Opening with nothing to pull

Everything on this site that folds is folded by somebody. A person, a press brake, a laminator, a robot: something applies a force where the pattern needs one, and if the pattern needs the force in two places at once it gets it in two places at once.

A leaf has none of that. It is packed inside a bud, and it opens because it grows — the tissue increases in area and the folded state stops being able to contain it. There is no hand, nothing to grip, and nothing that can pull one part of the leaf while pushing another.

That turns out to be a real constraint on the pattern, and it is one no folder ever has to think about.

What a growth-driven opening can and cannot do #

Growth increases area. That is the entire actuation available, and it has three properties worth listing because each rules something out.

It is monotone. Tissue does not un-grow. Whatever quantity is driving the unfolding increases and never decreases, so the motion runs one way along a path and cannot back up.

It is slow and distributed. There is no impulse, no snap and no single point where the force is applied. Every part of the sheet is doing a little of the work all the time.

And it is one-way in a stronger sense than a motor is: a motor that has run too far can run back. A leaf that has opened cannot repack itself, which means there is no recovery from a motion that goes wrong.

Together those say the pattern must have a path from packed to open along which the driving quantity increases the whole way. A pattern that has to get narrower before it gets wider needs something to narrow it, and nothing is available.

Why a folder never meets this constraint #

The contrast with human folding is sharp and it is worth spelling out, because it explains why this condition does not appear anywhere else on the site.

A person folding a model routinely does things that are locally backwards. A squash fold opens a flap in order to close it differently. A reverse fold pushes a point inside out. Half the traditional repertoire consists of moves that make the model temporarily larger, messier or less finished than it was, on the way to something better.

That is available because a hand can apply force anywhere, in any direction, in any order. The site has a whole field about what happens when that freedom is restricted — a machine that can only make one kind of move reaches a strictly smaller set of states than a hand does, and the restriction changes the complexity class.

Growth is the most restricted actuator in that hierarchy. It cannot choose where to act, cannot choose a direction, and cannot reverse. It is weaker than any machine model this site has studied, and the interesting thing is that it is nevertheless sufficient for the job a leaf needs done.

Checking the condition rather than asserting it #

The corrugation’s motion is easy to parameterise: one angle describes the whole state, and the exposed span is the number of panels times the sine of that angle.

The sine is monotone on the range in question, so the answer is obvious — which is exactly why the generator checks it step by step rather than appealing to the fact. The check costs nothing, it is stated in terms of the quantity the essay is about rather than the quantity the formula is about, and it would catch a sign error, an angle range that wrapped past 90°, or a future change to the model that introduced a non-monotone term.

The check that a claim survives is worth more when the claim is obvious than when it is surprising, because an obvious claim is the one nobody re-examines. This site has already been caught once by an assertion that restated its own derivation and could not fail; the discipline is to assert the thing the figure is about.

What this rules out, and it is a lot #

A great many perfectly good crease patterns fail this condition, and naming the classes is more useful than naming examples.

Anything with a locking state. A pattern can have a folded state that no continuous motion reaches — the flat state exists, the folded state exists, and the path between them does not. That is fatal for growth-driven deployment and merely inconvenient for a person, who can bend the material slightly and get past it.

Anything needing a sequence. A pattern whose creases must be closed in a particular order needs something that knows the order. Growth is a single quantity increasing; it carries no sequence information at all.

Anything with more than one degree of freedom. A pattern with several independent freedoms has a whole space of states rather than a path, and growth does not specify a direction through a space. That is the next rung and it is the sharpest of the three.

Anything that has to pass through a tighter state. A pattern that must first compress further to release a catch is asking for an actuator that can compress, and growth only expands.

Those four are not exotic. Between them they cover most of the patterns anybody would reach for if asked to design a deployable, which is a useful thing to know before designing one — and it is the reason the fold that growing things actually use is the plainest one available rather than the cleverest.

