A shrink is two numbers

Anybody who has folded an accordion knows something that the standard way of quoting a corrugation cannot express. The folded strip is much shorter than the sheet was and it is exactly as wide. Nothing happened in one of the two directions at all.

The number usually reported is how many times smaller the folded thing is, and that number is a product. Separating it into its two factors takes one measurement and changes what the whole family looks like.

The number that was already an average #

The measurement at the rung below folded five patterns and reported, for each, how many times smaller it got. That is a real quantity and it is not the wrong one; it is simply the product of two, and the two carry information it does not.

The separation is straightforward to make. A folded state is a set of panels placed in the plane, so it has an extent across and an extent down, and so does the flat sheet those panels came from. The ratio of the widths is one factor and the ratio of the heights is the other.

Done that way the accordion’s second factor comes out at 1.000 exactly, which is the arithmetic saying what a folder’s hands already knew.

Three places in the plane #

Once both factors are in hand the patterns sort themselves into three positions, and the positions turn out to have names.

A pattern that leaves one direction of the sheet alone has a second factor of exactly one, and sits on a horizontal line. That is a corrugation — parallel creases, nothing happening across them. The accordion is the pure case and is the only one of the five to sit there.

A pattern that draws in equally both ways sits on the diagonal. That is what a twist does: a square twist rotates a small polygon while the sheet closes symmetrically around it, and its two factors come out at 1.515 and 1.515.

Everything else is off both lines, and that is where the Miura is. Its 2.728 across and 1.748 down are not equal and neither is one. The anisotropy is not a detail of the Miura; it is what makes it a Miura rather than a pleat, and a pattern tuned until its two factors matched would have stopped being one.

The leaf corrugation is the instructive middle case, and it shows that the word does less work than the plane does. It is called a corrugation, it is drawn as one, and it is what a leaf packing into a bud uses — but its factors are 3.573 and 1.220, so its second direction is not left alone. The zigzag shifts sideways as it closes, which the accordion’s straight parallel creases never do, and 0.220 of drawing-in is what that shift costs. A name covers both patterns; the measurement separates them.

One caution, and it is the sort this site prefers to state rather than let a reader discover. Sitting on the diagonal does not make a pattern a twist. The Yoshimura sits there too, at 4.000 by 4.000, and it is a lattice of diamonds that closes into a tube with no twist anywhere in it. What the diagonal reports is isotropy in the drawing-in, and isotropy is a behaviour that patterns with nothing else in common can share.

The box and the paper are not the same area #

Here the measurement has to be read carefully, because two different areas are in play and this site has quoted both.

The two directional factors multiply to the ratio of two rectangles: the box the flat sheet occupies, over the box the folded object occupies. That is a statement about extents. It is not the same as the ratio of the sheet’s area to the area the folded object actually covers, because a folded object does not fill its own box.

The gap is large. The accordion fills its box completely — the folded strip is a rectangle — so its two accounts agree at 8.00. The Miura’s box is 4.77 times smaller than the flat sheet’s while the paper covers 8.14 times less area than it did, so the folded object occupies about 59% of the rectangle it needs. The square twist fills 77% of its box, the leaf corrugation 52%, and the Yoshimura almost exactly half.

The Yoshimura’s exact half is not a coincidence, and the layer map says why. Its folded outline is a triangle: full width along one edge, narrowing to a point at the other, because the offset courses close in on themselves from one end. A triangle takes exactly half of the rectangle drawn round it, whatever its proportions, so the fill fraction is a fact about the outline’s shape rather than a number that happened to land near a half.

Neither number is the honest one on its own. The area account is what the paper did; the box account is what the folded object demands of the world around it. Quoting one where the other is meant is an error of nearly a factor of two on the Yoshimura, and it is silent — both numbers are correct and both are called the shrink.

What a container actually constrains #

This is where the pair stops being a curiosity, because a container does not constrain an area.

An antenna that has to go into a cylindrical fairing has a diameter to fit inside and a length to fit along. Those are two separate inequalities against two separate extents, and a product satisfies neither of them. A pattern that halves one direction and quarters the other packs eight times smaller by area and may not fit at all.

