The direction that gets longer

The plane the five points were in swept six families of corrugation through their own dials and found four of them behaving as a reader would expect: both directional factors above one, the folded sheet smaller than the flat sheet in both directions, the product equal to the areal shrink. The other two cross below one over part of their range, and the crossing is worth an essay because it is not a failure of anything.

The leaf corrugation’s cross factor falls below one. At a zigzag angle of 0.78 radians it is 0.98555, which says the folded state’s extent across the pattern is a per cent and a half larger than the flat sheet’s. The window where it happens runs from about 0.70 radians to about 0.88, and at every angle in it the areal factor is above 1.47 — so the paper has not grown, the fold has not failed, and no measurement has gone wrong.

What a directional factor actually is #

The pair was defined by measuring the flat sheet’s extent along each axis, measuring the folded state’s extent along the same two axes, and dividing. That definition is the reason the number can go below one, and saying why needs one distinction that the word shrink obscures.

Area is conserved and extent is not. The area of a folded state’s footprint can only fall, because every panel keeps its own area through the fold and the panels can only pile up; that is the conservation law everything here is built on, and it is the reason the areal factor is above one for every pattern ever measured here. An extent is not an area. It is the width of the smallest axis-aligned box containing the thing, and a box is a property of the arrangement rather than of the material: rotate a long thin panel and the box it needs changes, with no paper going anywhere.

A corrugation folds by rotating its panels about parallel creases. Along the pattern that rotation brings panel after panel over one another and the extent falls hard — the leaf’s along factor runs from 4.78 down to 1.18 across its range. Across the pattern the creases are the wrong way round to bring anything together, so the cross extent is decided by the panels’ own tilt, and a tilt can take a corner further out than it started.

The leaf is not alone in it, and that is the first clue to the cause. The Yoshimura crosses the same line — at a row height of half a column its cross factor is 0.826, and it climbs through one before a row height of seven tenths. A Yoshimura has no taper, no unequal columns and nothing in common with a leaf corrugation but the fact that its creases run across the sheet.

The cause is not the one the picture suggests #

The obvious suspect is the taper, which is the one thing about the leaf that is unusual. It is the only one of the six families whose columns are not all the same width — its seven run 0.09, 0.15, 0.20, 0.22, 0.20, 0.15, 0.09, and that profile is what makes it a leaf rather than a plain corrugation. A tapered column’s zigzag carries its own shift across the pattern, the shifts do not cancel, and it is easy to tell a story in which the extreme panels reach further across than the sheet’s own edge.

The taper is not the cause. Drawn with seven equal columns at the same zigzag, the cross factor is 1.097 rather than 0.986 — lower than a straight corrugation might be expected to give, and comfortably above one. The taper is worth about a tenth of the effect. Something else supplies the rest of it.

What supplies it is the sheet’s own height, and the way it does so is exact.

The folded extent is a constant of the cell #

Fold a leaf corrugation at a zigzag of 0.78 radians and measure how far the folded state reaches across the pattern. At two rows it is 0.6900. At three rows it is 0.6900. At four, five, six, eight and ten rows it is 0.6900 — the same number at every row count, to every digit measured.

That is what a corrugation does, said in a way nothing here had said it. Folding brings the rows onto one another, so the folded state’s extent across them is set by one cell’s height and the angle it is folded to, and adding rows adds nothing to it. The flat sheet’s height, meanwhile, is the row count times the cell height and grows without limit.

So the cross factor is

$$s_{\perp} = \frac{\text{rows}\times h_{\text{cell}}}{0.6900}$$

which is a straight line through the origin — 0.2464 per row, to five figures at every count measured. It crosses one between four rows and five, and everything below that crossing is a sheet whose folded state is taller than it is.

That is why the window in the angle sweep exists and why it is where it is. The default leaf has four rows, which puts it just under the crossing; the folded extent varies with the zigzag angle, so the crossing point moves as the angle changes, and the window is the range of angles over which four rows is on the wrong side of it. Nothing about the window is a property of the leaf’s taper, and nothing about it is a property of four as a number. It is one line crossing another.

