The density a paper allows

How much line is on the paper measures a pattern’s crease density and the five of these essays after it read that measurement in different ways: where the length sits, what the shortest crease is, what survives a cut, what a band with no vertices carries, and what scale any of it is in. Every one of them is a quotient the pattern hands over.

None of them asks what the paper’s answer is, and the paper has one.

A crease is not a line. It is a rounded region a few sheet thicknesses across, and two creases laid closer together than that are not two creases — they are one wider fold, or a tear. So a sheet of a given thickness carries a largest density, and it is one division.

The ceiling, and why it is a division #

If a crease occupies a band $w$ wide, then a square metre of paper laid out in parallel creases at the closest spacing carries $1/w$ metres of crease, and no arrangement carries more. A two-directional pattern at that spacing is not denser — it is the same total length divided between two families at twice the spacing each.

So the ceiling is the reciprocal of the crease’s width, and the crease’s width is a few sheet thicknesses.

The six is stated rather than derived, and the essay says so wherever it is used. A crease’s rounded region is a few thicknesses across and nobody here has measured how many; what the figure is for is where the shelf sits against the ceiling, and that is robust to any factor between three and ten.

And the shelf is nowhere near it #

The printed patterns run from eleven metres of crease a square metre to eighty-nine.

The densest of them is a factor of nineteen below the ceiling of the thickest paper. On washi it is a factor of forty-seven. Nothing anybody prints comes anywhere near the material.

And the shelf’s own spread is small beside that gap. From eleven to eighty-nine is a factor of eight across eight patterns of five different kinds, and the whole of that range sits inside one twentieth of what the thickest usable paper allows. The patterns differ from each other by far less than any of them differs from the paper, which is the shape of a measurement whose limiting case is somewhere else entirely.

That is a deflationary result and it is worth having for exactly that reason. A quantity called crease density invites the reading that it is bounded by the paper, and it is — at a value that has nothing to do with anything on the shelf.

What a factor of nineteen means for the shelf #

It is worth converting the ceiling into a spacing, because a spacing is the thing a folder can look at.

The waterbomb at 89 metres a square metre is, laid out as parallel lines, one crease every 11 millimetres. Copier paper’s ceiling is one crease every 0.60 millimetres. So the densest pattern on the shelf has its creases eighteen crease-widths apart on average, and the sparsest has them a hundred and fifty apart.

That is a picture of how empty a crease pattern is. A sheet carrying the busiest pattern this site prints is, by area, more than ninety-four per cent unfolded paper — the creases themselves occupy under six per cent of it even counting their full rounded width.

Which is why nothing in this subject is ever described as running out of room, and it is worth stating because the language of the field suggests otherwise. A tessellation is called dense, a grid is called fine, and a pattern that cannot be folded is said to be too complicated. None of those is a statement about the paper being full.

Where the material would actually bind #

The way to find where it does bind is to make the pattern finer until it does, and the ceiling is far past what can be built: a Miura reaching it has tens of thousands of cells. So the answer is measured over a range and extrapolated.

Measured at four, eight, sixteen and twenty-four cells across, the density is 12.09 metres a square metre per cell, less 15.2 — a straight line to within 0.7 per cent, on a sheet a hundred and seventy millimetres across. The intercept is negative because a patch’s outermost creases are its boundary and are not counted.

Solve for each paper’s ceiling and the fineness comes out:

A hundred and thirty-nine cells across on copier paper, which is cells 1.22 millimetres wide. On kami it is a hundred and ninety-eight, at 0.86 millimetres. On washi three hundred and forty-five, at 0.49 millimetres. On foil-backed tissue five hundred and thirty-one, at 0.32.

The waterbomb family reaches the same ceilings at a coarser cell count, because it lays more crease into each cell — which is the difference between 89 and 36 metres a square metre on the shelf, read as a fineness rather than as a density. A denser family meets the material sooner, which is what “denser” means once there is a ceiling to meet.

