Patterns nobody designed

Stand a drink can on the floor and step on it. It does not crumple randomly. It collapses into a regular pattern of diamonds, and it does so every time.

The same pattern appears in crushed tubes, in the bark of some trees, in deployable booms, and in a great deal of packaging. It is a fold pattern that physics found.

Why a cylinder does this #

A thin-walled cylinder under axial compression has to shorten. It can do that by squashing uniformly, which costs a great deal of energy because the material must compress; or it can buckle, which costs bending energy instead.

Bending is much cheaper than stretching for a thin sheet — the stiffness against stretching scales with thickness and the stiffness against bending with thickness cubed — so the cylinder buckles.

Which buckling mode it chooses is the one with the lowest energy, and for a cylinder the answer is a diamond lattice: a pattern of inward and outward folds arranged in offset rows. That is the Yoshimura pattern.

The critical point is that the mode is selected by minimisation rather than by design. Nothing chose the diamonds; they are simply cheaper than anything else.

Why it satisfies the theorems #

There is an argument here that is worth making carefully, because it is a good demonstration that the flat-folding conditions describe sheets rather than folders.

The crushed can folded flat. Therefore its crease pattern is flat-foldable. Therefore Maekawa holds at every interior vertex — the mountains and valleys differ by exactly two — and so does Kawasaki.

Nobody applied the theorems. Nobody knew them. The can obeys them because they are consequences of being a sheet that reached a flat state, and the sheet reached one.

The same argument covers a crumpled ball of paper, a rumpled shirt and a leaf in a bud. Every one of them is a flat-foldable crease pattern, verified by the fact that it folded.

Deliberate and found patterns meet #

Yoshimura’s diamonds and the Miura fold are closely related, and the relationship is not a coincidence.

Both are periodic tilings of a degree-four vertex. Both are rigid-foldable. Both collapse along essentially one path. The Miura is a shear-like arrangement suited to a flat sheet; the Yoshimura is a cylindrical arrangement suited to a tube.

Miura himself worked on both, and the connection runs the other way too: the Miura fold’s discovery came partly from studying how cylinders buckle. A deployable structure and a failure mode turn out to be the same geometry looked at with different intentions.

That is a recurring pattern in this subject. The efficient way to collapse a sheet is a small set of arrangements, and physics and designers keep arriving at them independently.

The theorems have no length and the can does #

There is a gap in the argument above that is worth opening rather than stepping over, because it says exactly what the flat-folding conditions contribute and what they cannot.

Kawasaki reads angles, Maekawa counts letters, and neither contains a length. So the theorems are satisfied by a diamond lattice of any pitch — a can with eight diamonds round it and a can with sixty are equally admissible, and nothing in the local theory prefers either.

The can prefers one, and the quantity that decides is a length the theorems do not have. Buckling picks the wavelength that balances bending against the stretching a curved sheet cannot avoid, and for a cylinder of radius $R$ and wall thickness $t$ that scale is

$$\lambda \sim \sqrt{Rt}.$$

So the number of diamonds around the circumference goes as $2\pi R/\lambda \propto \sqrt{R/t}$: a thinner wall gives more diamonds and smaller ones, and the same can in a heavier gauge collapses into a coarser lattice. That is a prediction anybody can check on a drink can against a food tin, and it is invisible to every condition in this subject.

And the cheapness has an exact size #

The same length settles the essay’s other assertion. Bending is cheaper than stretching for a thin sheet, and the factor is not merely large — it is the thinness ratio itself.

The cost of bending at wavelength $\lambda$ scales as $(t/\lambda)^2$ against stretching, and putting $\lambda \sim \sqrt{Rt}$ into that gives $t^2/Rt = t/R$. Buckling beats squashing by a factor of $t/R$, which for a drink can with a 0.1 mm wall and a 33 mm radius is about one in three hundred and thirty.

So the two halves of the account divide cleanly. Flat-foldability says which lattices are available to a sheet that reaches a flat state, and it is a combinatorial statement with no scale in it. Energy says which one is taken, and it is a statement about $R$ and $t$ and nothing else. Neither answers the other’s question, and the can is where they meet.

Crumpling, which is different #

It is worth separating the regular found patterns from genuine crumpling, because they behave completely differently.

The Yoshimura pattern is periodic. Crumpling a sheet of paper into a ball is not: the crease network is disordered, with a broad distribution of facet sizes and a characteristic concentration of energy into point-like vertices and line-like ridges.

Crumpling has its own physics — the energy concentrates in a vanishing fraction of the material, the stiffness of a crumpled ball scales with a power law in its density, and the whole thing is a well-studied problem in statistical mechanics.

