The paper that will not hold a crease

Three of these essays are about thinning. The paper had to arrive first computes what a stack of sixty-four layers is in ordinary copier paper and finds six and a half millimetres of it. A sheet has a size as well finds the finished size shrinking as the square root of the layer count. Which ceiling is binding puts the two together and finds a crossing that moves with the thickness.

Every one of those is a bound that improves as the paper gets thinner, and the essays here therefore reads as a story with a direction in it: better paper is thinner paper, and a tradition’s reach follows its mills.

There is a bound that does not improve, and it stops the story.

What a crease is made of #

A crease is not a line and it is not a zero-radius bend either. It is a region of the sheet a millimetre or so across in which the paper has been bent past what it can recover from, and the recovering is the point: a fold that springs back is not a crease.

What does not recover is the fibre structure. A sheet of paper is a mat of cellulose fibres, each of them twenty or thirty microns across and a millimetre or two long, felted together and held by hydrogen bonds where they touch. Bending the sheet sharply stretches the fibres on the outside of the bend and compresses the ones on the inside; past a certain curvature the bonds between them break and the fibres slip, and the slipped arrangement is the new shape. The crease is the damage.

That gives the thickness a job. The stretching and the compressing happen because the outside of the bend is further from the neutral surface than the inside, and how far depends on the thickness. A sheet with through-thickness structure — fibres above other fibres — has an outside and an inside to damage. A sheet one fibre thick has neither: every fibre is the whole thickness, there is nothing for it to slip against, and bending it bends a mat of loose fibres that springs back when released.

So there is a floor, and it is at some small number of fibre diameters.

Where each paper sits #

The fibre is about twenty-five microns across. Divide.

The four papers a folder creases unbacked run from four fibres down to 1.6, and stop there. Washi at forty microns is the thinnest paper in general use for complex folding, and it is a fibre and a half thick. Below it the two entries in this collection’s shelf are both tissues, and one of them is in the list as foil-backed — which is to say the tradition’s answer to a paper too thin to hold a crease is to laminate it to something that will.

That laminate is the finding read from the other side. Foil holds a crease because a metal foil is a plastic hinge by nature, and backing a tissue with it puts the hinge in the metal and lets the paper supply only the surface. A technique that exists to solve a problem is evidence the problem is real, and this one exists.

What the stack ceiling was promising #

It is worth writing out what the stack argument says about the papers below the floor, because the numbers are large and they are the reason the floor matters.

A stack becomes unworkable when it approaches the smallest feature a folder is working to — three millimetres is the working figure these essays uses. Copier paper at a hundred microns reaches thirty layers before that; washi at forty reaches seventy-five; unryu tissue at eighteen reaches a hundred and sixty-six.

A hundred and sixty-six layers is a different subject. It is more than twice what the most elaborate models in the tradition reach, and if it were available the whole of the first of these essays’ argument — that the complex tradition is downstream of a manufacturing achievement — would have a further chapter waiting in the mills. The thickness arithmetic says it is there.

It is not there, and nothing in the thickness arithmetic says why. A ceiling that rises is only useful while the floor stays below it, and this is the first thing in the essays here that looks at the floor at all.

The ceiling that stops being reachable #

Set the floor beside the ceilings and these essays’ arithmetic comes out differently.

Of forty-two combinations of thickness and sheet size, twenty-seven are stopped by the stack, fourteen by the crease, and one by the hand’s precision — where without the crease floor, six would have been.

That is the result. Which ceiling is binding computes the crossing between the stack and the grid, and concludes that on classical papers the substrate is the limit and that “only at tissue weights does the hand take over”. The tissue weights are exactly where the crease floor sits. The regime the third of these essays predicted is almost entirely inside the regime where the paper will not hold a crease at all, and the one cell of forty-two that survives is the corner of the map: a sheet smaller than a postcard, on a paper right at the floor.

