The corrugation that curves

Take a single flat-foldable vertex and tile the plane with it, and the sheet stops being a sheet and becomes a material — with a stiffness, a packing behaviour and a Poisson’s ratio the paper never had. The Miura is the standing example and it is a flat material: it starts flat, it ends flat, and everywhere in between it is a slab.

Open its straight creases into a fan and it is not a slab any more.

The construction, as a tessellation #

A Miura is usually described as a grid of identical parallelograms. It is more useful here to describe it as two families of crease: one family straight, running the length of the sheet, and one family zigzag, reflecting in each straight crease it crosses.

Written that way the generalisation is obvious and nobody had to invent it: let the straight family fan out instead of running parallel. The zigzag family still reflects in every line it crosses, so every vertex is still developable and still satisfies Kawasaki identically — the two row creases sit at equal and opposite angles to a crease running straight through, and two pairs of supplementary angles have equal alternating sums whatever the angles are.

The repetition is what makes it a tessellation, and the repetition here is not a translation. Rows are scaled copies of one another about the point the fan meets at, so the pattern is periodic in the way a logarithmic spiral is periodic rather than the way a grid is: the same shape, at a different size, over and over.

There is a small piece of arithmetic in the construction that is worth having, because it is what makes the drawing possible at all. Going along a row, the reflection rule and the fan’s angle between rays fix the angle at every subsequent vertex: if the row leaves one ray at an angle β, it arrives at the next at π − Δφ − β, and then leaves at β again. So the angles alternate between two values and the distances from the fan’s centre alternate in the ratio sin β ⁄ sin(β + Δφ). A row is a zigzag between two radii, going round the fan — which is a thing a person could draw with a protractor and a pair of dividers.

What it folds onto #

The folded shape is the surprise, and it is a good deal more specific than “curved”.

Take a fan member, fold it, and extend its straight creases as lines in space. They meet. Not nearly: the largest distance from a common point to any of them is 1.8 × 10⁻¹⁵ of a sheet-width, which is the arithmetic’s own noise. The folded corrugation lies on a cone.

That is not something a corrugation had to do. A sheet of straight creases could bend onto a cylinder, or onto a general developable surface, or onto nothing in particular. This one lands on a cone, exactly, and the reason is in the construction: the straight creases were concurrent in the flat sheet — that is what a fan is — and folding is a rigid motion of each panel, which cannot make concurrent lines stop being concurrent within one panel and evidently does not do so across the whole sheet either.

The apex moves. At a fifth of the way through the fold it is a fraction of a sheet-width from the flat sheet’s own plane; near the end it is two thirds of a sheet-width away. So the cone the sheet wraps is a different cone at every moment, opening and travelling as the corrugation closes.

There is a way to see why a cone and not something else, and it is worth having because it makes the measurement predictable rather than lucky. A corrugation of this kind is ruled by its straight creases: those creases stay straight through the fold, because a straight crease is the boundary of two rigid panels and rigid panels do not bend. A surface made of straight lines that all meet at one point is a cone, and a surface made of straight lines that are all parallel is a cylinder. The Miura’s straight creases are parallel and it stays flat — which is the degenerate case of a cylinder with no curvature at all. Fan them and the only thing they can be is a cone.

The fan closes as the sheet folds #

The second measurable is how wide the cone is, and it does not stay put either.

Measured on the straight creases — which is the right place, because a corrugation’s panels zigzag and comparing panel orientations reports the zigzag rather than the shape — a fan of 9.2° between neighbouring columns spans 55.0° flat and 47.3° when folded to about half. At the widest fan tried, 13.8°, it is 82.5° flat and falls to 57.1° near the end of the motion.

So the pattern’s fan angle is not the folded cone’s angle, and reading one off the other is a mistake. What the pattern fixes is a family of cones, one per stage of the fold, and the sheet visits all of them on the way to being packed.

What it is for #

A corrugation that wraps a cone is a useful object and the reason is worth stating plainly.

