The crease has a radius

Three of the four idealisations get regular attention. Zero thickness has an essay of its own and a whole engineering literature; inextensibility is worked around deliberately; perfect memory is obviously false to anybody who has unfolded something. The fourth is quieter, and it is the one that decides whether a large grid comes out square.

What a crease is made of #

Fold a sheet and look at the fold end-on. The paper does not turn a corner. It runs straight, curves through an arc, and runs straight again, and the arc has a radius $\rho$ that depends on the paper’s thickness, its fibre, how much moisture is in it and how hard it was pressed.

For ordinary kami $\rho$ is around a tenth of a millimetre. For heavy watercolour paper it is closer to half a millimetre; for tissue foil it is small enough to be hard to measure.

A fold through $180°$ carries the paper round a half-circle. The arc length is

$$\pi\rho,$$

and the two arms of the fold end up separated by

$$2\rho.$$

The ideal fold, through a line, uses no paper at all and separates the arms by nothing. So the real fold consumes $\pi\rho$ of the sheet’s length and returns $2\rho$ of stack width, and the difference

$$(\pi - 2)\rho \approx 1.14\rho$$

is length that has disappeared into the crease.

One crease is nothing #

For a tenth-of-a-millimetre radius that is about $0.11$ mm per crease. On a single fold it is invisible; it is a tenth of the width of the crease itself.

The interest is entirely in the accumulation, and the accumulation is worse than it first looks because it happens along every line of a grid rather than once.

A $32 \times 32$ box-pleated grid has thirty-one creases in each direction. Along any line across the sheet, thirty-one creases each swallow $1.14\rho$, so the sheet is short by about $3.5$ mm in a $150$ mm sheet — a little over two per cent. In both directions at once.

Two per cent is not a rounding error in a design whose flaps are sized to the millimetre. It is the reason a folder’s grid does not line up at the far corner, and the reason a complex model’s last few collapses have to be forced.

The grid at which the loss is a whole cell #

There is a single number that turns all of this into a design limit, and it comes out of setting the accumulated shortfall against the thing it has to be measured in.

A grid of $g$ divisions on a sheet of side $L$ has $g-1$ creases along any line, each losing $1.14\rho$, so the shortfall is about $1.14\rho g$. A cell is $L/g$ across. Setting the two equal:

$$g \approx \sqrt{\frac{L}{1.14\,\rho}}$$

At that grid the far corner is a whole cell out of place, which is not a tolerance to work around but a pattern that has stopped meaning what it says.

For ordinary kami at a tenth of a millimetre on a 150 mm sheet, that grid is thirty-six. For a paper twice as thick it is twenty-six. For tissue foil at five hundredths it is fifty-one.

Which is exactly where box pleating sits #

Thirty-two is the grid the essay’s own example uses and the grid a great deal of complex work is drawn on, and thirty-six is where the sheet gives up.

So a folder working at thirty-two on kami is at ninety per cent of the paper’s limit, with the far corner already nine tenths of a cell adrift — which is the practical meaning of “the grid does not line up at the corner” and why the last collapses have to be forced.

At sixty-four the shortfall is 7.2 mm against a cell of 2.3 mm: three cells out. The pattern cannot be folded on that paper at that size at all, by any amount of care.

And it prices the thin-paper move exactly #

The limit goes as one over the square root of the radius, so the exchange rate between paper and resolution is fixed:

Halving the paper’s thickness multiplies the usable grid by $\sqrt2$. To go from a thirty-two grid to a sixty-four grid takes paper four times thinner, not twice.

That is the whole of the tissue-foil argument in one line, and it explains why the progression in complex folding has been so steep on the materials side. Each doubling of the grid — which is one more level of detail — costs a factor of four in thickness, and there are not many factors of four available between kami and the thinnest laminate anybody can fold.

It also says the alternative. $g$ goes as the square root of $L$ too, so a folder can buy the same factor of $\sqrt2$ by doubling the sheet, which is why competition pieces are folded from squares a metre across.

Why thin paper, not strong paper #

This explains a preference that looks like snobbery from outside.

A complex design puts great demands on the paper: it is folded and unfolded repeatedly, layers are pulled through one another, and thin paper tears. The obvious response is to use stronger, and therefore thicker, paper. Folders do the opposite, and go thinner and thinner as designs get more complex — tissue foil, glassine, laminates of tissue and foil a few hundredths of a millimetre thick.