So the third of these essays’ conclusion needs a qualification rather than a correction. Its arithmetic is right; the regime it identifies is real in the arithmetic and is almost empty in paper.

The escape the tradition found #

Foil backing deserves more than a mention, because it is an engineering answer to exactly this and it changes what the arithmetic is about.

A foil-backed tissue is a sheet of aluminium a few microns thick with tissue laminated to one or both faces. The foil is the hinge: a metal bent past yield stays bent, with no fibres and no bonds in the argument at all. The tissue is there to give the surface a colour and a texture and to stop the foil looking like foil.

So the laminate has the thinness of a tissue and the crease of a metal, which is exactly the combination the floor forbids in a single material — and the entry in this collection’s shelf sits at twenty-six microns and a fibre thick. Something a fibre thick that holds a crease is not a counterexample to the argument; it is a composite built to evade it.

What it costs is everything else about paper. A foil-backed sheet does not spring, does not soften when damp, does not take a wet fold, and creases permanently on the first accident — so a model in it is a model that cannot be adjusted. The paper is all still there is about what a folded sheet conserves, and a laminate conserves it differently: the two layers have different neutral surfaces and the crease radius is set by the foil rather than by the stack.

Two properties, one thickness, opposite directions #

This is the second time these essays have found a design bounded by two properties of one material, and the shape is different from the first time.

A sheet has a size as well reports thickness and size bounding different things, both improving with a different property, and crossing. Here the same property — thickness — bounds two things in opposite directions: thinner is better for the stack and worse for the crease. There is no crossing to compute because they do not both improve; there is an optimum, and the optimum is the floor.

A design wanting the most layers should use the thinnest paper that still holds a crease, which is the floor exactly, and nothing below it is available at any price. On the numbers here that is somewhere near forty microns for an unbacked paper — which is washi, and which is what the tradition uses.

That is the closest these essays comes to explaining a choice rather than describing one. Washi is not the thinnest paper a mill can make; it is not the thinnest paper in this collection’s shelf; and it is the thinnest one that folds. A material chosen at the boundary of two opposing constraints looks, from the inside, like a tradition’s taste.

The floor is not a folder’s problem alone #

The same argument bounds anything that folds a thin sheet, and the two places it shows up outside the tradition are worth naming because they are where the numbers are checked.

A deployable structure folds a membrane and wants it thin, for the reason every essay here gives: a thinner sheet packs more layers into the same volume. What it does not want is a permanent crease — a deployable is supposed to open again — so the fold is a hinge rather than a crease, and the material is chosen to avoid the yielding this essay is about. The fold count sets the spring is the same trade in a structure that has to recover.

And a folded metal is the opposite case. Foil has no fibres and no floor of this kind; what limits a metal foil is tearing rather than springing back, and the failure arrives from above rather than from below. So the two materials a folder might use have their limits at opposite ends of the same axis, and a laminate of the two has neither.

That is the strongest evidence available here that the floor is a property of fibres rather than of thin sheets in general. If a sheet a micron thick could not hold a crease because it was thin, foil would not hold one either, and it does.

A floor is not a ceiling turned over #

The three bounds in these essays are not three of the same thing, and the difference decides what each one is evidence about.

The stack and the sheet are ceilings on a design: given a material, they say how elaborate a model may be, and a folder meets them by being ambitious. The crease floor is a bound on the material: it says which papers exist to be used at all, and nobody meets it by folding anything.

So the first two are about what a tradition made and the third is about what a tradition could buy — which is why the third is the one that explains a choice. A ceiling says what was out of reach; a floor says what the range of available materials actually was, and a material chosen at the bottom of that range looks deliberate only once the bottom is known.

What the fibre count is and is not #

The twenty-five microns is stated, not measured here. It is a representative width for a cellulose fibre and the real figure varies by plant and by processing — a softwood fibre is thicker than a hardwood one, and the bast fibres of the mulberry used for washi are thinner and much longer. Every count in the table would move with it, and the ordering would not.

