The plane the five points were in

A shrink is two numbers separated a quantity that had been quoted as one. How much smaller a folded sheet gets is a pair — how far it draws in along the pattern and how far across it — and the single figure everybody uses is the two multiplied together. Measured on five patterns, the pair put an accordion on a line where one direction is untouched, a twist on the diagonal where both draw in equally, and a Miura between the two.

That essay ended by admitting what its own picture was. Five points is not a survey. The corrugations sat on a line and the twists on a diagonal, and everything else in the plane was blank, with no way to tell a region that no pattern can reach from a region nobody had tried.

The gap is cheap to close, and the reason it stayed open is worth naming first. Every one of those five patterns was measured at the setting its builder happens to default to — a slant of 0.35 radians for the Miura, a zigzag of 0.42 for the leaf, a radius of 0.17 for the twist. None of those numbers is a property of the family. Each is a dial, and a dial that is never turned reads as a constant.

Two families are confined to a line, exactly #

The accordion’s cross factor is one at every fold count measured — two folds, three, four, six, eight, ten, twelve, sixteen — and it is exactly one, not one to within a tolerance. That is not a surprise once it is said out loud: an accordion’s creases are all parallel, so nothing in the pattern reaches across them, and the sheet’s width is untouched by any amount of folding. But it changes the object. The accordion is not a point on the lower edge of the plane; it is the whole of the lower edge, visited at integer positions, and its along-factor is the fold count itself.

The waterbomb does something similar in two directions at once. Swept through its column count, its factors come out at exactly the column count and exactly four, every time — a row of points rather than a curve, because the only dial it has is a count. Two of the six families reach a set of measure zero in the plane and no more.

That is worth holding on to, because it separates two kinds of dial. A dial that changes a pattern’s shape moves it through the plane; a dial that changes how many cells it has moves it along a line the shape already fixed. What a corrugation costs measured its whole table at one size per family, and the accordion and the waterbomb are the two entries where that decision cost nothing.

One more thing about the accordion is worth stating because it is the cleanest case of the pair carrying information the product destroys. Its areal factor is the fold count, the same number as its along factor, because the second factor is one. So the accordion is the pattern for which the old single-number quotation was correct — and it was correct by accident, in the sense that the number it reported happened to coincide with one of the pair rather than being a summary of both. Every other family in this table has a product that is nothing like either of its factors, and reading it as though it were a length is where the confusion the pair was invented to settle comes from.

The twist’s diagonal belongs to the square #

The square twist is on the diagonal at every radius from 0.06 to 0.33 — the two factors equal to every digit, eight settings, a range of radii over which the twist grows from a sixth of the sheet to a third of it. So the diagonal is not a place a twist happens to have been measured at; it is where a square twist lives.

It is not where a twist lives. A twist can be built around any regular polygon, which is the finding which polygons twist rests on, and measured at the same radius the other polygons are not on the diagonal at all. A triangle twist draws in 1.082 along and 1.168 across. A hexagon twist draws in 1.287 and 1.168 — the same cross factor as the triangle and a different along factor. A pentagon is 1.232 and 1.253, a heptagon 1.374 and 1.369.

The reason is a symmetry, and it is the one the plane can see. A pattern whose folded state is carried onto itself by a quarter turn has to draw in by the same factor in the two directions the quarter turn exchanges; a pattern with a threefold or sixfold symmetry has no such pair of perpendicular directions to exchange, and a hexagon twist’s two factors are measured along axes the pattern has no reason to treat alike. So the diagonal is a statement about fourfold symmetry rather than about twisting, and every square-based pattern in the library should be on it while nothing else need be.

That is a testable consequence rather than an explanation, and it is not tested here: the Miura is built on a square grid and is nowhere near the diagonal, which the symmetry reading has to account for and does, since a Miura’s slant destroys the quarter turn that would have forced it.

There is a second reading of the polygon twists’ numbers and it is the odder one. The triangle twist and the hexagon twist have the same cross factor to four figures — 1.168 for both — and different along factors, 1.082 and 1.287. Two patterns whose polygons have three and six sides, at the same radius on the same sheet, drawing in identically in one direction and not in the other, is not something the symmetry reading predicts and it is not obviously a coincidence either. A hexagon is a triangle’s polygon with its corners cut, and how that operation leaves one extent alone is a question about the pleats rather than about the polygon.