The accounting above is why. Strength scales with thickness; the crease loss scales with thickness too, and it scales multiplied by the number of creases, which for a complex design is in the hundreds. A design with four hundred creases along its longest line loses $456\rho$, which for a paper twice as thick is twice as much. Past a certain complexity, the paper that is strong enough to survive the folding is too thick for the design to close.

The resolution is a laminate: tissue for strength, foil for hold, and a total thickness lower than either would give alone. That is a materials answer to a geometry problem, and the geometry is the equation above.

Where the loss actually goes #

It is worth being precise about what “the sheet is short” means, because there are two different effects and only one of them is the crease radius.

The first is the one computed above: paper consumed by the arcs. It is a property of a single crease and it does not depend on how many layers are underneath.

The second is that a crease made through a stack of $n$ layers has a much larger radius than one made through a single sheet, because the outer layers have to go round everything inside them. The radius is roughly $n$ times the sheet thickness, and the loss for that crease is $1.14 n t$ rather than $1.14 t$. In a box-pleated base the creases near the centre pass through dozens of layers, and their losses dwarf everything at the edges.

So the estimate above is a lower bound, computed as though every crease were made in a single sheet. The real loss is larger, unevenly distributed, and concentrated exactly where the design is tightest.

Measuring it on a real sheet #

The quantity is not hard to get at, and it is worth describing how because the method is the sort of thing that makes an idealisation concrete.

Fold a strip in half, then in half again, and again, $n$ times, and measure the folded stack’s length. In the ideal the length is the original divided by $2^n$. In practice it is shorter, by the accumulated crease loss, and dividing the discrepancy by the number of creases gives $1.14\rho$ directly.

The measurement is dominated by the last fold’s creases, since those pass through the most layers, so the honest procedure is to do it for several $n$ and fit — which recovers a per-sheet radius and a per-layer term separately.

A folder does something like this by feel constantly, which is what “the paper does not want to do that” means. The value of writing it down is that the feel does not transfer between papers, and the number does.

Which theorem was checked, and how #

There is no theorem here; there is a piece of geometry and an arithmetic consequence, and the figure checks that its own arithmetic agrees with itself.

The arc length is computed as $\pi\rho$ from the half-circle and separately as the difference between the consumed and advanced lengths, and the generator throws if the two disagree by more than a part in $10^{12}$. That is a check on the code rather than on the world, which is the right kind of check for a figure whose content is an identity.

The figure also refuses a crease radius of zero, which is the idealisation rather than a crease, and refuses a grid list whose loss does not grow with the grid — since growing with the grid is the whole point.

What is not checked is the value of $\rho$, and it cannot be: it is a measured property of a particular paper folded by a particular person, and the figure takes it as a parameter with a defensible default rather than pretending to a number.

What it means for a tessellation #

Tessellations are where this bites hardest, and for a reason that is separate from the count of creases.

A twist tessellation is folded by collapsing a pre-creased grid, and the collapse only works if the grid lines meet where the pattern says they do. An error of two per cent at the far corner of a thirty-two grid is larger than the twist itself at the scale most people work at, and the collapse simply does not close.

The response in that community is thin paper, and beyond thin paper, wet-folding the grid so the creases relax and the sheet can be coaxed into agreement — which trades one idealisation for another, since a wetted sheet stretches.

That is a nice illustration of how the four idealisations interact rather than sitting in separate boxes. The crease radius is a consequence of thickness; the fix for the crease radius is to abandon inextensibility; and the cost of abandoning inextensibility is that the finished piece holds a shape it was not designed to hold.

What the picture cannot show #

The cross-section shows a crease as a circular arc joining two straight arms. Real creases are not circular. The paper’s fibres break unevenly, the inner surface is compressed while the outer is stretched, and the resulting profile is closer to an elastica than a circle — a curve with continuously varying curvature.

Using a circle overestimates the loss slightly for a sharp crease and underestimates it for a soft one. The order of magnitude is right and the second decimal place is not, which is the appropriate ambition for a quantity that depends on how hard somebody pressed.

The figure also cannot show what the loss does to a shape. Two per cent short is not the same as two per cent smaller: the loss is concentrated at the creases, so a folded base is short in the directions that have creases and correct in the directions that do not, and the distortion is anisotropic.

