What a corrugation costs

The reason to fold a sheet into a corrugation is to make it smaller, and the number that gets quoted is how much smaller. It is the wrong number on its own, and this essay is about what else has to go beside it.

What is measured, and how #

Every pattern is folded by composing reflections, which places each panel exactly where the fold puts it. Then three quantities are read off the result.

The footprint is the area the folded object covers, sampled on a grid. The shrinkage is the sheet’s area divided by that. The depth is how many panels lie over each point, reported as an average and as a maximum.

The first two are the same quantity, and that is the conservation law: footprint times average depth is the area of the sheet, exactly, because the paper has nowhere else to be. So the shrink factor is the average layer count, and the figure reports their product against the sheet as a check rather than as a result. It is one for every pattern in the table.

The fourth column is different in kind. Crease length is measured in sheet-widths — total creased length divided by the square root of the sheet’s area — so that patterns built on sheets of different sizes can be compared. It is not a property of the folded state at all. It is what a folder has to do.

The ranking, and the ranking it replaces #

By shrinkage alone the order is: Yoshimura at 32.0 times, waterbomb at 31.8, leaf corrugation at 8.3, Miura at 8.1, preliminary base at 8.0, square twist at 3.0.

By shrinkage per sheet-width of creasing the order changes completely: Yoshimura 2.71, waterbomb 2.22, preliminary base 1.65, Miura 1.31, leaf corrugation 0.93, square twist 0.64.

Two things move. The preliminary base rises from fifth to third — it folds eight times smaller for under five sheet-widths of creasing, which is a better rate than the Miura’s and far better than the twist’s. And the square twist falls to last: it costs about as much creasing as the preliminary base and delivers three times rather than eight.

The preliminary base’s position needs a qualification and it is an important one. It is a base, not a corrugation: it folds one square of paper into a smaller stack and it does not cover an area. A deployable needs a pattern that can be made large and still folds, and the bases do not scale — a preliminary base of a square metre folds a square metre into an eighth of one, and that is all it ever does. The corrugations scale, and scaling is the entire point of a corrugation.

Which is the honest reading of the table: the rate column ranks the patterns and the scaling decides which of them are candidates at all.

Why bigger patterns look better #

The rate improves as a corrugation grows, and that is a fact about the arithmetic rather than about the patterns.

Shrinkage rises roughly with the number of cells, because each cell contributes a layer to the stack. Crease length rises roughly with the linear size of the pattern times the cell count — that is, with the total length of the grid lines. So the ratio of the two climbs as the grid is refined, and a fine enough corrugation looks arbitrarily good.

It does not stay good, and the reason is not in any of these figures.

A real crease has a radius and consumes a fixed length of sheet whatever the pattern says. Refining the grid multiplies the number of creases, so the paper lost to the creases themselves grows while the cells shrink, and past some grid size the pattern will not close. Every number in this essay is computed on a sheet of zero thickness whose creases are lines, and that idealisation is exactly the one that decides how fine a corrugation is worth making.

The rate grows linearly, which makes the ranking size-dependent #

The climb can be written down, and writing it down changes how the table should be read.

At $n$ divisions across a fixed sheet the cells number $n^2$ and each contributes to the stack, so the shrinkage goes as $n^2$. The creasing is the grid’s total line length, which on a fixed sheet goes as $n$. So

$$\text{rate} \;\propto\; \frac{n^2}{n} \;=\; n.$$

The Yoshimura at four across bears it out: eleven sheet-widths of creasing, thirty-two-fold shrinkage, a rate of 2.71 against the $n/2$ the count alone predicts.

So the rate column is not a property of a pattern. It is a property of a pattern at a size, and every corrugation in the table climbs the same ladder at the same slope.

Which is what the base’s third place actually means #

That settles the qualification the essay makes about the preliminary base, and settles it more sharply than “it does not scale”.

A base has one size. Its rate of 1.65 is the only rate it has, at every square it is ever folded from. The corrugations’ rates rise with $n$ without limit until the paper stops them, so any refinement whatever puts every corrugation above every base, and the crossing has already happened for two of them at four divisions.

The preliminary base is therefore not third. It is third at $n = 4$, second at $n = 3$, and last at any grid a deployable would actually use — and the same is true of every comparison in the column that crosses the two families.