The two patterns already measured make the reversal concrete. Read each folded box as a fraction of the sheet it came from. The accordion’s is an eighth of the sheet’s width by the whole of its height — the second factor is one, so that dimension never moved. The Miura’s is 0.367 of its width by 0.572 of its height. By box area the accordion wins comfortably, 8.00 against 4.77, and it is the accordion that will not go into the canister, because its longest folded dimension is still a full side of the original sheet while the Miura’s is a little over half of one. The pattern with the better shrink needs the longer container. Nothing in the single number can report that, and reversals of this kind are not rare: they happen whenever a pattern’s advantage sits in the direction the container was generous about anyway.

The same reasoning runs through the biological version of the problem. A bud selects the number of folds a leaf uses by trading strip width against stack thickness, and both of those are extents. Nothing in that argument is about area either, and it could not be run on a single shrink factor.

The gap between a folding result and the hardware that uses it is partly made of exactly this. A pattern arrives with a packing figure; a fairing arrives with a drawing; and turning one into the other is a piece of work that the packing figure was never going to do by itself.

The figure of merit is the smaller factor #

The container argument can be finished rather than illustrated, because a container that is a scaled copy of the sheet turns the pair into a single honest number — a different single number.

Both directions have to fit, so the scale a pattern achieves against such a container is limited by whichever direction shrank least. The figure of merit is $\min(f_1, f_2)$, not $f_1 f_2$.

Applied to the five, that re-ranks them completely:

The accordion falls from second to last, and it falls to exactly one — which is the arithmetic saying that against a container demanding both directions, an accordion has not folded at all. The square twist, worst by product, comes third.

And the plot measures the size of the error #

The two summaries are related, and the relation is the essay’s own picture read as arithmetic.

$$\frac{\sqrt{f_1 f_2}}{\min(f_1, f_2)} = \sqrt{f_1/f_2},$$

so the product overstates a pattern’s container performance by exactly the square root of its anisotropy — the distance from the diagonal, in the plot at the head of this essay, is the size of the mistake.

For the patterns on the diagonal it is one: the square twist’s and the Yoshimura’s single numbers are honest, because there is no direction for the summary to hide in. For the Miura it is $\sqrt{2.728/1.748} = 1.25$. For the accordion it is $\sqrt{8} = 2.83$, which is where a factor of nearly three of apparent packing goes.

So the plot is not merely a classification. A pattern’s horizontal distance from the diagonal is how much its own headline number is lying about it, and the two positions the essay names — the untouched-direction line and the diagonal — are the worst case and the only honest one.

The other half of the same measurement #

The area account has a second life, and it is the result this ladder started from.

The paper has nowhere else to be, so the area the folded object covers, multiplied by the average number of layers over it, is the area of the sheet. The shrink measured by area is therefore the average layer count, exactly — not approximately, and not as an empirical relation.

So the depth of a stack is not an extra property of a corrugation. It is the shrink, counted the other way up, and a pattern cannot be tuned for one without moving the other.

Putting the two halves together gives the shape of the whole finding. A folded pattern has a pair of directional factors, which say what box it needs; the product of that pair is a box ratio; and a different areal ratio, the one about the paper rather than the rectangle, is the average layer count. Three numbers, two of them commonly called the same thing.

What was checked, and what would have broken it #

The identity between the two factors and the box ratio is arithmetic — a width ratio times a height ratio is an area ratio, and no measurement could contradict it. So what the check is actually testing is the machinery rather than the geometry, and it is worth being exact about that rather than presenting a tautology as a result.

The two factors are read off the extents of the folded panels. The box ratio is accumulated separately, from the corner coordinates, by the same routine that measures every panel’s area. Two different pieces of code, one reading extents and one summing a polygon, and their answers agree to 8.9 parts in a thousand million million. A sign error in the folded coordinates, a panel left out of the extent sweep, or a shoelace that mishandled a rectangle would all show up here, and none of them would show up in the picture.

