Corrugation — where it appears
Named by 36 essays across 6 fields — each of them below, with the objects they name alongside it.
A leaf packs by corrugating
A corrugation is the cheapest fold there is — parallel creases, no interior vertex to think about — and a leaf that uses one has to taper it, because a leaf is broad in the middle. Which direction the taper is allowed to run turns out not to be a matter of taste.
The bud chooses the pattern
Nothing in the geometry of a corrugation says how many folds it should have. The container does: too few folds is a strip too wide to fit, too many is a stack too thick to fit, and the window that fits at all is narrow and has a best point in the middle of it.
Opening with nothing to pull
A leaf is not opened by a hand, a hinge or a motor. It opens because it keeps growing — which puts a condition on the pattern that no folder ever has to satisfy, because a person can always push.
A wing that folds into nothing
A beetle stows a wing longer than its body under a case a fraction of that length, and the ratio is the whole engineering problem. What a fold achieves is computable from the pattern alone, and the four geometries available are not close to each other.
How much surface fits in a body
An organ whose whole job is to have area — a gut, a gill, a cortex — is solving a packing problem in reverse. Folding buys surface inside a fixed volume, and with a sheet of zero thickness it buys an unlimited amount. With a thickness the curve turns over and then falls to nothing.
The same corrugation in four places
A leaf, a wing, a crushed cylinder and a solar array arrive at nearly the same fold, and none of them copied any of the others. Convergence stories are cheap; this one is checkable, because the constraint that forces it can be computed rather than admired.
What a corrugation costs
Every tessellation this repository can fold, measured the same way: how much smaller it gets, how deep the stack becomes, and how much creasing was needed to buy it. The last column is the one nobody quotes and the one a folder feels.
A shrink is two numbers
How much smaller a folded sheet gets is quoted as a single factor, and that factor is a product. Measured along each axis separately, an accordion turns out to leave one direction of the paper exactly alone, a twist draws in equally both ways, and the Miura does neither — which is the whole of what makes it a Miura.
The family the Miura belongs to
Move one vertex of a Miura and the sheet has no rigid folded position at all — which leaves the obvious question unanswered. What else moves? A row of paper reflected in each of a fan of lines is flat-foldable for nothing at all, and whether it also folds rigidly turns out to be a condition on a table of cosines: it has to be a column of numbers times a row of numbers.
The corrugation that curves
A Miura is a flat sheet that becomes a flat slab. Open its straight creases into a fan and the same construction gives a corrugation that wraps a cone — exactly a cone, with every straight crease passing through one point to fifteen decimal places, at every moment of the fold, with the apex travelling as the sheet closes.
Four finders, one option
Four unrelated lineages arriving at the same corrugation is read as evidence that the corrugation is good. It is at least as much evidence that there was nothing else to arrive at: how far a folded sheet shrinks is exactly its average layer count, so a lineage choosing a packing ratio is choosing a number of layers and nothing else — and the quantity that is genuinely free turns out to be almost uncorrelated with it.
A corrugation agrees with itself
A Miura fold of forty-eight panels and a twist tessellation patch of forty-nine have almost exactly the same number of independent closed chains for their letters to contradict themselves round — thirty-five against thirty-six. Sixty-four per cent of the Miura's drawn letterings are consistent and thirteen per cent of the patch's. A Yoshimura at thirty-three chains manages ninety-three. The room to fail sets the scale; the construction decides where in it a pattern lands.
The taper decides nothing
A leaf's corrugation narrows toward its margin, and the taper is what the pattern is for. It has no effect whatever on how often the pattern's letters agree with themselves: four width profiles from perfectly even to strongly tapered give a hundred and seventy-four consistent letterings of two hundred, identically. What moves the number is the count of rows, and on that measure a leaf tracks a Miura rather than the corrugation it most resembles.
Sixty-four rules, sixteen fold
The Miura fold's letters are usually given as a recipe: rows one way, columns changing at every row. Write down every rule of that shape — the letter on a crease depending only on which row and which column it is in — and there are sixty-four. Sixteen fold flat. They are exactly the ones whose columns change at every row, the row letters do not matter at all, and every one of the forty-eight refusals is the counting theorem's alone.