The paper’s own limit is where the climb stops, and the area version of it is generous: the creases consume the sheet when their total length times a crease radius reaches the sheet’s area, which for 0.1 mm creases on a 150 mm square is some hundreds of divisions. Every other limit in this subject binds first — the stack’s depth against the cell’s width, the hand’s accuracy, the layers’ thickness — which is why no corrugation is folded anywhere near the rate this arithmetic permits.

The stack, which is the other half of the price #

Shrinkage and depth are the same number, so a pattern that packs thirty-two times smaller is thirty-two layers deep. That is not a neutral fact.

The distinction between the average and the maximum sorts the patterns into two families without being asked to.

The corrugations put nearly all of their footprint at nearly the deepest count. The Yoshimura at four cells across averages 32.0 layers with a maximum of 36; the waterbomb averages 31.6 with a maximum of 32. Their whole purpose is to bring every cell to the same place, and they do.

The Miura does not. At four by four it averages 8.1 layers with a maximum of 16 — the average is half the maximum, because the folded Miura is not a single tidy stack but a footprint with deep and shallow regions.

For hardware the maximum is what matters and the average is what gets quoted. A stowed array has to fit through an aperture, and the aperture has to admit the thickest part.

The winner of the rate column is worth looking at, because of where it came from.

The pattern that comes last, and why it is still used #

The square twist finishes last on both columns — three times smaller for 4.7 sheet-widths of creasing, a rate of 0.64 — and it is nonetheless one of the most folded patterns in the subject. Explaining that is a good test of whether the table is being read correctly.

The twist is not a packing device. What it does is rotate a small polygon while the sheet closes around it, and the interest is in the rotation and in what it does to the surrounding pleats. As a way of making a sheet smaller it is poor; as a unit that can be tiled into patterns with an enormous variety of appearance it is the most productive object in the field.

So the rate column measures a price and does not measure what was bought. A pattern folded for its look, or for a mechanical property, or because it tiles interestingly, is not being bought for its footprint at all, and a table that ranks it last on footprint per unit of work has said nothing about whether it was worth folding.

The same caution applies at the other end. The Yoshimura tops the rate column and is the pattern nobody chooses for a deployable, because it arrived by accident — it is the buckling mode of a crushed cylinder, and its excellence at packing is a consequence of being what a cylinder does when it collapses rather than of anybody optimising it. It is superb at making a tube smaller and has almost no other properties to recommend it.

There are properties a corrugation is chosen for that no packing measurement reaches.

Is the rate a rate at all #

There is a real objection to the last column and it is worth stating rather than hiding, because it limits what the column can be used for.

Shrinkage is not linear in anything. Adding a row of cells to a corrugation multiplies the layer count rather than adding to it, while adding a row of cells adds a fixed amount of crease length. So a ratio of the two is not a constant of the pattern; it is a number that grows as the pattern does, and comparing two patterns at different grid sizes compares their sizes as much as their designs.

Every pattern in the main table is therefore built on the same grid, four cells across, and the second table repeats the comparison at five by four to show what moving the grid does. The corrugations improve and the bases do not, because the bases have no grid to enlarge.

What the column can be used for is a comparison at a fixed size, which is the comparison a designer actually faces: given a sheet and a target, which pattern gets there for the least creasing. What it cannot be used for is a claim that one pattern is intrinsically more efficient than another, because that claim would need a limit and the limit is where the paper runs out rather than where the arithmetic stops.

What a partly folded state does instead #

Everything above is about the fully folded state, and a deployable spends almost none of its life there.

The shrinkage quoted here is a limit: the value the ratio reaches when every crease is fully folded. A partly folded Miura is partly packed, and the relationship between the fold angle and the footprint is the pattern’s kinematics rather than its flat state.

That matters because the two questions have different answers. A pattern with an excellent flat packing ratio may reach it only at the very end of its motion, so a mechanism that stops ninety per cent of the way is nowhere near ninety per cent of the ratio. And a rigidly foldable pattern has to reach its state through a continuous motion, which is a stronger requirement than the state existing.

The engineering figures for the same quantity are worth putting beside the computed ones.

Which theorem was checked, and how #

The measurement is a sample and the check is exact.