The dial that turns round #

The Miura’s slant is the clearest case of a dial doing something a single number cannot report. Ran from 0.15 radians to 1.25, the along factor falls steadily — 3.47 down to 1.06 — which is what everybody would expect, since a steeper slant means a shallower zigzag along the pattern.

The cross factor does not. It falls from 2.51 to a minimum near a slant of 0.80 and then climbs back, reaching 1.97 by 1.30 radians. There is a slant at which a Miura draws in least across itself, and steepening past it costs cross-shrink while continuing to buy along-shrink.

The square twist’s radius does the same thing and more sharply. Its shared factor rises from 1.136 at a radius of 0.06 to 1.923 at 0.26 and then falls to 1.515 by 0.33. A larger twist does not mean a smaller folded sheet. Past a certain size the twist has eaten the paper its own pleats were folded out of, and the sheet stops closing further because there is less left to close.

Both turning points are facts about a family rather than about a pattern, and neither is visible from any single measurement. A designer choosing a slant to maximise compaction has a maximum to find, and the two factors do not find it in the same place.

The Yoshimura arrives at the diagonal and stops #

The Yoshimura’s dial is its row height, and it is the one family here whose dial has a fence that is not about paper running out. Its row height is fixed by the big-little-big lemma rather than by Kawasaki: at the equilateral proportion no sector is strictly smallest and the lemma is vacuous, and above it one sector is strictly smallest at every interior vertex and the lemma refuses the pattern outright while developability, Kawasaki and Maekawa all go on holding.

Swept up to that fence, the Yoshimura’s along factor barely moves — 3.27 at a row height of half a column, wandering between 3.9 and 4.3, and 4.00 at the equilateral value — while its cross factor climbs the whole way, from 0.83 to 4.00. It arrives at the diagonal exactly where the lemma stops it. The pattern that draws in equally both ways is the last one the conditions permit, and one step further is a drawing rather than a fold.

That is the sharpest instance here of a dial whose limit is a theorem rather than a geometry. It also fixes where a real object sits: the cylinder the pattern chooses shows that a Yoshimura folds to a tube whose diameter is spent the moment the columns are drawn, and a tube is a pattern that has closed round on itself in one direction — which is the direction whose factor climbs all the way to four here. Fenced at both ends found the twist’s turn bounded above by paper and below by the assignment disappearing, and only one of those two fences was geometric. Here the geometric fence is nowhere near: a Yoshimura at a row height of two would draw perfectly well and fold at no vertex at all.

Two of the curves go below the line #

The accordion’s exact one looks like a floor, and it is not. Two of the six families cross it, and what the crossing means is a question of its own.

The leaf corrugation is the clearer case. Swept through its zigzag angle from 0.2 radians to 1.1, its along factor falls from 4.78 to 1.18 — a bigger range than any other family here — and its cross factor falls from 1.83 to below one, reaching 0.986, and comes back. The Yoshimura does it at the other end of its own dial: at a row height of half a column its cross factor is 0.826, and it climbs through one somewhere before a row height of seven tenths.

A factor below one is a direction in which the folded sheet is larger than the flat one, which the word shrink cannot describe. It is not a failure of the measurement — the areal factor stays above one in both cases, so no paper is created — and it is not a rounding artefact, since the leaf’s dip reaches almost a per cent and a half and the Yoshimura’s is seventeen per cent. What the two have in common is not their shape, and finding out what it is is a question of its own.

What a curve is, and what a point was #

It is worth being precise about the change in object, because the two pictures answer different questions and only one of them was ever asked out loud.

A point in this plane says: this pattern, drawn this way, does this. It is a complete answer about an instance and it supports exactly one kind of comparison — this pattern against that one, at the settings each happened to be drawn at. Every comparison what a corrugation costs makes is of that kind, and every one of them is correct.

A curve says something the points cannot: this family, whatever it is set to, stays here. That supports a different comparison — between what a family can do and what it cannot — and it is the comparison a designer needs, because a designer chooses the setting. The accordion’s line and the square twist’s diagonal are the two strongest statements in this whole picture and neither is about a pattern. They are about what a shape forbids, and a point cannot make a statement of that form at all.

