The shortest crease is not a crease

How much line is on the paper is the natural first measurement of how demanding a crease pattern is: add up every crease’s length, divide by the area of the sheet, and compare. It is a real quantity, it separates a Miura from a fold-and-cut pattern immediately, and it is what a manufacturer wants to know about a corrugation.

It is also a total, and a total discards the distribution it came from. Two patterns with the same crease density can be built of ten long creases or a thousand short ones, and a hand asked to fold them is not facing the same task at all.

What the distribution says #

On the hexagonal tessellation patch the lengths run from 7.9 × 10⁻⁶ of the sheet’s side to 0.198 — a range of four and a half orders of magnitude on a pattern that looks completely uniform.

The lengths are not spread through that range. They form two groups with an enormous vacancy between them: twelve creases below 6 × 10⁻⁵ and a hundred and thirty above 2.9 × 10⁻², with nothing at all in between. The gap is a factor of five hundred, and no crease anywhere in a sweep of a hundred and twenty patches falls inside it.

A total cannot show that. The twelve short creases contribute about six millionths of the patch’s crease length, so the density is unchanged to five decimal places whether they are present or absent. The quantity most likely to be quoted about the pattern is exactly the quantity that is blind to the most peculiar thing about it.

There is a second thing a total hides, and it is the more common one. Crease density is a ratio, so it is unchanged by scaling the sheet — a Miura at fifteen centimetres and the same Miura at a metre have the same density, and one of them can be folded by hand and one of them cannot. Any statement about how demanding a pattern is has to name a physical size somewhere, and the density does not have a place to put one.

A floor set by the hand, not by a theorem #

Every theorem in this subject is scale-free. Kawasaki reads angles; Maekawa counts letters; the big-little-big lemma compares sectors. None of them contains a length, so none of them has any opinion about a crease being short.

That is right for the mathematics and wrong for paper. A crease has a physical floor, and the floor has several components, none of them from geometry:

The fold itself has a radius. Paper does not turn a corner — it bends through a small arc whose radius depends on the stock, typically a tenth of a millimetre for office paper and rather more for anything heavier. A crease shorter than a few times that radius is not a fold, it is a dimple.

The printing has a resolution. A laser printer’s finest addressable dot is around twenty micrometres, so a crease shorter than that has no representation on the sheet at all.

The hand has a resolution too, and it is the coarsest of the three. Aligning two points to better than half a millimetre by eye is difficult, and a crease shorter than the error in placing its endpoints is a crease whose direction is not determined by anything the folder can do.

The halving limit is the same observation applied to repeated folding, and it lands in the same place: the sheet stops cooperating long before the geometry does.

Twelve creases below every floor #

On a sheet fifteen centimetres square — the size the collection prints its patterns at — the twelve shortest creases of the hexagonal patch are between 1.2 and 9 micrometres.

That is below all three floors by a wide margin. The shortest is a sixteenth of a printer’s finest addressable dot, about an eightieth of a fold’s own radius, and four hundred times smaller than the half-millimetre a hand can place.

And they are in every count the collection takes of that patch — its crease count, its interior vertex count, its arc count — because a crease list has no length threshold in it and never has. What they are and where they come from is a separate matter; what they are here is a demonstration that a pattern’s crease lengths are worth reading rather than summing.

It is worth noticing which of the three floors is binding, because it is not the one a mathematician would guess. The printer and the fold radius are both around a tenth of a millimetre; the hand is around half. So the binding constraint on how short a crease may usefully be is the accuracy with which a person can place its endpoints, and that is a fact about people rather than about paper or about geometry.

That has a consequence for machine folding that is easy to state and hard to exploit. A machine placing creases to ten micrometres moves the floor by a factor of fifty, which makes patterns available that no hand could execute — and none of the theorems change, because none of them ever mentioned a length. The mathematics of folding is entirely indifferent to the scale at which folding is done, and the whole of the difference between hand and machine work is in floors like these.