What it shares with the ordered case is only the flat-foldability argument: the ball can be flattened, so its pattern is flat-foldable. Beyond that, ordered and disordered folding are different subjects.

Where it is used deliberately #

Once the pattern is understood as a mechanism rather than as a failure, it becomes a design.

Deployable booms. A Yoshimura-patterned tube collapses axially in a controlled way and springs back, which makes a mast that packs short and extends long.

Crash structures. A tube that buckles into a known pattern absorbs energy predictably. Vehicle crumple zones are designed so that the collapse mode is the one the engineer chose rather than whichever one the accident found.

Packaging and stents. Anything that needs to compress radially or axially along a known path.

The move in each case is to encourage the pattern the structure would find anyway, by scoring or perforating along its lines, so that the collapse is repeatable rather than merely likely.

The pattern in living things #

Folded structures in biology are common, and it is unclear how often the mathematics is doing the work.

Insect wings fold along patterns close to known tessellations, and some are close to the Miura. Leaves emerge from buds in corrugated arrangements that unfold as they grow. Beech and hornbeam leaves in particular have a fold structure very like a Miura, and this is frequently cited.

The honest position is that the resemblance is real and the causation is unsettled. A leaf packing into a bud faces the same problem as a solar array packing into a fairing — a flat thing must fit a small volume and open reliably — and it would be surprising if similar geometry did not appear.

Whether the leaf is “using the Miura fold” or has arrived at something similar by a different route is not usually determined, and the popular accounts are more confident than the literature.

What makes a pattern findable #

There is a question underneath all of this: why do so few patterns keep turning up?

Part of the answer is that the constraints are severe. A pattern that tiles, folds flat, is rigid-foldable and has one degree of freedom is a small set — the vertices must be identical, the sectors must satisfy Kawasaki, the assignment must satisfy Maekawa, and the whole thing must actually collapse.

Part of it is energy. A buckling sheet finds a low-energy mode, and low-energy modes are regular, because regularity is what lets bending distribute rather than concentrate.

So both the designer and the physics are searching a small space, from different directions, for the same thing: an arrangement that collapses cheaply and repeatably. It is unsurprising that they meet.

Encouraging the pattern #

The engineering move is not to invent a collapse mode but to make the one physics prefers repeatable, and the techniques are simple.

Scoring. Weaken the material along the intended crease lines, so the buckle initiates there rather than wherever the largest imperfection happens to be.

Perforation. The same idea with holes, used where scoring would compromise the surface.

Embossing. Pre-form a shallow version of the pattern so the structure is already slightly folded and the mode is biased from the start.

Triggering. Add a deliberate imperfection at one end so the collapse starts at a known place and propagates predictably.

All four are ways of removing the randomness from a buckling event. An unscored tube collapses into the Yoshimura pattern with the diamonds in an arbitrary rotational position and sometimes into a different mode entirely; a scored one collapses the same way every time.

That repeatability is the entire product for a crash structure. The energy absorbed by a controlled collapse is calculable; the energy absorbed by whatever happens is not.

The line between found and designed #

There is a general point here that reaches past this pattern.

A designed fold is a set of creases chosen to produce a behaviour. A found fold is a set of creases the material chose to minimise energy. The two coincide surprisingly often, and the reason is that both are searching a small space of arrangements that collapse cheaply and repeatably.

Tessellations that work are those with identical vertices, satisfying the local conditions, with one degree of freedom. Buckling modes that appear are those with low bending energy and regular structure. Those two descriptions pick out overlapping sets, which is why a crushed can looks like something somebody folded.

The practical consequence is that a designer looking for a collapse pattern is often better served by crushing something and looking at the result than by starting from theory — which is a slightly uncomfortable thing to say on a site built around computing patterns from first principles, and is true.

Where the model stops #

Idealised buckling. The account of why a cylinder finds this mode is qualitative. The real analysis is a nonlinear shell problem, the mode depends on the radius-to-thickness ratio, and short thick tubes do something else entirely.

The pattern is drawn flat. The Yoshimura pattern is intrinsically cylindrical, and drawing it as a flat lattice is a development of the cylinder rather than the thing itself. The figure is honest about the connectivity and silent about the curvature.

Material matters. Aluminium buckles differently from paper, which is one of the idealisations failing, which buckles differently from Mylar. The pattern is a family, and which member appears depends on the material.

Biology is asserted, not shown. The leaf and insect-wing claims are reported here as reported elsewhere. No figure on this page is a measurement of anything living.

Crumpling is a different subject. It is mentioned and not treated, and almost nothing said here about periodic patterns transfers to it.