The floor at one and a half fibres is a reading rather than a derivation. Nothing here computes at what thickness a crease stops holding; what is computed is how many fibre diameters each paper is, and the observation is that the papers a tradition folds unbacked stop at 1.6 and the ones below are backed. That is evidence about where the floor is and it is not a measurement of it.

And nothing here models a fibre. No modulus, no bond strength, no slip criterion. The crease has a radius is the nearest this collection comes to the mechanics and it is a geometry rather than a material argument. The claim made here is structural: a plastic hinge needs something to be plastic, and a single layer of fibres has no through-thickness structure to be plastic with.

The shelf is six papers. They are the ones measured here and they were chosen to span the range rather than to be a survey. A seventh paper at thirty microns that folded unbacked would move the floor and would not remove it.

A fibre is not a cube. It is long and thin, so a sheet a fibre thick is nothing like a sheet one grain thick in a granular material — the fibres overlap along their length even when they do not stack, and a tissue holds together for that reason. The argument is about what happens through the sheet, which is the one direction the overlapping does not help in.

Reading the essays here backwards #

With the floor in place these essays’ four of these essays make a different shape, and it is worth stating because the first of these essays’ headline survives it and its mechanism does not.

The headline is that the complex tradition is downstream of a manufacturing achievement. That stands: a design of sixty-four layers is unfoldable in a paper of a hundred microns and foldable in one of forty, and the difference is what a mill did.

The mechanism was that thinner is better without limit, and it is not. The achievement was not thinning the paper as far as possible; it was thinning it to about forty microns and stopping, which the floor says is where it had to stop and which reads, without the floor, as a mill that could have done better and did not.

The census at a narrower fibre is the check on that. Mulberry bast, which washi is made from, has thinner and much longer fibres than wood pulp — so a washi sheet of forty microns is more fibres thick than the table’s twenty-five-micron division says, and the floor for that plant is at a thinner sheet than for another. A tradition’s floor is a property of its plant, which is a testable statement about why one paper-making tradition reached thinner sheets than another and is not a statement about skill.

Still open: where the floor actually is #

The floor is bracketed by a shelf of six papers and it deserves a measurement.

The experiment is simple and nobody in this collection can run it. Take a sequence of papers of the same fibre and the same finish at descending thicknesses, crease each one, release it, and measure the angle it holds. The curve of held angle against thickness would have a knee, and the knee is the floor. What would make it a result rather than an observation is doing it in fibre diameters rather than in microns — the same sequence in a long-fibre washi and a short-fibre wood pulp, to see whether the knee lands at the same number of fibres or at the same thickness. If it is fibres, the floor is a property of the plant; if it is microns, it is a property of something else and the argument here is wrong.

And the foil laminate has an arithmetic of its own. A backed sheet is two materials, its thickness is the sum, and its crease is held by the one that yields — so the design question is the thinnest foil that holds a crease plus the thinnest paper that carries a surface, which is a two-variable minimisation nobody has written down. The interesting part is that it has no floor of the same kind: a metal foil has no fibres, so whatever limits it is not this, and the laminate may not have a floor at all until it meets something else entirely.

Sideways from here, the floor puts a number on the largest model anybody can fold. Take the thinnest unbacked paper that holds a crease, take the stack ceiling it allows, and take the sheet two arms can make: the three together bound the layer count and the finished size at once, and the bound is a single region rather than three separate ceilings.

The floor is what fixes the horizontal line, and it is the only one of the three bounds that no manufacturing improvement moves. The sheet can be made larger by two people or by joining, and the stack ceiling follows whatever paper is chosen; the crease floor is a property of the plant the fibre came from.

The habit worth carrying is about improvements with a direction. When every constraint in an argument improves with the same change, look for the one that does not — an argument all of whose terms point one way is usually an argument that has left a term out, and the missing one is often the thing that stops the improvement being free.