Where the crossing is, and what decides it #

The crossing has a closed form and it is worth writing out, because it turns a curiosity into a rule a reader can apply without measuring anything.

The flat sheet’s height is the row count $n$ times the cell height $h$. The folded extent is some length $E$ that depends on the cell’s own shape and on the zigzag angle and on nothing else. So the cross factor is $nh/E$, and it is below one exactly when

$$n < \frac{E}{h}$$

At the zigzag measured here that ratio is $0.6900/0.17 = 4.06$, so four rows is under and five is over. Both numbers are properties of a single cell, which means the threshold can be computed from one row of the pattern and applied to any sheet of it.

That also says what happens to a family of corrugations as it is made larger. A wide, shallow sheet of a few rows folds to something taller than itself; the same pattern taken to twenty rows folds to something five times shorter than itself; and the factor between those two is the row count and nothing else. A single number quoted for the family would be a number quoted for whichever sheet somebody drew.

What a corrugation costs measured its whole table at one size per family, which was the right decision for the comparison it was making and is exactly the decision this line makes visible. Where a factor is proportional to a count, a table at one size is a table at one point of a line.

What goes wrong with the word #

The pair was introduced to replace a single number that hid a distinction, and it succeeded. What it inherited from that single number is the word, and the word does not survive the window.

A factor of 0.9858 is not a shrink of 0.9858. It is a growth of one and a half per cent, reported in a unit built to count shrinking, and every sentence of the form the pattern draws in by is false of it. Nothing in the arithmetic is wrong — the product of 1.6554 and 0.98582 is 1.6319, which is the areal factor measured independently, to nine figures — but a reader told that a corrugation draws in 1.66 one way and 0.99 the other has to notice the second number’s size to understand it, and a reader told a single areal figure of 1.63 would never suspect anything.

And the pair’s own geometry is convention. The two axes are the drawing’s, chosen because a corrugation is drawn with its creases along one of them. A folded state that turns — and a twist tessellation’s turns by a definite angle — is measured against a box aligned with nothing it contains, so its two factors are about the box rather than about the object. The leaf does not turn, which is why its window is a clean statement about a real thing; a pattern that turned could produce a factor below one for a reason that was purely about the axes.

What the window is not #

It is not a thickness effect. Every fold here is of a sheet with no thickness, so the folded state’s panels lie exactly on one another and the extents are computed from their corners. A panel is not the unit of depth is where the thickness question belongs, and a real corrugation folded out of paper would stand taller than either number here for reasons that have nothing to do with the geometry. A real leaf corrugation folded out of paper would be thicker where the panels pile up and that would change the footprint, but it is not what produces the window.

It is not an artefact of the flat sheet’s shape. The leaf’s flat outline is not a rectangle — the taper makes it a stepped shape — but the flat extent used is the box around it, which is the same convention used for every other pattern here, and it is constant at 0.68 across the whole angle sweep at four rows. What varies is the folded extent, which climbs to 0.6900.

And it is not about the leaf. Every corrugation whose creases run across the sheet has a folded extent set by its cell, so every one of them has a row count below which its cross factor is under one. The Yoshimura is the measured second witness, and it is a cleaner one than the leaf: at a row height of one its cross factor is exactly half the row count — 1.000 at two rows, 2.000 at four, 3.000 at six, 4.000 at eight — with its along factor sitting at exactly the column count throughout. Same line, same origin, a different constant. The accordion is the exception that shows the rule: its creases are parallel to the direction being measured, so there is no collapse across them at all and the factor is exactly one at every size. What makes the leaf the one this was found on is that the sheet it is usually drawn at happens to have four rows.

It is not an artefact of the sample. The dip is measured at every angle drawn between 0.60 and 1.00 in steps of 0.02, and it is smooth: below one from 0.70 through 0.88, with a minimum at 0.78, returning to 1.0053 by 0.90. A one-point dip would be suspicious; twelve points are a curve.

And it is not a failure of the fold. Every state measured closes: the reflections compose consistently around every vertex, the panels keep their areas, and the layer count integrated over the footprint returns the sheet. A pattern that had failed any of those would have been refused rather than measured.