Reading a failure as a design #

There is a habit of mind this subject encourages, and the Yoshimura pattern is the clearest case of it.

An engineer looking at a crushed tube sees a failure. The structure was supposed to carry load and it collapsed, and the collapse pattern is a description of how it went wrong.

A folder looking at the same tube sees a mechanism. The pattern is a repeatable, low-energy way of converting a long cylinder into a short one, which is exactly what a deployable wants, and the fact that it happens spontaneously is a feature rather than a fault.

Both readings are correct and they lead to opposite actions. The engineer strengthens the tube to prevent the pattern; the designer scores it to encourage the pattern. Crash structures do the second deliberately — a crumple zone is a component designed to fail in a specified mode, and specifying the mode is what makes the energy absorption calculable.

That inversion — treating a failure mode as a mechanism — is where a good deal of the engineering value in this subject comes from, and it requires knowing that the pattern is a pattern rather than a mess.

Why regularity is cheap #

The energy argument deserves a little more, because “the lowest-energy mode is regular” is not obvious.

A thin sheet resists stretching far more than it resists bending — the stiffnesses differ by a factor of order $(L/t)^2$, which for a drink can is around ten thousand. So any deformation that avoids stretching is enormously cheaper than one that does not.

Deformations that avoid stretching are the developable ones: the sheet bends but does not change any distance measured within it. A collapse made of flat facets joined at creases is developable everywhere except at the creases and vertices, and at a vertex the angles still sum to a full turn.

So the cheap collapses are precisely the ones that look like origami — and among those, the regular ones distribute the bending evenly rather than concentrating it, which is cheaper again.

That is the whole reason a crushed can looks designed. It is not that physics has taste; it is that the cheap deformations of a thin sheet are a small and highly structured set, and everybody searching that set finds the same things.

Looking for patterns by breaking things #

There is a research method implied here that is worth stating, because it inverts the usual direction.

The conventional route is to derive a pattern from requirements: decide what the structure must do, work out the geometry, verify it folds. That is what design algorithms do and it works for representational folding.

The other route is to load a sheet and see what it does. Crush a tube, compress a cylinder, buckle a plate, and record the crease network that results. Whatever appears is by construction low-energy, repeatable and flat-foldable, because it just happened.

That is how the Yoshimura pattern was found, and it is how several deployable patterns have been found since. It is also a reasonable way to search a space that is hard to characterise: physics does the optimisation, and the experimenter reads the answer off the debris.

The limitation is that it only finds patterns for the loading applied. A crushed cylinder gives patterns for axial collapse and nothing for anything else, and the method has no way to aim.

The same shapes keep appearing #

Collecting the coincidences makes the point better than any of them alone.

A crushed drink can finds the Yoshimura pattern. A compressed thin-walled tube finds it too. Tree bark under growth stress approximates it. Deployable booms are built with it deliberately.

The Miura fold appears in a spacecraft array, in a road map, in the wings of some insects and in leaves emerging from buds.

Crumpled paper, a rumpled shirt and a leaf all satisfy Maekawa’s theorem at every interior vertex, because all of them folded flat.

The common cause is that the cheap ways to collapse a thin sheet form a small, structured set, and every process that collapses one — a designer, a buckling instability, a growing plant — is drawing from it. The recurrence is not mysterious and it is not evidence of anything beyond that.

The pattern predates the name #

A closing note on attribution, because this pattern has an unusually clear case of it.

Yoshimura described the diamond lattice in 1955, in a technical report on the buckling of cylindrical shells. The report is about structural failure. It contains no origami, no folding, and no suggestion that the pattern might be useful.

The pattern itself is older than any description. Every crushed tube in history had it. Bark had it. Insects had it. What Yoshimura contributed was the observation that it is regular and reproducible, which is what turned it from debris into a subject.

That sequence — the thing exists, somebody notices it is structured, it becomes a design — is the same one the Miura fold followed and the same one circle packing followed. This field advances mostly by noticing rather than by inventing, which is worth saying because it is unusual.

The ladder from here #

Later rungs: cylinder buckling analysed properly. The Yoshimura pattern’s geometry and its relation to the Miura. Crumpling, ridges and the concentration of energy. Deployable booms. Crash structures and controlled collapse. Folding in insect wings, measured. Leaf packing. Bistability in found patterns. And the classification question — which periodic flat-foldable tilings of a single vertex exist, which is a finite and largely answered problem for degree four.

Yoshimura published in 1955, on the buckling of cylindrical shells, in a report for the Japanese aeronautical research institute. The paper is about structural failure and contains no origami at all, and the pattern it describes is now folded on purpose by people who have never read it.