Each folded state is verified three ways before anything is measured: every panel reachable by two routes is placed in the same spot both times, which is Kawasaki arriving from a direction the construction does not compute; every panel keeps its area exactly, which is what “the paper does not stretch” means; and the sampled layer count integrates back to the sheet’s area.

That last one is the check on the sampling itself, and the figure publishes it. The final column of the table is footprint times average depth divided by the sheet, and it comes out at 1.000 or within a thousandth of it for every pattern. A grid too coarse to resolve the footprint would show up there immediately, and the generator refuses to draw if any pattern misses by more than five per cent.

The crease lengths are exact and need no check: a crease pattern’s total creased length is a sum over its edges.

The first thing a real material does to the table is to take the thickness column seriously: a panel with a depth and no accommodation at its creases does not fold at all, so every shrinkage figure in this essay is a statement about a surface rather than about anything anybody could build out of aluminium.

What the table does not rank #

Four things a deployable is chosen on that no column here measures.

Degrees of freedom. The Miura moves in exactly one way and the waterbomb does not, and a mechanism with one freedom needs one thing to pull it. That is a decisive engineering property and it is invisible in a packing ratio.

Rigid foldability. Every number above is about paper, which bends. Whether the same pattern works with rigid panels is a stronger and separate question.

Thickness. Every panel here is a surface, and getting real thickness round a corner is the central problem of turning any of this into hardware.

Layer order. The folded states computed here place the panels and do not order them. Two patterns with identical footprints and identical depths can have entirely different orderings, and deciding whether an ordering exists is the intractable part.

Who measures it this way, and when #

Packing ratio as an engineering quantity is old and arrived from hardware: a stowed array has a volume, a deployed one has an area, and the ratio is what a launch is priced on. It was being measured with rulers long before anybody computed one from a crease pattern.

What computation adds is not accuracy — a folded array can be measured — but the ability to ask the question of a pattern nobody has built, which is the only way to compare a hundred candidates. That capability arrived with the interchange formats and the folded-state algorithms of the last two decades.

The crease-length column is this repository’s own addition to the comparison and it is the one that reflects the practical tradition rather than the engineering one. A folder choosing between two tessellations is not choosing on packing ratio; they are choosing on how long it takes, and the length of crease is that in one number.

It is a fair proxy for effort and not a perfect one, and the ways it is imperfect are worth naming. Two creases of the same length are not the same work if one runs edge to edge on flat paper and the other has to be made through eight accumulated layers. A crease made against a straightedge is quicker than one made by aligning two points. And a pattern whose creases can be made in long uninterrupted passes — the Yoshimura’s courses, the Miura’s rows — is much faster than one of the same total length made in short segments, which is part of why the waterbomb feels harder than its number suggests.

A better measure would weight each crease by the depth of the stack it has to pass through, which is a quantity the folded state already knows. That is a natural extension and it is not made here, because it would need a model of how a folder sequences the work and there is no such model in this repository. The unweighted length is what can be computed from the pattern alone, and saying which of the two is being reported is more useful than quietly reporting the easier one.

What would have to change to rank them properly #

A ranking that a deployable could actually be chosen from would need three more columns, and it is worth saying what they are so that this table is not mistaken for one.

Degrees of freedom, because a mechanism with one input is a different engineering proposition from one with many, and the difference is decisive rather than marginal.

Rigid foldability, because everything here is measured on paper, which bends, and a pattern that only reaches its folded state by letting panels flex is not a candidate for panelled hardware at all.

Thickness accommodation, because a stack of thirty-two layers of anything real is a solid object, and how the pattern responds to being given depth decides whether the packing ratio survives contact with a material.

None of the three is measurable from a flat folded state, which is why none of them is here. What this table is for is the narrower question it answers exactly: given a sheet and a pattern, how much smaller does it get, how deep does it become, and how much creasing did it take. Those three are computed rather than estimated, and they are the floor a comparison starts from.

Where the ladder goes next #

The natural continuation is downward, into what the idealisations were hiding: the crease that has a radius sets the finest useful grid, and the sheet that has a thickness sets what a stack can do once it is not paper.

The other direction is the patterns themselves, where the waterbomb’s price — the highest crease length in the table — is explained by what its geometry has to do.