The second assertion is not arithmetic and could genuinely fail. The layer map is sampled — a grid laid over the folded state, counting how many panels cover each cell — and the sheet’s area is computed exactly. Requiring their product to come to the whole sheet is a demand that the sampling be fine enough to see the pattern, and it is a demand that can be refused. Asked for a coarse grid, the machinery reports the accordion putting back 0.9600 of its own sheet and stops. The tolerance is 2%, the worst of the five at the working resolution is 1.0%, and the margin between those two numbers is the only reason the figure draws at all.

That is the site’s standing habit applied to its own arithmetic: one claim that cannot fail and is checked anyway because the route to it can, and one claim that can fail and does when the conditions are wrong.

Where the model stops #

Every pair here is the fully folded state. A Miura has a motion, and its two factors move along it — at a quarter closed they are near one, and the values quoted are the endpoint. A single pair is a snapshot of a pattern at full fold, not a property of the pattern across its range.

That connection is worth one more sentence, because it makes the accordion’s exact 1.000 less of a coincidence. A ratio of zero and a second shrink factor of exactly one are the same statement about the accordion, differentiated and integrated: nothing happens across the creases at any point of the motion, so nothing has happened across them at the end. The Miura’s surface of ratios is the same relation for a pattern where something does.

The box is axis-aligned. The extents are measured along the axes the folded state happens to arrive in. A folded object turned forty-five degrees might need a smaller rectangle, and nothing here searches for the best orientation. Every box quoted is an upper bound of a particular kind.

The sheet has no thickness. A stack thirty-six layers deep is thirty-six thicknesses of real material, and what that adds is not in any of these numbers. The pair is a geometric quantity and the packed height of a real deployable is not.

Nothing here is a claim about force. How hard the pattern is to close, what holds it shut, and what happens to it under load are outside this measurement entirely. Every figure above is a length, an angle or an area.

Who quoted one number, and why #

The single figure has an honest origin. A packing ratio is what a proposal needs — one number, comparable across candidates, small is good — and the deployable literature quotes stowed volume against deployed area because that is what a mass and volume budget is written in.

It is also what the folding literature can supply cheaply. The areal shrink falls out of the layer count and the layer count falls out of the folded state, so it is the number a pattern hands over without being asked. The directional pair takes a decision about which axes to measure along, and a decision is exactly what a comparison table is trying to avoid.

There is a third reason, and it is the one this site is most exposed to. A single number can be printed in a column and sorted. Two numbers cannot: a table of pairs has no order, because there is no way to rank a pattern that draws in eight times one way and not at all the other against one that draws in three both ways without first saying what the container looks like. The pressure toward one number is a pressure toward a ranking, and a ranking is what a comparison is usually for. The rung below this one ran straight into the same thing from the other side, where introducing a fourth column reordered the table completely.

The people who never used the single number are the ones building the hardware. A deployable’s requirement has always been written as a set of dimensions, and the engineer converting a pattern into a canister was doing this separation by hand. What is new here is only that both factors are computed from the crease pattern rather than measured off a prototype, so the separation can be made before anything is built and on patterns nobody has folded.

Where the ladder goes next #

The immediate continuation is the pattern that hides the most behind its single number. The Yoshimura folds sixteen times smaller by box and thirty-two times smaller by area, exactly half its rectangle empty, because what it actually makes is a tube and a flattened ring is half a rectangle. The pair says nothing about that; the geometry of the closure does.

The other direction is the motion the pairs were read at the end of. A pattern’s position in the shrink plane traces a path as the sheet closes, starting at one-by-one on the flat sheet and ending at the point quoted here, and which route it takes between them is a fact about the pattern that no endpoint captures.

And there is a design question left open by the plane itself. The corrugations sit on a line, the twists on a diagonal, and the space between and beyond them is mostly empty in the pattern library — five points is not a survey. Whether the gaps are geometrically forbidden or merely unvisited is not settled by any measurement here, and it is the kind of question a wider census could answer.