, which is what it has been given. The surviving half is worth stating here because it is the half that was doing the work: the pattern that gets built is not the pattern with the best packing rate, which is true at 0.67 as it was at 4.26, and the Yoshimura still converts crease into layers nearly three times better while being the pattern nobody deploys.

Which numbers a reader should keep #

It is worth separating the three kinds of number these essays report, because the correction lands on exactly one of them and the other two are what most of its findings rest on.

A raw sum is a length in coordinates nobody should quote. It is the output of the summation and it has no unit until a scale is supplied, and every error above is an instance of quoting it as though it did.

A ratio within one drawing — length per unit area, length per crease, length per layer of packing, the share of length in a band — is dimensionless or nearly so, and the scale cancels. Those are the pattern’s own properties and none of them moves with the sheet.

A physical length — millimetres of crease on a sheet of a stated size — needs the scale and is a property of the sheet as much as of the pattern.

The first of these essays says this, in a section about what changes with the size of the sheet: the density and the mean crease length, quoted in sheet-widths instead of millimetres, do not move at all… the sheet-width figures are properties of the pattern and the millimetre figures are properties of a decision about paper. That is exactly right and it is the sentence the error contradicts, because the figures it calls sheet-widths were in cell-widths on two of the eight patterns and were therefore properties of a builder’s argument list.

Why it lasted four of these essays #

Three things kept it hidden and each of them is a general hazard.

The unit was implicit and the label was explicit. Every figure that printed the raw sum called it “sheet-widths of folding”. The label claimed the scale; the number did not carry it; and a reader — including the one writing the next essay — has no way to check a claim of that kind against anything.

The affected patterns were a minority and not a random one. Six of eight agreed, so any spot check had a three-in-four chance of landing on a pattern where the bug is invisible. Worse, the two that disagree are both generated patterns rather than drawn ones, and a generated pattern is the kind whose builder takes a cell count — so the error correlates with exactly the patterns nobody hand-checks.

And the right answer was already on the page. The printable version of every pattern — the one a reader downloads and folds — computes its crease length by dividing by the pattern’s own width, and has since it was written. So the Miura’s downloadable sheet has said 1,049 millimetres while the figure beside it said 6,679, and nothing compared the two. That is the shape of this defect exactly: not a calculation nobody could do, but two calculations nobody put side by side.

What did not move #

The ratios did not move, and that is most of these essays.

Where the length sits divides each pattern’s folding into bands by distance from its own edge and reports length per unit of paper in each. Both halves of that quotient scale together, so every number in it is unchanged.

The shortest crease is not a crease reads the crease lengths individually and finds twelve on a printed patch shorter than a wavelength of light. That one is a physical length and it was computed on the printed patch directly rather than through the shelf, so it stands.

A count is not a length cuts a rectangle out of a tessellation and finds the crease count converging from above while the crease length per unit area is exact at every size. Both quantities there are measured on one pattern at one scale, and the finding is about what survives a cut rather than about millimetres.

So the essays that measure ratios are untouched and the essays that quote millimetres are not — which is the distinction the first of these essays itself draws, in a section headed what changes with the size of the sheet, and then does not apply to its own tables.

The correction #

The fix is one divisor in one place, and it is now a named function rather than an expression: the crease length in sheet widths is the raw sum divided by the pattern’s own width, and the printed length is that times the printed size. Everything that reports millimetres or sheet-widths goes through it.

The pattern’s own width is the x-extent, because that is what the printable sheet is scaled to at the stated size — the sheet is printed at mm across whatever its proportion, and the y-extent follows. Using the larger of the two extents would be a different convention and would disagree with the printed sheet, which is the thing being described.

And the two calculations are now required to agree. The figure that compares them stops if the number of patterns whose extent is one is zero or is all of them — the first would mean the correction is doing nothing on this shelf, the second that there is nothing to correct — which is a check that this essay’s subject exists rather than a check on the arithmetic.

What a check would have looked like #

The site’s own habit is to compute a quantity two ways that share no code and require them to agree, and these essays had two such computations and never compared them.

That is worth stating precisely, because “it should have been checked” is not a finding. The printable sheet and the shelf figure are not two computations of the same quantity by two methods; they are two computations of different quantities — one a printed length, one a raw sum — with the same label. So a check comparing them would have had to know they should be equal, which is the fact that was missing.

