The paper is all still there

Folding makes things smaller. That is most of why anybody does it to a solar array, an airbag or a map, and it is the one property of a crease pattern that a person who has never met a crease pattern immediately wants a number for.

The number is available, and it comes with an obligation attached.

What is being measured #

A flat folded state is a map from the sheet to the plane that is an isometry on every panel. Nothing stretches; each panel arrives the same size and shape it left, somewhere else and possibly turned over.

Two quantities follow immediately and neither needs anything else.

The footprint is the region of the plane the folded object covers — the union of all the panels’ new positions. It is what the object measures on a table.

The layer count at a point of the footprint is how many panels lie over it. It is one near the edges of most folded objects and considerably more in the middle, and it is what a folder feels in their fingers when a stack becomes too thick to crease.

The obligation is that these two are not independent. Integrate the layer count over the footprint and the answer is the area of the sheet, because every scrap of paper is under exactly as many points as it covers. So

footprint × average layers = area of the sheet

exactly, for every flat folded state of every pattern. Rearranged, the factor by which a pattern shrinks is its average layer count. Not proportional to it, not roughly it — the same number.

Where the panels go #

The construction that produces the folded state is one line long and the line is worth reading slowly.

Folding flat about a crease is reflection in that crease. So a panel on the far side of a crease carries the near panel’s motion followed by that reflection, and starting from any one panel a breadth-first walk over the panel adjacency places every other panel in the pattern. Each motion is a composition of reflections in the sheet’s own coordinates: exact, with no angles solved for and nothing iterated.

One detail of the construction decides everything that follows. The walk visits each panel once, along a spanning tree of the panel adjacency, and a spanning tree has no cycles in it — so the walk can never contradict itself and never notices whether the pattern was foldable. It would place the panels of a pattern that cannot fold at all, cheerfully and wrongly, and report nothing.

What catches that is the creases the tree did not use. Every shared crease outside the tree is a second route to a panel that already has a position, and the two routes agree only if the reflections around each interior vertex compose to the identity.

That condition is Kawasaki’s. Going once around a vertex, the product of the reflections in its creases is a rotation through twice the alternating sum of its sectors, and it is the identity exactly when that alternating sum is zero.

So a construction that computes no angles rediscovers the angle condition. On the Miura fold above there are six panels reachable two ways and the two routes agree to within three parts in a quadrillion; on a five-by-four Miura it is twelve panels; on the waterbomb tessellation twenty-five. On a corrugation tapered the wrong way, which fails Kawasaki by six and a half degrees at every interior vertex, the routes disagree immediately and the folded state cannot be built at all.

Which theorem was checked, and how #

Three claims, each checked against something the construction was not given.

Consistency, which is Kawasaki arriving from a direction nothing here computes. Reported as the largest disagreement between two routes to any panel; anything above a part in a billion refuses the pattern.

Isometry, which is what “the paper does not stretch” means and is the only thing separating a fold from a squash. Every panel’s folded area is compared with its flat area, and the two must be equal — not to a tolerance chosen for convenience, but to the last bit, since a composition of reflections is exact arithmetic.

Conservation, which is the claim this essay is about. The layer count is sampled on a grid over the folded footprint, the samples are added up, and the total has to come back as the sheet’s area. It is the one check with real error in it, because it is a sample rather than an overlay, and the assertion allows two per cent of slack and typically sees a tenth of that.

The three are deliberately different in kind. The first is exact and structural, the second is exact and geometric, the third is numerical and could be wrong by an amount that shrinks as the grid refines. A single check that passed would prove much less than three checks that could each fail for different reasons.

The census #

Five patterns, all folded the same way and measured the same way.

The numbers separate the patterns more sharply than their pictures do. A square twist folds three times smaller and is nine layers at its deepest. A four-by-three Miura folds six times smaller. The preliminary base — one vertex, eight creases — folds eight times smaller, and it is worth noticing that a base with a single interior vertex beats a Miura fold with six.