The state that exists and cannot be reached #

The first of those four classes deserves a picture, because it is the one that is genuinely surprising and the one this site established with its own machinery.

A folded state is a configuration of the sheet. A motion is a continuous path of configurations. Those are different objects, and it is entirely possible for a pattern to have a folded state that no motion arrives at: the flat sheet is one configuration, the folded state is another, and every continuous path between them passes through something the panels cannot do.

A person meets this and shrugs. Paper bends; a slight flex of a panel gets past the obstruction and the model reaches the state anyway, which is why a great many traditional models are not rigid-foldable and nobody notices. The distinction between the two properties only becomes urgent when panels are involved.

Growth cannot shrug. There is no flex to apply and nothing to apply it with, and a leaf whose pattern had an unreachable open state would simply stay shut. So the biological case sits at the strict end of a distinction that is usually academic.

Two properties that are easy to confuse #

It is worth putting the two side by side, because a deployable needs both and they are independent.

Flat-foldability is about a destination: does a state exist in which the sheet lies in a plane. Rigid-foldability is about a journey: is there a continuous motion, with the panels staying flat, from the open state to the folded one.

Monotonicity, which is what this essay is about, is a third thing again — a condition on the journey’s direction rather than on its existence. A pattern can be rigid-foldable and non-monotone in every quantity a grower could drive.

That the corrugation satisfies all three is why it is the fold that shows up in growing things. It is not the best packer, it is not the most elegant, and it is the one whose motion asks for the least.

The ordering of the three is worth holding on to, because it is a hierarchy of increasingly demanding questions and most discussions of deployable folding slide between them. Does a folded state exist? Can the panels get there? Can they get there under a driver that only pushes one way? Each answer is a subset of the last, each subset is strictly smaller, and a sheet with one freedom is the rare case that clears all three at once and is therefore worth its reputation.

The practical consequence for anybody reading a claim about a deployable pattern is to ask which of the three has actually been established. A published crease pattern that folds flat has cleared the first. A published simulation that closes it has cleared the second. The third is usually left to the hardware to discover.

The engineering version of the same constraint #

Self-deploying hardware has exactly this problem and solves it in ways a leaf cannot.

A pattern driven by stored elastic energy releases in whatever direction lowers the energy, which can be non-monotone in any particular geometric quantity. A pattern driven by a shape-memory alloy can be driven both ways by heating and cooling. A pattern driven by a chemical or thermal stimulus in the material itself is closest to the biological case and still has the advantage that the stimulus can be patterned — applied to some creases and not others.

The comparison is useful because it locates precisely what is special about the biological case. It is not that the actuation is in the material; engineered self-folding sheets have that too. It is that the actuation is a single scalar that only increases, and that is a much stronger restriction than any of the engineered mechanisms accept.

What monotone does not mean #

Two clarifications, because the word is doing a lot of work.

Monotone in the exposed span does not mean monotone in everything. The depth of the corrugation falls throughout the same motion — that is what opening means — and there is no contradiction, because the driving quantity is the area of tissue and the constraint is on the path rather than on every measurable feature of it.

And monotone does not mean the motion is uniform. The span rises quickly at first and slowly near the end, which is the sine flattening out, and that is why the last few degrees of a leaf’s opening take a disproportionate share of the growth. Anybody who has watched a leaf finish unfurling has watched the flat part of that curve.

That flattening has a consequence worth drawing out, because it inverts the intuition about where the difficulty is. The hard part of a growth-driven deployment is not getting started — the packed state is where the geometry gives the most span for the least motion. The hard part is finishing, because near the open state a great deal of growth buys very little extra span, and any residual fold that has not opened is expensive to remove.

That is the geometric reason a leaf can spend a long time looking almost open, and it is also why creases in leaves are often still faintly visible in a mature specimen. The last few percent of the motion is the part the driver is worst at, and there is nothing available to finish the job by hand.