What would have caught it is cruder and would have worked: a length quoted in millimetres against a drawing of the sheet at that size. Six point seven metres of crease on a square 170 millimetres across is 231 metres of line per square metre of paper, which is a crease every four millimetres over the whole sheet, on a pattern with thirty-eight creases in it. The number is absurd on its face and nobody put it in a form where its absurdity showed.

Which is what computing the density against a physical ceiling did. A crease occupies a band a few sheet thicknesses across, so a sheet has a largest density it can carry, and comparing the shelf against it put the Miura above the ceiling of every paper — which is the next essay and is what made the arithmetic look at itself.

Everywhere else the number appears #

Four other essays quote the affected figures, and all four have been corrected rather than annotated.

The patterns nobody own uses the shelf to ask whether the published patterns are the elaborate ones, and reports the Miura alone as forty per cent of the shelf’s folding. It is eleven per cent, and the finding it supported — that publication does not track elaboration — is now carried by the Yoshimura rather than by the Miura.

What a corrugation costs compares the rate across the shelf and has the preliminary base beating the Miura. It still does, by less.

Taught with a wrong reason lists “the Miura is used because it packs well” among the claims a reader meets and refuses it with the rate. The refusal stands and its number is now 0.67 against 0.23 rather than 4.26 against 0.23 — which is a weaker refusal and is still a refusal.

And the worst rate on the shelf was about the number itself and has been rewritten.

Four essays and one rewrite from one missing divisor, which is the measure of how much a sequence’s later essays rest on its first one’s arithmetic. Nothing in any of them was reasoned badly; they were all reasoning correctly from a number.

What this does not fix #

Every number quoted in a physical unit has a scale in it. This is one quantity in one module, found because a second quantity was computed beside it. Nothing here audits the others, and the reason to write the essay rather than only the fix is that the audit is the work this suggests and has not been done.

The printed sizes themselves are a decision. A pattern is printed at 150 or 170 millimetres because somebody chose a sheet, so every millimetre figure in these essays is a property of that choice and not of the pattern. The sheet-width figures are the pattern’s own.

Nor does it touch the length distribution. The shortest crease is not a crease reads individual crease lengths on a printed patch and finds twelve shorter than a wavelength of light; that measurement takes its scale from the patch rather than from the shelf, and a systematic factor on the total would have moved every one of those lengths together without changing which were shortest.

And the corrected numbers are not more precisely measured, only more correctly scaled. The crease length itself is a sum over coordinates and was always right; what was wrong was the units it was reported in, which is a different kind of error from an imprecise measurement and is the kind no amount of care in the summing would have caught.

What a reader should do with a corrected essay #

The essays affected by this have been corrected rather than annotated, and that is a decision worth stating.

An annotation — a note saying a number used to be something else — is honest and makes an essay worse to read, because every reader after the correction has to carry a fact about the collection’s history in order to read a fact about paper. A correction leaves the essay saying what is true.

What is left behind instead is this essay, which is where the history lives. The record of a mistake belongs in one place rather than in every place it touched, and the places it touched are listed above so the trail is not lost.

Still open: the rest of the scales #

The defect is a class rather than an instance, and the class has members nobody has looked for.

Every quantity reported here in a physical unit has a scale in it, and the scales come from the same place: a pattern’s coordinates, a printed size, and a divisor that is one on most patterns. Crease length is the one with a physical unit in its name. Area, flap length, river width and packing radius are the others, and each is quoted somewhere in millimetres or as a share of a sheet. An audit would be one script: for every quantity reported in a physical unit, find the two computations of it and require them to agree, which is what caught this one and caught it by accident.

The deeper fix is to make the scale impossible to omit. A crease length is not a number, it is a number and a unit, and a module that returned a pair — the sum and the extent it is in — would make the missing divisor a type error rather than a silent factor of six. That is a change to how every pattern is handed around and it is the kind of change that is cheap now and expensive later.

The habit worth carrying is about labels that claim what a number does not carry. A quantity called “sheet-widths” is only in sheet-widths if something divided by a sheet width, and a label is not that something. The two readings of the Miura sat in the same repository for four earlier essays, one in a figure and one in the sheet the figure’s own caption invites a reader to print, and the only thing that would have caught it sooner is putting a third number beside them.