The Yoshimura pattern and the waterbomb tessellation both fold about thirty-two times smaller at the sizes drawn here, and both are thirty-two-odd layers deep, and that is the same fact said twice.

The obvious consequence is the one a designer should carry: a pattern that packs better is thicker, and there is no pattern that is not. Any drawing that claims a large packing ratio has committed to a stack, whether or not its author mentions one.

It is worth saying what this does not rule out, because the law is easy to over-read. It does not say that all patterns are equally good. Two patterns that shrink by the same factor can differ in how much crease length they need to do it, in whether they fold rigidly or only flat, in how many independent motions they have, and in whether the stack is even or peaked. The conservation statement fixes exactly one relationship and leaves every other question open — which is what a conservation law usually does, and why it is a starting point for a comparison rather than the end of one.

It also says nothing about how the object gets there. A folded state existing and a motion reaching it are different questions, and the numbers here are all about the destination.

The other end of the range is worth seeing beside it.

Folding conserves volume exactly #

The law has a consequence that the phrase folding makes things smaller actively hides, and it falls out by multiplying both sides by a thickness.

Give the sheet a thickness $t$. Its volume is its area times $t$. The folded object’s bounding volume is its footprint times its stack depth times $t$ — and the law says footprint times depth is the sheet’s area.

$$\text{folded volume} = \text{footprint} \times \text{depth} \times t = \text{sheet area} \times t = \text{sheet volume}$$

So folding conserves volume exactly, for every pattern, at every packing ratio. A sheet that folds thirty-two times smaller in footprint is thirty-two times thicker, and the product is what it always was.

There is no packing ratio in volume. There never was one, and no pattern anybody could design would produce one.

Which says what a deployable is actually buying #

That reframes every number in the census, and it reframes them in the direction that matches what deployables are for.

A ten-metre square membrane a tenth of a millimetre thick has a volume of ten litres. Folded to a metre square it is ten millimetres thick and its volume is ten litres. Nothing has been saved.

What has changed is the shape. A launch fairing is a box a couple of metres across, and a ten-metre sheet does not fit in it at any thickness while a one-metre stack does. A bud is a cylinder a few millimetres wide; a leaf’s blade is not, and a corrugated bundle is.

So the whole value of folding is converting a wide thin object into a narrow thick one at constant volume, and the packing ratio is a measure of how far that conversion has been taken rather than of anything being made smaller.

That also explains why the thickness corrections bite so hard rather than being a refinement. In the ideal model the volume is conserved with nothing to spare; a real stack has gaps, crease radii and layers that will not compress, so its volume is strictly larger than the sheet’s. Every departure from the idealisation makes the folded object bigger, and there is no term anywhere that could make it smaller.

A deployable’s designer is therefore not fighting for volume, which is fixed. They are fighting to keep the real object’s volume close to the sheet’s, and every layer added to the stack is another place to lose that fight.

The average is not the experience #

The law is about an average and the paper is not average anywhere. A folded object is thin at its edges and thickest wherever the most panels have arrived, and the difference between those two is what decides whether a pattern can be folded at all rather than merely described.

Two patterns can shrink by the same factor and be quite different to fold. One that puts a uniform stack over its whole footprint is easier: the layers accumulate evenly, and the last creases are as hard as the first. One with a sharp peak has a region where the paper runs out early, and that region is where a real attempt stops.

The profiles measured here divide the patterns into two families without being asked to. The corrugations — Miura, Yoshimura, waterbomb — put nearly all of their footprint at nearly the deepest count, because a corrugation’s whole purpose is to bring every cell to the same place. The bases do not: the preliminary base has a thin outer region at fewer layers and a core at the full eight, which is why its outer flaps stay workable long after its centre has become a lump.

The deepest count is also the one number in all of this that a folder can predict without any computation. It is how many cells the pattern has, when the pattern is a corrugation that closes completely — twelve for a four-by-three Miura, twenty-seven for that Yoshimura, thirty-two for a four-by-four waterbomb. The average is slightly less because the margins are shallower, and the gap between the deepest and the average is a measure of how much of the folded object is edge.