A Miura packs a flat sheet into a flat packet, which is exactly what a map or a solar array wants. What neither wants is a curved surface — but a great many other things do: a reflector, a shroud, anything that has to be a section of a cone when deployed and a small package when not.

The usual way to get a curved deployable is to accept panels that are not flat, or to build a curved thing out of flat pieces and hinge them. This family gives it out of a single uncut sheet with straight creases, which is the cheapest kind of thing to manufacture, and it gives it with a single degree of freedom so that one actuator drives the whole surface.

There is one more number worth quoting for a maker, and it is a caution rather than a selling point. The fan closes by roughly a quarter over the motion, but not linearly: most of the closing happens in the last third of the fold. A mechanism designed around the deployed geometry and driven by a single actuator will therefore spend most of its travel doing very little to the cone and the last part of it doing a great deal, which is exactly the behaviour a control system dislikes.

What is the same and what is not #

It is worth separating the properties this family inherits from the Miura from the ones it does not.

Inherited. Every vertex is a Miura vertex — one straight crease with two symmetric creases across it — so everything local is unchanged: the same degree, the same sector structure, the same gearing between the two fold angles at a vertex. A patch small enough to contain one vertex cannot tell the difference.

Not inherited. The global behaviour. The Miura’s two in-plane dimensions shrink together, which is what its negative Poisson’s ratio means; a fan member’s “in-plane dimensions” are not well defined, because the folded object is not in a plane. Whatever the fan members do instead, it is not that.

That is the distinction between a unit and a tessellation arriving in a new family and pointing the other way. There, a patch too small to contain all the pattern’s vertices could not see the rules the pattern really had. Here, a patch of any size sees the correct local rules and cannot see the fan at all — because the fan is a property of how the copies are related and not of any copy.

What a folder can make of it #

The pattern is drawable by hand and worth drawing, because the folded object is not what the flat one looks like it will be.

Mark a point off one corner of the sheet and draw a fan of rays from it — seven or eight, a few degrees apart. Then draw the zigzag: start somewhere on the first ray, go to the second, and at each ray reflect, which in practice means measuring the angle the line arrived at and leaving at the same angle on the other side. Do that for four or five rows, each starting further out along the first ray in the same ratio as the last. Score the rays as one assignment and the zigzags alternating, and collapse.

What comes off the table is a curved shell rather than a pleated slab, and the curvature is not an accident of how firmly it was pressed: it is in the pattern, and it will be there every time. That is worth ninety seconds of anybody’s evening, and it is the sort of thing this subject does better than any other — the argument is the artefact, and the artefact can be held.

The zigzag is a logarithmic spiral #

The construction’s arithmetic is given above as a rule for angles and a ratio for distances, and one step of the sine rule turns it into an identification of the curve the zigzag traces.

Take three consecutive points of a row, on three consecutive rays. The triangle they make with the fan’s centre has an angle of $\Delta\varphi$ at the centre, an angle of $\beta$ where the row leaves one ray, and $\pi - \Delta\varphi - \beta$ where it arrives at the next. The sine rule gives the distances from the centre as

$$\frac{d_{k+1}}{d_k} = \frac{\sin\beta}{\sin(\beta + \Delta\varphi)}.$$

And the reflection rule sends the row out of the next ray at $\beta$ again, so that ratio is the same at every step. The distances from the centre form a geometric sequence, and a zigzag whose distance from a point multiplies by a constant for every constant turn about it is a discrete logarithmic spiral. The essay’s remark about being periodic the way a logarithmic spiral is periodic turns out to be an identification rather than a simile.

Which says how many rows a sheet holds #

The consequence is about the sheet rather than the curve, and it is sharper than more folding near the middle.

The ratio is below one whenever the fan opens — $\sin\beta$ against $\sin(\beta + \Delta\varphi)$ with both angles under a right angle — so a row marches inward by a constant factor at every ray. Over a fan of $n$ rays the row’s distance from the centre falls by that factor to the $n$-th power, which is exponential in the number of columns, not linear.