The idealisation underneath #

This essay is about an idealisation, so the interesting question is what idealisations it makes in turn, and it makes two.

Paper is treated as having a single characteristic crease radius. It does not: the radius depends on the fold’s direction relative to the grain, and a sheet folded along the grain creases more sharply than one folded across it. A grid therefore loses different amounts in its two directions, which turns an isotropic shortfall into a slightly rectangular one — and folders orient their paper for exactly this reason.

And the loss is treated as though it happened at the moment of folding and then stopped. Paper relaxes: a crease made and left overnight opens slightly, its radius grows, and the loss grows with it. Perfect memory is the fourth idealisation, and it interacts with this one — a model that fitted when it was folded does not necessarily fit a week later.

Why the design software says nothing about it #

Every piece of origami design software works in the idealisation, and it is worth asking whether that is a shortcoming or a choice.

It is mostly a choice, and a defensible one. The crease loss depends on a material property the software cannot know, on the folding sequence — which creases are made through how many layers, and in what order — and on the folder. A tool that guessed at all three would produce a compensated pattern that was wrong in a new and less predictable way.

What a tool could usefully do is report the number the correction depends on — the maximum layer count along each crease, which it knows exactly and which the folder can multiply by their own paper’s thickness. That number is available in every design program and displayed by none of them, which is a small missed opportunity rather than a scandal.

The design method is in this sense complete and incompletely delivered: it produces a correct pattern for a sheet that does not exist, and hands the translation to a person.

The surprising connection #

The same quantity, under a different name, decides whether a piece of sheet metal can be bent.

A bend in metal has a bend allowance: the length of material consumed by the bend, computed from the bend radius, the included angle and a factor accounting for where in the thickness the material neither stretches nor compresses. Every sheet-metal drawing carries these numbers, and a flat pattern that ignores them produces a part that is the wrong size.

The equation is the same equation. What differs is that the metalworkers have had it standardised for a century, with tables of the neutral-axis factor for each alloy and radius, while folders have it as tacit knowledge — “use thinner paper for a bigger grid” — and no tables at all.

That is a genuine gap rather than a curiosity. A folder designing a $64 \times 64$ grid is doing a bend-allowance calculation by feel, and the reason it is done by feel is that nobody has published the tables.

The other direction: when the radius is wanted #

The loss is a cost when the aim is a shape the pattern predicted. There is a whole practice in which it is the point.

A soft, large-radius crease is what gives wet-folded work its character: the fold is a gentle bend rather than a line, the surface between folds curves, and the finished piece reads as sculpture rather than as geometry. Folders working this way choose thick paper because its crease radius is large, which is the exact inverse of the tessellation folder’s reasoning.

The same is true in hardware. A living hinge in a moulded part is a deliberate large-radius crease, sized so the material’s strain stays below yield as it bends — too sharp and it cracks, too soft and it will not hold a position. The radius is a design variable with a specified value.

So the fourth idealisation is not simply false; it is a modelling choice that is right in one regime and wrong in another, and the regimes are separated by what the work is for. That is worth saying because the other three idealisations are not like this. Zero thickness is never wanted. Inextensibility is occasionally abandoned on purpose and never wanted for its own sake. A sharp crease is genuinely undesirable half the time.

Who noticed, and when #

The idealisation has been named in the mathematical literature since flat-foldability was formalised — every statement of the theory says “zero-thickness” somewhere near the top — and the crease radius specifically appears in the thick-panel engineering work of the 2000s, where it becomes a hinge design parameter rather than a nuisance.

Among folders it is older and unattributed. The advice to use thinner paper for more complex work is as old as complex work, which is to say the 1970s and the tsujiura generation, and the reasoning above is a reconstruction of what that advice is tracking rather than a citation of anybody’s argument.

The sheet-metal version, by contrast, has an author and a date for every table.

The ladder from here #

This rung prices one idealisation. Above it sits the question of what to do about it: a designer can compensate by scaling the pattern slightly, and the compensation is not uniform because the loss is not, so the corrected pattern is a distorted one — which is a real technique in box pleating and which nobody has written down properly.

Below it, four things that are not true names all four idealisations at once, and paper that stretches on purpose takes a different one and makes it useful instead of costly.