The measurements above are all of the completely folded state, and that is a limit rather than a working condition.

What the model does not answer #

The construction places every panel. It does not say which panel lies above which, and that omission is not an oversight — it is the hard half of the subject.

Layer ordering is a separate object with its own rules, the non-crossing conditions are statements about a cross-section rather than about a vertex, and deciding whether a consistent order exists for a general pattern is the NP-hard part of flat-foldability. A folded state as computed here is a set of positions, and every valid ordering of those positions is a different physical object with the same footprint and the same layer count.

So the conservation law is blind to the thing that actually stops a pattern folding, and says so. It is a statement about where the paper is, not about whether the paper can get there.

Two further idealisations, both standing ones on this site. The paper has no thickness, so a thirty-two-layer stack is drawn as thirty-two lines with nothing between them. And a crease is a line, so no paper is consumed turning the corner.

Both are false, and the second is the one that bites first.

A real stack of thirty-two layers of ordinary copier paper is about three millimetres thick, and the outermost layers of the stack have to travel around all of that at every crease. The geometry says the footprint is one cell; the paper says it is one cell plus several millimetres of accumulated arc. How much a crease costs is a separate essay and the answer is a fraction of a millimetre per fold, which is negligible once and ruinous three hundred times.

The layer count is also where the idealisation is most expensive, and the cheapest demonstration of that is the oldest one.

The same law from the design side #

The conservation statement has a second reading that has nothing to do with corrugations.

Anything a design wants to show has to come out of the sheet, and it comes out of the sheet at the rate the law dictates. The most compact case is a single fold: turning a flap of the paper back over itself puts the reverse side on show, and the arithmetic is immediate.

Widen the flap by a hundredth of the sheet and the reverse side gains a hundredth while the front loses two. That is the conservation law applied to a two-panel pattern, and what it means for a design is that visible reverse is the most expensive material in origami.

The same reading explains why circle packing has the shape it does. A flap of a given length claims every point of the sheet within that distance of it, which is an area; the areas cannot overlap; and the whole design problem is the accounting. Nothing about that is a coincidence of two subjects — it is one conservation statement showing up in the two places a designer meets it.

And where the stack stops being an abstraction is where hardware begins.

Who worked it out, and when #

The conservation statement itself is arithmetic and belongs to nobody. What is recent is the ability to compute the folded state rather than draw it.

The interchange format that makes it routine dates from the 2010s, and its arrival changed what a crease pattern is: not a picture with a caption but a graph with coordinates and assignments, which a program can fold. Before it, the folded state of a pattern was something a person produced by folding, and comparing two patterns’ packing meant folding both.

The measurement here is on the easy side of that boundary and it is worth saying which side that is. Placing the panels is a spanning-tree walk and takes no search at all. Ordering them is the hard problem, and everything this essay reports is from the easy half — which is exactly why the numbers can be quoted for five patterns at once without any of them having been folded.

The packing ratio as an engineering quantity is older than any of that and arrived from the other end. A stowed solar array has a volume and a deployed area, the ratio between them is what a launch is priced on, and it was being measured on hardware decades before anybody computed one from a crease pattern. What the computation adds is not accuracy — a folded array can be measured with a ruler — but the ability to ask the question of a pattern that has never been built, which is the only way to compare a hundred candidates.

The two halves also disagree about what the number means, in a way worth keeping in view. To an engineer the packing ratio is a volume ratio and includes the thickness; to the geometry it is an area ratio and the thickness is exactly what has been idealised away. The geometric number is an upper bound on the engineering one and can be a generous bound: where the panels have somewhere to go is a subject in its own right precisely because real hardware never reaches the ratio its pattern promises.

Where the ladder goes next #

The immediate continuation is the design one: a colour change costs twice what it shows, and that is this law with two panels in it.

The engineering continuation is what a corrugation costs, where the same measurement is made across every tessellation this repository can fold and the result is a table a deployable can be chosen from.

And the sharp limit is the layer ordering, which the construction here deliberately does not attempt. Everything above describes a folded state completely except for the one question that decides whether it exists.