What else is measured as a length #

The reading that separates the two kinds of quantity is worth applying to every other measurement made here, because it sorts them cleanly and the sorting has not been done.

Safe, because it is an area or derived from one. The areal factor. The footprint. The average stack depth, which is the areal factor read backwards and is why the paper is all still there can be stated as an identity rather than as an observation. Crease length per unit area. Anything measured by integrating over the sheet.

Not safe, because it is an extent. Every width, height, diameter and radius. The cylinder the pattern chooses is the clearest case: its whole argument is about the diameter of the tube a Yoshimura folds to, and a diameter is an extent of a folded state. That argument happens to be about a closure rather than a bounding box — the course of diamonds has to go round exactly once, which is a length along the paper — so it survives, but the survival is a fact about that argument and not about diameters.

And the two periods are a third case. The turn a column costs measures a folded period, which is neither an area nor an extent but a translation: the smallest slide that carries the folded state onto itself. A translation is a property of the object rather than of any box around it, so it is immune to the whole of this, and it is the one quantity here that would still be the right number if every drawing were rotated.

That sorting also explains why the effect had to be found in this particular column. The pair of directional factors is the only measurement here that is defined as a ratio of extents. Everything else either integrates or counts.

What this costs the table #

Three statements elsewhere quote a directional factor and describe it as a shrink. None of them is about the leaf inside its window, so none of them is wrong. What changes is the form of words available in future: a directional factor is a ratio of extents, and a ratio of extents is not bounded below by one.

It also changes what the pair is good for. Its purpose was to separate an accordion, which leaves one direction alone, from a twist, which draws in equally, from a Miura, which does neither — and for that purpose it works, because all three of those statements are about the shape of a family and are read from the ratio of the two factors rather than from either. The ratio is dimensionless in the row count as well: multiply the sheet’s rows and only one of the two moves, so a family’s position in the plane moves along a line and its character moves with it. A pattern on the diagonal at one size is not on the diagonal at another unless both of its factors scale together, which is a condition nothing has checked.

The practical consequence is for anybody choosing a corrugation for a shape rather than for compaction. A designer who needs a panel that packs along its length and stays exactly as wide has the accordion, which is exactly one across, forever. A designer who needs one that packs along its length and gets slightly wider has the leaf at a zigzag near 0.78 — a requirement nobody would think to state, because the vocabulary says it is impossible.

Still open: how far below one a fold can go #

The line through the origin says the factor can be made as small as wanted by taking rows away, and the sheet runs out at two rows, where the measured value is 0.4928 — a folded state twice as tall as its own sheet. That is the extreme available by shrinking the sheet, and it is not very interesting, because a two-row corrugation is barely a corrugation.

The interesting question is the other constant. $E$ is set by the cell’s shape, and the leaf’s cell was drawn with one taper profile out of infinitely many. How large $E$ can be made for a given cell height is an optimisation with a definite answer, and it decides the threshold row count for the whole family — a cell whose $E/h$ were ten would give sheets of nine rows that fold taller than they are.

There is a boundary on it that comes free. The areal factor is the product of the two and cannot fall below one, so a cross factor of $c<1$ forces an along factor above $1/c$. A corrugation that grew ten per cent across would have to draw in at least ten per cent more along than its area alone requires, which is a constraint rather than a trade, since the along factor is what a corrugation is for.

Sideways from here, the line through the origin is a shape of result worth recognising. A quantity that is a ratio of two things, one of which grows with the sheet and one of which does not, is a quantity with a threshold in it — and a threshold is invisible to any measurement taken at one size. One vertex, repeated is the argument that a tiling becomes a material once it is large enough for the pattern rather than the sheet to decide its behaviour; a factor proportional to the row count is the other end of the same observation, and says how large “large enough” has to be before any quoted number means what it appears to.

The habit worth carrying is about quantities with a floor that was never proved. When a measurement is described by a word that implies a direction, check whether the direction is a theorem or a habit. Here it was a habit, the check was a sweep already being run for another purpose, and the window fell out of a column of numbers nobody had a reason to read twice.