What the flattening costs, in numbers #

The observation that the last part of the motion is the expensive part can be made exact, and the exactness is what turns it from a remark into an explanation.

The exposed span is the sheet’s length times the sine of the fold angle, so the span gained per degree of opening is proportional to the cosine — largest at the packed end and zero at the open one.

Run the arithmetic the other way, which is the way a leaf experiences it. Reaching 90 per cent of the full span takes the angle to 64°, which is 71 per cent of the travel. Reaching 99 per cent takes it to 82°, which is 91 per cent. The final one per cent of span costs nine per cent of the whole motion, and the final tenth costs nearly a third.

So a leaf that has used most of its available growth is a leaf that looks open and is not finished, and the part it has not finished is the part its driver is worst at delivering.

Why the creases stay visible #

The same relation, read backwards, explains something the essay only asserts.

Near the open end, one minus the sine of the angle is very nearly half the square of the angle from flat. Invert it: a fractional shortfall $\delta$ in span corresponds to a residual fold angle of about

$$\sqrt{2\delta}$$

in radians. A square root, which is an amplification. A leaf that reaches 99 per cent of its span still has its creases sitting at eight degrees from flat. At 99.9 per cent they are still at two and a half degrees.

That is why a mature leaf shows its folds. There is no failure involved and no arrested development: a fold angle small enough to be invisible needs a span accurate to a part in ten thousand, and the last fraction of a per cent is exactly the part the cosine has made unaffordable.

Two facts about the same curve, pulling against each other. The driver’s efficiency vanishes as the sine flattens, and the visibility of the residual grows as a square root. Between them they guarantee that a growth-driven opening ends with a visible crease, whatever the leaf does.

An engineered deployable escapes this by finishing with something other than the driver — a latch, a preload, a panel stop — which is a fourth thing growth does not have. The corrugation is the right pattern for a leaf and the ending is the part the pattern cannot help with.

What the picture cannot show #

The figure plots a geometric quantity against a fold angle. It does not plot anything against time, and it must not be read as a rate.

The relationship between the fold angle and the amount of growth that has happened is not modelled here at all. It depends on where the growth is, how the folded state constrains it, and what the tissue does when it is confined — none of which is modelled here. So the curve says the motion is available in the order the leaf needs; it says nothing about how long any part of it takes.

The three zigzags drawn beneath the curve are sampled states of the same motion, drawn at true angle, and they are the whole of the geometry. There is no attempt to draw a leaf, and the panels are flat because the model’s panels are flat.

The idealisation, named #

The panels are rigid and the creases are the only things that move. A real leaf bends everywhere, and a substantial part of a real unfolding is the panels themselves flattening out rather than the fold angles opening.

That matters for the monotonicity argument in an interesting way: a sheet that can bend has more paths available, not fewer, so a pattern that fails the rigid test might still open for a leaf. The condition computed here is therefore sufficient rather than necessary — a pattern that passes it can be opened by a monotone driver, and a pattern that fails it might still manage with some panel bending. The honest statement is the one-directional one.

The corrugation is also treated as having a single angle, which assumes every fold opens by the same amount at the same time. Real corrugated leaves open from the tip or from the base, with a wave of opening travelling along them, and that is a motion with a position in it rather than a single number.

Where this ladder goes next #

The leaf ladder ends here, and the wings ladder begins from the same place with the opposite advantage.

An insect deploying a hindwing has muscles, but not at every crease — the pattern has to do most of the work itself, and the condition that makes that possible is a count rather than a monotonicity. One degree of freedom means one number determines the whole state, which means one actuator suffices, which is why the same property shows up in everything anybody builds to deploy.

The connection worth carrying is that these are two different requirements often confused. Monotonicity is about the path; a single degree of freedom is about the space. A pattern can have one freedom and a non-monotone path, or a monotone path through a multi-dimensional space that happens to be the one taken. A deployable driven by growth needs both, which is a narrower requirement than either.