The difference shows most clearly where the two disagree. At the radius it is usually drawn at, the square twist sits at 1.515 on the diagonal, and read as a point that is a modest compaction, worse than the Miura’s product and far worse than the accordion’s. Read as a curve it reaches 1.923 both ways, which is a product of 3.70 and puts it above the Miura’s default — and the setting that does it is a radius of a quarter of the sheet, which is a number nobody had cause to write down. The family was being judged on a number its builder chose.

What the sweeps are measuring #

Everything is read off a folded state. Each pattern is folded by composing reflections across its creases, and the two factors are the ratios of the flat extent to the folded extent along each axis. Nothing is predicted from the rule and nothing is inferred from a formula, so a pattern whose panels failed to close would be reported as a failure rather than as a number.

One size per family. The Miura is four columns by four rows throughout, the leaf has its own seven column widths, the Yoshimura is four by four. Where the count is itself the dial — the accordion and the waterbomb — the size varies and nothing else does. A family’s curve is therefore its shape’s curve at a fixed size, and the accordion is the demonstration that size can matter.

The extents are axis-aligned. The two factors are measured along the drawing’s own axes, which is the convention the pair was defined with. A pattern whose folded state turns — and a twist tessellation’s does, by a definite angle — has extents measured across a box that is not aligned with anything in the folded object.

And the twist here is a unit, not a tessellation. A single twist on a square of paper with its pleats running to the edge is a different object from the plane version, and its factors include the uncreased paper around it. A unit that folds is not a tessellation is the standing statement of that distinction, and the curve labelled as the twist’s here is the unit’s curve; the plane version’s would be measured on a patch and would carry the patch’s rim with it.

What the picture does not decide #

It does not say the blank regions are forbidden. Six families is more than five points and it is still six families. A region no curve passes through may be unreachable, may be reachable by a family not in the library, or may be reachable by one of these families under a dial nobody has identified as a dial.

It does not fill the plane in the useful direction. Every curve here runs upward and to the right of one-by-one, and the interesting question for a designer is the opposite corner — how much of the plane is reachable at a given areal shrink, which is a curve of constant product and cuts across all of these.

It does not separate shape from size where both move. The waterbomb’s dial is a count and the accordion’s is a count, so those two families’ curves are size curves; the others are shape curves at fixed size. A family whose shape and size both mattered would need a surface rather than a curve, and nothing here would have noticed.

It says nothing about depth. A pattern’s stack depth is the areal factor read backwards, which is one of the two halves of what a corrugation costs; two patterns at the same point of this plane have the same depth and may be nothing like each other to handle.

And the fourfold reading is an observation rather than a proof. That a quarter turn forces the two factors equal is an argument about symmetry; that nothing else forces them equal is not established, and a pattern on the diagonal with no fourfold symmetry would settle it at once.

Still open: the curve of constant product #

The plane has a family of curves drawn through it that nothing here has used, and they are the ones a designer would ask for. The areal factor is the product of the two, so a hyperbola in this plane is a set of patterns that compact the same amount and distribute it differently — one that takes it all along the pattern, one that splits it evenly, and everything between.

Reading the measurements against those hyperbolas rather than against the axes turns the picture into a comparison: at a fixed amount of compaction, which families are available, and what does each of them do to the shape of what is left? The Yoshimura at its fence and the square twist at its best radius have very different products and neither is obviously the better answer to any real question, because the question has never been posed in the units the answer lives in.

The other open thing is a family that is not in the library. Every curve here belongs to a pattern somebody drew because somebody folds it, and the patterns nobody designed are the reminder that a sheet can arrive at a corrugation without anybody choosing it. A buckled cylinder’s diamonds are a Yoshimura at whatever proportion the cylinder’s own dimensions force, which is a point on that family’s curve chosen by a physical process rather than by a designer — and where on the curve it lands is a question this measurement makes askable and does not answer.

Sideways from here, every dial here deserves the treatment these six have had. A drawing rule takes a default; a default reads as a property; and a property that is really a default is invisible in exactly the way a measurement cannot catch, because the measurement is correct about the pattern it was handed. The check is to sweep and see whether anything moves, and it costs one loop.

The habit worth carrying is about constants that are not. Ask of every number in a table which of them the builder chose, and turn those. Five of the numbers in the original pair-table were the builders’ defaults, and the two that could not have been — the accordion’s exact one and the twist’s exact equality — are the two that turned out to be properties rather than readings.