Those are not fineness limits anybody has ever met, because they are below what a hand can place. The finest divisions a folder can lay in are about a millimetre, which is the figure the substrate’s grid ceiling is computed with, and every one of the numbers above is at or below it.

Why it took an error to look #

The ceiling is one division and nobody in seven of these essays performed it, which deserves a sentence because the reason is instructive rather than embarrassing.

A quotient of two measured quantities feels like a complete answer. Length over area is a density, a density is a number, and a number invites comparison with other numbers of the same kind — which is exactly what six of these essays did, comparing patterns against patterns. Nothing in the practice of making that comparison suggests asking what the largest possible value is, because the patterns are the subject and the paper is the background.

What made it happen was an arithmetic that had gone wrong. The Miura’s density was reported at 231 metres a square metre, and a pattern with thirty-eight creases on a sheet the size of a postcard cannot carry a crease every four millimetres — but the absurdity is only visible against a scale, and the scale is this ceiling. Computing the bound is what showed the measurement was broken, and the broken measurement is why anybody computed the bound.

That is a cheaper check than it sounds and it generalises. A quantity with a physical unit has a physical maximum, the maximum is usually one division, and a measurement sitting near it or above it is a measurement to distrust.

So the hand binds, and on every paper #

Put the two limits side by side on a hundred-and-seventy-millimetre sheet.

The hand can place a division no finer than about a millimetre, so a Miura on that sheet stops at about a hundred and seventy cells whatever the paper is.

The paper stops at a hundred and thirty-nine cells on copier, a hundred and ninety-eight on kami, three hundred and forty-five on washi and five hundred and thirty-one on foil-backed tissue. Those four numbers are the thickness ratio and nothing else: the ceiling is the reciprocal of the thickness, the fineness is linear in the ceiling, so a paper half as thick carries a pattern twice as fine, exactly.

So the paper binds on the thickest sheet in the list and the hand binds on the other three — and on the thickest, the two are within twenty per cent of each other. The crossing between the two is inside the range of papers a folder actually uses, which is the interesting thing here and is not what the shelf suggested: the shelf is a factor of nineteen below the ceiling, and the extrapolation says the ceiling is nonetheless reachable at a fineness the hand nearly reaches.

That is a different statement from either “the paper limits the density” or “the paper never limits it”. It is: the two limits are the same size, and neither of them is anywhere near the patterns anybody prints.

The mean is not the number that meets the ceiling #

A ceiling on a density is met by a maximum and the shelf’s densities are means, so the comparison above is generous to the pattern.

Where the length sits measures exactly this: divide a pattern into bands by distance from its own edge and report the length per unit of paper in each. A traditional base carries five times its share in the middle four per cent of the sheet; a tessellation carries between 0.9 and 1.4 everywhere.

So the peak local density of a pattern is between 1.4 and 5 times its mean, depending on kind. Apply the worst of those to the densest pattern and the waterbomb’s peak is under 130 metres a square metre against copier’s 1,667 — still a factor of thirteen. The correction moves the numbers and not the conclusion, which is what a factor of nineteen has room for.

Where it would matter is at the fineness the extrapolation reaches. A Miura at a hundred and thirty cells is uniform, so its peak is its mean and the ceiling is met everywhere at once; a base at that fineness would meet the ceiling in its middle four per cent long before its mean got close. A pattern that concentrates its folding reaches the material limit earlier than its density says, and the concentration is a property these essays already measures.

The two kinds of pattern, measured against the material #

Splitting the shelf by how concentrated its folding is puts the two kinds of pattern in different relations to the ceiling, and the difference is larger than the density column suggests.

A tessellation is uniform. The Miura, the waterbomb, the Yoshimura and the corrugations carry between 0.9 and 1.4 times their share of the folding in every band, so their peak is within forty per cent of their mean. At the fineness the extrapolation reaches, such a pattern meets the material limit everywhere at once, and the density column is the right number to compare.