What size the patch would have to be #

The floors are lengths and the crease is a ratio, so the two can be divided, and the quotient is the sheet size at which the pattern stops being a drawing.

The shortest crease is $7.9 \times 10^{-6}$ of the sheet’s side. Setting that equal to each floor in turn gives the side required:

So the patch is printable at two and a half metres, physically creasable at thirteen, and foldable by a person at sixty-three metres square — which is not a size of paper, it is a size of building.

That is the useful form of the finding, and it is a different statement from the crease is very short. A ratio invites the reply that the sheet could always be made bigger; the table answers it. The pattern is not marginal at any scale anybody works at, and the factor between the printed size and the hand’s requirement is four hundred and twenty.

Which is what a shortest-crease check is really reporting #

It also gives the check its natural units. Dividing the floor by the shortest ratio turns a pass-or-fail into a number with a meaning: the smallest sheet on which this pattern is a set of folds rather than a picture of some.

For the ten patches in the sweep that number runs above a metre. For the other hundred and ten it is a few centimetres, which is to say no constraint at all — the same 15 cm sheet the collection already prints on.

There is nothing in between, for the same reason the length distribution has nothing in between. A clipped pattern either has a fragment or it does not, and the quantity is a threshold artefact rather than a continuous property of the drawing.

How common that is #

Ten patches of a hundred and twenty carry a crease shorter than a thousandth of a sheet.

That rate is not a property of one careless construction. It is what happens whenever an infinite pattern is fitted into a finite region: somewhere in such a construction is a test of whether a piece is inside the region, the test is a boolean derived from a continuous quantity, and any piece sitting close to the threshold produces something very small and perfectly legal.

So a length distribution is worth reading on any clipped pattern, and reading it costs a single sweep over the crease list. The shortest crease of a patch is a one-number summary that catches every case above, and it is not a number anybody was in the habit of looking at.

The vacancy is the argument #

Before any of that, the shape of the distribution deserves one more sentence, because it is what turns an observation into a measurement.

A threshold on a continuous quantity is an opinion. Somebody decides that a crease under a thousandth of a sheet does not count, somebody else prefers a ten-thousandth, and there is no fact to settle it — which is why quantities like this usually generate argument rather than results.

Here there is nothing to argue about. Every threshold anywhere between 6 × 10⁻⁵ and 2.9 × 10⁻² catches exactly the same twelve creases, so the number chosen carries no information and cannot be wrong. A vacancy of that width in a measured distribution is the strongest evidence available that two different things are being measured, and it arrived without anybody looking for it.

What a better summary would be #

If a total is the wrong one-number summary, the question is what a right one would be, and the honest answer is that no single number will do — but three of them together do most of the work.

The total stays, because it is what a manufacturer needs: how much scoring, how much material fatigue, how much time on a machine that moves at a fixed speed.

The shortest crease is the feasibility check, and it costs one pass over the crease list. Below a stated floor the pattern cannot be executed at the stated size, whatever its total says, and every one of the ten patches in the sweep would have been caught by it.

The ratio of longest to shortest describes how uniform the demand is. A pattern whose creases are all within a factor of ten of each other can be folded at a steady rhythm; one spanning four orders of magnitude cannot, and the difficulty is concentrated wherever the small creases are.

None of the three is new and none of them is difficult. What is new is reading them together, and the reason nobody had is that the total answers the question anybody thought to ask.

Where the length actually sits #

The other end of the distribution is worth a moment too, because it is where the manufacturing cost is.

Where the length sits established that a corrugation’s crease length is dominated by its longest creases — the pleats that run the width of the sheet — and that the count of creases and the total length behave differently as a pattern is refined. Halving the pitch of a Miura doubles the number of creases and roughly doubles the total length; halving the pitch of a twist tessellation quadruples the count and does something else again, because the polygons’ perimeters and the pleats scale differently.

Reading both ends together gives a summary a total does not: the longest crease says what the pattern costs to fold, the shortest says whether it can be folded at all, and the ratio between them says how uniform the demand is. On a Miura that ratio is under ten. On the hexagonal patch it is twenty-five thousand.