So the crease density does not merely vary across the sheet: it varies geometrically. A pattern drawn with a comfortable cell at the outside has cells a fixed fraction smaller at each step inward, and after enough steps they are below anything a hand can fold. The number of columns a sheet can carry is therefore the logarithm of the ratio between its largest usable cell and its smallest, divided by the logarithm of that one ratio — a formula with no free choices in it once the fan angle and the row angle are fixed.

That is a real constraint on the family and it is the one a maker meets first. A wide fan closes the ratio faster: $\Delta\varphi$ appears inside the sine on the denominator, so opening the fan makes each step contract more, and the widest fan tried here is also the one with the fewest usable columns. The fan angle buys curvature and spends columns, and the exchange rate is a logarithm.

It also explains the awkwardness the previous section names without resolving. A translation-periodic tessellation can be continued indefinitely in both directions because its cells are all the same size. This one cannot be continued in either: outward the cells grow without bound and outrun the paper, inward they shrink without bound and outrun the crease. A pattern with a centre and an outside is a pattern with two ends, and both of them are set by the same ratio.

A note on what “tessellation” is doing here #

The word deserves scrutiny, because a fan corrugation is not periodic in the way the rest of this field’s patterns are.

A Miura repeats by translation: shift the pattern one cell along and it lands on itself. A fan member repeats by scaling — shift a row outward and it lands on the next one only after being enlarged by a fixed factor. That is a perfectly good repetition and it is the reason the vertex geometry is the same everywhere, which is what makes the sheet behave as a material rather than as a shape.

It also means the pattern has a centre and an outside, which no translation-periodic tessellation has. Rows near the fan’s centre are small and rows far from it are large, so the crease density varies across the sheet — a lot of folding near the middle and much less at the rim. A folder feels that immediately and a translation-periodic pattern never does it.

So the family sits awkwardly between two of this collection’s categories — it is one vertex repeated, which is the definition of a tessellation here, and the repetition is not the one the word usually implies. Both facts are worth carrying, and the second is what makes the folded object a cone rather than a slab.

Where the model stops #

One fan, one shape. Every member here has straight creases that are concurrent in the plane. Straight creases that are neither parallel nor concurrent are a different family and it is not built.

The cone is measured, not derived. The concurrency of the folded creases is a measurement at a set of drive angles, on a set of fan angles. The argument that it must hold is a paragraph that has not been written; what is reported is that it does, to fifteen decimal places, everywhere it was tested.

Rigid panels and no material. Every panel is rigid by declaration. A real sheet has a crease with a radius and panels that bend, and a real corrugation of this kind would depart from its cone by whatever the material gives.

No packing measurement. How efficiently a fan member packs — the quantity that makes the Miura useful — is not computed, because the footprint of a curved object is a projection and the right measure is not obvious.

The pattern is a patch. The construction produces a finite fan. Continuing it round to a full turn is possible and would close on itself only for particular fan angles, and that is not investigated.

Who found it, and when #

Corrugations that fold onto curved surfaces are not new; they are a substantial thread in the folding-engineering literature of the past two decades, principally through Tomohiro Tachi’s work on generalised and freeform quadrilateral meshes, and conical corrugations in particular are a known object with known applications.

What this site has is the measurement rather than the family: the concurrency at 10⁻¹⁵, the apex’s travel through the motion, and the fan closing by a quarter between the flat sheet and the packed one. All three are the kind of number that gets asserted qualitatively in a description and is worth having as a figure.

Where the ladder goes next #

The obvious continuation is the surface. A cone is one surface and a designer wants many, so the question is which surfaces are reachable by relaxing the construction — letting the straight family be a general set of lines, or letting the rows be something other than scaled copies. The rows have to be copies for the sheet to fold at all, so the freedom is in the fan and the question is what a fan of arbitrary lines gives.

The other direction is the packing. The Miura is used because it packs, and nobody has said what a conical corrugation packs into. The answer is presumably a sector of an annulus rather than a rectangle, and knowing how efficiently would decide whether this family is a curiosity or a component.