A base is not. The preliminary base carries five times its share in the middle four per cent of the sheet, because its four creases all pass through the centre. Its mean density is 32 metres a square metre and the middle of it is at 160, which is five times nearer the ceiling than the mean says — and a base made five times finer would meet the material in its centre with the rest of the sheet almost empty.

So the honest statement about the material limit has two forms. For a repeating pattern it is a bound on the whole sheet and is met all at once. For a pattern with a hub it is a bound on the hub, and the rest of the design has nothing to do with it. The band profile is the measurement that tells them apart, and it was made two of these essays before anybody had a ceiling to compare it to.

What the ceiling is not #

It is not a fold-count limit. The ceiling is on length per area and says nothing about how many creases that length is divided into. A pattern with a thousand short creases and one with ten long ones can have the same density, and the material has the same opinion of both.

It is not about stacking. Once folded, layers pile up and a stack has its own limit — eighty layers of the thinnest paper that holds a crease — and that is a bound on the folded object where this is a bound on the flat one. A pattern can be well inside the density ceiling and impossible to fold because its stack is too thick.

It is not a bound on a folded pattern. Once a sheet has creases in it, a new crease crossing them is being laid into paper that is no longer flat, and the width the new crease needs is set by what it is crossing rather than by the sheet’s thickness alone. That is a much tighter constraint and it is not this one; what this bounds is the drawing.

And it is not a bound anybody has hit, which is the finding. The nearest thing to a reported instance is a pattern so fine that its creases interfere during folding, and interference during folding is a different phenomenon from two creases overlapping on the flat sheet — the first is about the order of operations and the second is about geometry.

Nor is the crease’s width measured here. Six thicknesses is a stated figure, the whole result is reported as a bracket around it, and the crease’s radius is the nearest thing here to a measurement of it — a geometry rather than a material property.

One number from two subjects #

The ceiling is a material fact multiplied by a geometric one, and both halves come from elsewhere in this collection.

The material half is the crease’s width, which is a few sheet thicknesses and is the subject of the idealisation this field is built on. The geometric half is that parallel lines at spacing $w$ give $1/w$ of length per unit area, which is true of any family of lines on any surface and has nothing to do with paper.

So the ceiling is one line of geometry taking one number from a material, which is why it costs a division and why nobody had to measure anything new to get it. Every quantity of that shape is available for the price of noticing it is available.

Still open: what does bind #

The result is that neither the paper nor the shelf is where the limit on crease density lives, which leaves the question of where it does.

The candidate is interference during folding. The first of these essays ends by saying that laying the last crease into a sheet that already carries all the others is not the same operation as laying the first, and that nothing measures that interference. It is still nothing. What would measure it is a count of how many already-folded creases a new one has to cross, as a function of density — a quantity computable from the pattern alone and never computed.

And the concentration deserves the same treatment as the mean. A pattern’s peak band density is measured and its mean is measured and nobody has asked what the maximum over any disc of a crease’s width is, which is the quantity a material ceiling is actually met by. That is a local maximum over a sliding window rather than over concentric bands, it is one pass over the pattern, and it would say which of the printed patterns is nearest the material anywhere rather than on average.

Sideways from here, the same division bounds something the site measures often and never bounds. The shortest crease on a printed patch is shorter than a wavelength of light, which is a statement about a length and not about a spacing — and the same crease-width argument says a crease shorter than a crease is wide is not a crease either. The width bounds the length as well as the spacing, and applying it to that essay’s distribution would say how many of a pattern’s creases are creases rather than how many are short.

The habit worth carrying is about limits nobody has computed. Compute the physical bound on a quantity under measurement, even when it is expected to be far away — it took one division to find that the shelf sits a factor of nineteen inside it, and the same division found that the shelf’s own arithmetic was out by a factor of six, because a number nineteen times too small is easy to overlook and a number above a ceiling is not.