What a folder actually notices #

The gap between the three floors and the mathematics is not abstract, and a folder meets it in an ordinary way.

Given a pattern with creases of very unequal lengths, the short ones are folded last and badly. A long crease can be made by aligning two edges and running a fingernail; a short one has to be placed relative to creases already made, and every error in those accumulates into it. So the difficulty of a short crease is not its length — it is that it inherits the errors of everything that positioned it, which is how error propagates through a construction in a form a hand can feel.

The twelve creases on the printed patch escape all of that by being unfoldable. A folder does not make them, notices nothing, and produces the correct folded object, because a fold of zero length is not a fold anybody performs and the two panels it separates are one piece of paper in the hand.

Which theorem was checked, and how #

Nothing here is a theorem about folding, and it is worth saying so plainly: crease lengths are measurements taken off the drawn pattern, and the only checked claim is the gap.

That claim is asserted rather than described. On the printed hexagonal patch, exactly twelve creases fall below a thousandth of the sheet, the shortest crease above them is more than a hundred times the longest crease below them, and both halves have to hold. A construction change that closed the gap would make the threshold a judgement rather than a reading, and the assertion is there so that it says so.

The patch itself is checked by the usual three instruments — the four conditions at every interior vertex, the panels’ closure computed by composing reflections, and a lettering searched for and verified — and it passes all of them with the twelve in place. Nothing about being unfoldably short makes a crease illegal.

What the picture cannot show #

The short creases. That is unavoidable and it is the point: any figure that could draw one would be a figure at a magnification where nothing else fits on the page. The length plot shows them as positions on an axis, which is a picture of the measurement rather than of the ink.

Nor does a length distribution capture the whole of what a hand faces. Two creases of the same length differ in difficulty according to what they have to be aligned with, how many layers are already stacked where they fall, and whether they are made early or late in a sequence. A sheet has a thickness and the fold has a radius, and a crease length is one input to it rather than a summary of it.

And the floors quoted are for office paper and a domestic printer. A different stock, a scoring machine or a laser cutter moves all three, and a pattern that is unfoldable at fifteen centimetres may be perfectly reasonable at a metre — which is the ordinary reason a manufactured corrugation and a folded one are different objects.

The measurement this suggests everywhere #

The general form is worth stating because it applies well outside crease patterns and it is cheap.

Any quantity summarised as a total over many parts is worth plotting as a distribution once, at the start, and the reason is not thoroughness. It is that a total is a sum, and a sum is dominated by its largest terms while feasibility is decided by its smallest — so the summary statistic and the binding constraint are at opposite ends of the same data, and a total is systematically blind to the end that decides whether the thing can be done.

Here the two ends are five orders of magnitude apart and the blindness is total: the twelve creases that make the pattern peculiar contribute six millionths of the number that describes it. That is an extreme case and it is not a rare shape, and one plot at the beginning would have found it.

What this does not fix #

The three-number summary above catches a crease that is too short. It does not catch a crease that is too short for where it is, and that is a different and harder quantity.

A crease three millimetres long in the open middle of a sheet is comfortable. The same crease made through eleven accumulated layers, at a place where two pleats already cross, is not — the paper is thick there, the fold radius is larger, and the alignment marks it would be placed against are themselves folded. Nothing in a length distribution knows about layers, and how deep the folded state gets at its worst point is a separate measurement that would have to be joined to this one.

Joining them is straightforward and has not been done. What it would give is a per-crease difficulty rather than a per-pattern one — for each crease, its length against the thickness of paper it has to turn — and that is much closer to what a folder actually experiences than any total.

Where the ladder goes next #

A length distribution is one of several readings that a total conceals, and the same shape recurs wherever a pattern is summarised by a single number. The nearest neighbour is what happens to a crumple as it deepens, where the summary quantity is a search cost rather than a length and the conclusion is the same: the aggregate moves smoothly and the thing underneath it does not.