The loop is in the rule
Of the forty-eight repeating rules that do not fold a grid corrugation, thirty-eight send four panels round in a circle and ten merely fail the count. Which is which can be read off three of the rule's six bits, without building the pattern, folding it or walking a single arrow — and the closed form agrees with the arrows on all sixty-four rules of both grid families.
Where a rule can close a loop
Three corrugation families have repeating rules whose letters send four panels round in a circle, and one has none at all. The one that has none is the one whose vertices are all of degree six — and the reason is that a straight line through a point carries a single letter under any repeating rule, while a strict alternation round six creases needs the two halves of that line to differ.
Half the recipe is decoration
Every account of the Miura fold gives its letters as two instructions: the rows go one way, and the columns change letter every time they cross a row. Enumerate all sixty-four repeating rules and the second instruction is the whole of the condition — all four ways of writing the rows appear among the sixteen that fold, in every combination. The first instruction has never constrained anything.
The leaf's rules are the Miura's
A plicate leaf packs into a corrugation that is broad in the middle and narrow at both ends. Enumerate every repeating mountain-valley rule it admits and the table is the Miura fold's table, rule for rule, number for number — and it stays that table when the leaf is redrawn with even columns, a violent taper, taller rows or a steeper zigzag. The plant's geometry cannot reach its own letters.
A corrugation never backtracks
As a box-pleating grid goes from two divisions to sixteen, the share of random letterings that agree with themselves falls from a hundred in a hundred to one. The cost of finding one that does stays at exactly one step per panel — four, nine, sixteen, twenty-five, and two hundred and fifty-six — with not a single wrong guess anywhere in the family.
The plant's pattern is not a hard case
A hornbeam leaf packs into its bud by corrugating, and the pattern it uses gives up a consistent lettering at nine, twelve, fifteen, eighteen, twenty and twenty-four steps on nine, twelve, fifteen, eighteen, twenty and twenty-four panels. Nothing about the plant's problem is combinatorially difficult, and saying so is worth as much as finding a case that is.
One step per panel is a table size
Four families of crease pattern search at exactly one step per panel — a grid at nine sizes, a leaf, a Miura, six crumples — and it was read as a law about patterns that fill their own sheet. It is a number: the conditions at each of their vertices admit eight labellings. Where the conditions admit four, the cost is half. Where they admit thirty, it moves again, and the same pattern at two proportions demonstrates it with everything else held still.
Six creases and the same straight line
The one family here whose vertices are degree six was said to break the arithmetic that every other family obeys, on the strength of a single pattern. Built as a family — six sizes from twenty-one panels to a hundred and nineteen — the Yoshimura is exactly as linear as a grid, with no decision ever withdrawn. What degree changes is the constant, and it changes it in both directions depending on one angle.
A knife edge nine decimals wide
Draw the Yoshimura with its rows 1.7320508 half-columns tall and each vertex admits thirty labellings and the pattern costs fifty-seven steps. Draw it at 1.7320509 and each admits eight and it costs nineteen. The number between them is √3, which is the proportion everybody draws — and below it the sectors are unequal and the lemma is still silent, because the small ones sit next to each other.
The edge was not what made it hard
Five families of pattern searched at one step per panel and a tessellation patch did not, and the property left standing after four alternatives were killed was having a rim. Measured under a fixed letter order the patches cost between a half and two-thirds of a step per panel, at every tiling and every size — below the line rather than above it, and the rim is why.
The designer's grid is the dearest thing here
Two hundred and fifty-six panels of box-pleating grid take two hundred and fifty-six search steps to letter — exactly one per panel, at every size from two divisions to sixteen, with not one decision withdrawn. That is the most any pattern in this collection costs per panel of paper. A twist tessellation costs half of it, and a tilted corrugation a quarter.
Nothing grown was cut out of anything
A leaf's corrugation costs twelve steps on twelve panels, sixteen on sixteen, twenty on twenty, twenty-four on twenty-four — exactly one per panel at every geometry, which is the most any pattern here costs. A tessellation patch costs half that, and the reason is that somebody cut it out of something. A leaf's creases stop at the margin because the plant stopped there.
The corrugation that closes on itself
A Miura cell crosses one crease per period in one direction and four in the other. So one of its two directions has a parity condition that half the sizes fail, and the other direction cannot fail at any size — the same sheet, the same drawing, and two gluings that behave completely differently.
The turn a column costs
The Yoshimura's drawing repeats every column. Folded flat, it does not: the fold carries one column onto the next by a turn of two hundred and forty degrees, so the folded state repeats every third column and not before. A pattern has two periods and only one of them has ever been written down.
The census returns one
The rung below this one asked for a census: every pattern reaching a stated packing ratio while opening from a single input, with its crease density. The census is makeable for the corrugations and it comes back with one member. A corrugation piles its panels over a footprint as wide as its longest panel, so its ratio is the total length divided by that longest one — and that equals the panel count only when every panel is the same.
Four materials, four optima
The convergence argument gets its pattern and stops there. A hinge has a radius, the radius takes a fixed length of surface out of every fold, and the fold count that gets the most packing out of a sheet is inversely proportional to it — so a leaf, a wing, a gut lining and a metal array agreeing on a corrugation still disagree by an order of magnitude about how many creases to put in one.
The number is the angle
Every packing ratio worked out so far is computed at a fold closed all the way, and a folded wing is not closed all the way. At zero thickness the ratio runs away as the fold shuts, so the size of a quoted number says how far the fold got and not what the pattern is — and the pattern contributes only an exponent, which makes the same quoted ratio mean two quite different angles depending on which geometry produced it.
The surface has to be supplied
The curve that turns over does so because the sheet's own thickness fills the box it is folding into. A surface in a body has to be reached as well as fitted, and the channel that reaches it takes depth out of the same box on exactly the same terms — so the best fold count and the surface it delivers both fall by the ratio of the sheet's thickness to the sheet and its supply together.
The fold count sets the spring
A corrugation sweeps the same span at every fold count, and the count decides only how much room the zigzag needs while it does it — which was counted as a gain with nothing pushing back. Something does push back. Every hinge is a spring, a finer corrugation has proportionally more of them, and the force to hold a given span rises exactly as the clearance falls: the product of the two is the same number whatever the count, and at each material's own best count the spring goes as one over the square of the hinge radius.
How far open is a question about the grip
A corrugation of hinges that rest flat is loaded when it is shut, so it opens by itself and the force in the held-state calculation is a restraint rather than a drive. Followed from shut to flat that restraint only ever falls, and by exactly π over two — so every partly open state a structure can occupy is squeezed into a band a third wide, and a grip that weakens by a third leaves the sheet nine tenths open.
Springs that disagree do not offer a choice
A corrugation of hinges that remember different angles was expected to have more than one position in which nothing pushes. It has exactly one, at the stiffness-weighted mean of what they remember, because a sum of parabolas in one variable is a parabola. The second resting place comes from somewhere else entirely — the mirror pattern — and the flat sheet is the barrier between them, which is also why a creased sheet cannot be pulled flat at all.
Holding a fold moves the force
A creased hinge held at an angle slowly comes to remember that angle, so a leaf or a wing packed in a bud for a season is gradually holding itself and the bud has less to do. The force is not used up in the process. The torque the container must supply falls as e^(−T⁄τ), the torque later needed to open the structure rises by exactly the same amount, and the two sum to the same total at every moment of the holding. Held for three relaxation times, a corrugation creased to 0.8 radians and packed to 2.4 needs 4 per cent of the total from its container and 96 per cent from whatever opens it — and the barrier to its mirror image has grown more than eightfold. A packing that lasts buys independence from its container with a harder unfolding.
Named alongside it
The objects these essays reach for when they reach for this one.
AssignmentPacking ratioConvergenceConstraint propagationLeaf foldingSearch costDegree-fourMiura-oriUnit cellCrease radiusDeploymentMaekawa's theorem