One cut short of falling apart

The site’s two essays on cutting have both been about what a cut buys. A cut changes the angle at a point where a fold cannot, and that is the whole of what makes kirigami a different subject; a slit array reaches a Poisson’s ratio of exactly −1 where a folded sheet only approaches one, and the price is that half of it is hole.

There is a third quantity, and it is the one a person holding the sheet is aware of. Everything the cut sheet does is transmitted through the material still joining one side of a cut line to the other, and there is not very much of it.

The ligament, and where it is #

Cuts of length ℓ sit on a lattice of pitch one, with every other row shifted by half a period. Between the end of one cut and the start of the next, along the same line, there is a strip of uncut sheet: the ligament, of width exactly one minus ℓ.

That is all there is holding the two sides of a cut line together. A sheet six rows deep has five internal cut lines, and each of them is bridged by a row of ligaments and by nothing else. Cut to 85% of the pitch and each ligament is 0.15 of a pitch wide; cut to 97% and it is 0.03; cut to 99.9% and it is a thousandth.

At each of those the sheet is one piece.

A count is not a length #

The reason that sentence is worth writing down is that two quantities are being confused whenever a cut sheet is described as getting weaker.

The ligament is a length and it falls linearly: 0.60, 0.40, 0.25, 0.15, 0.08, 0.03, 0.00 across the seven cut lengths in the figure. Nothing about it is a threshold; it is a straight line to zero.

The piece count is an integer and it is 1, 1, 1, 1, 1, 1, and then 6. Nothing about it is gradual.

Both descriptions are correct and they are about different questions. How much load the sheet can carry is about the ligament and falls smoothly. Whether the sheet is one object is about the count and does not move at all. A sentence like “the sheet gets progressively weaker until it falls apart” runs the two together, and the second half of it is false: the sheet does not gradually fall apart, it is one thing and then it is six.

The mechanism model has no ligament at all #

Here the two models the site now carries have to be separated, because they are models of different things and only one of them can be asked this question.

The rotating-squares array of the rung below is a mechanism: squares of side a on a lattice, neighbours turning opposite ways, joined where two corners coincide. Requiring the corners to coincide is one equation and it fixes the pitch at a(cos θ + sin θ). That model gives the −1 exactly and it gives it for a reason — the same function of θ describes both directions — and it contains no material whatever. Its hinges are points.

A real cut sheet is the other model. Its hinges are ligaments with a width, and the width is what the tiles rotate about. Everything the mechanism does idealises that width away, which is the correct thing to do when the question is the motion and the wrong thing when the question is the sheet.

The relationship between them is the ordinary one between a mechanism and the thing it is a mechanism of. As the ligament narrows, the real sheet’s motion approaches the mechanism’s; the mechanism is the limit, and the limit is unreachable, since at zero ligament there is no sheet.

The count, worked out rather than measured #

Counting the pieces is done two ways here on purpose, and the second way is the interesting one.

The direct route is an argument about intervals. Along each cut line, merge the cut intervals and ask whether anything is left; the sheet is severed there exactly when nothing is. The number of pieces is one plus the number of lines cut the whole way across. That is exact, it is cheap, and it is what the figures use.

The other route is a flood fill: lay a raster over the sheet, join neighbouring cells when the segment between their centres does not cross a cut, and count the regions. That is the dumbest possible method and it exists to be a second opinion.

The two agree — when the raster can see the ligament. Run the flood fill on a sheet cut to 90% with cells wider than the tenth of a pitch that is left, and it confidently reports the sheet in pieces. Nothing is wrong with the sheet; the measurement cannot resolve what holds it together, so it reports a gap where there is material.

That failure is deliberately provoked and asserted rather than avoided, because it is the same failure as measuring a hinge with a ruler too coarse for it, and it is exactly what a photograph of a cut sheet does to a reader. A measurement that cannot see a ligament reports a sheet that has fallen apart.

What the ligaments cost #

The ligament’s width is the sheet’s whole connection to itself, and there is a second number worth having: how much of the cut line it accounts for.

At 85% of the pitch, each ligament is 0.15 of a pitch, and a six-column sheet has seven of them per cut line, so 1.05 of sheet width bridges a cut line that is six wide — about a sixth. At 97% it is 0.21 of six, about a thirtieth. The material joining the halves of a cut sheet is a small fraction of the sheet even when the cuts are modest, and it is a small fraction concentrated in a few narrow places.

That concentration is the practical fact. A cut sheet does not distribute its work; it puts all of it into a small number of narrow bridges, which is why real kirigami sheets fail at the ends of the cuts and why cut patterns intended to be used are drawn with rounded cut ends rather than square ones. Rounding does nothing geometric — the ligament is the same width — and it changes where the material tears, which is a question about stress and belongs to structural-engineering-statics.com.

The same arithmetic on a different cut #

The slit array is one cut pattern and the step is not special to it — but the covering rule is, and it is worth saying exactly how far it reaches before it is carried anywhere else.

For cuts lying on a family of parallel lines that span the sheet, separation happens exactly when one line’s intervals cover it. That is the argument above and it is correct for this pattern. It is sufficient and not necessary in general: four cuts arranged around a square region separate that region from the rest without covering any single line, and so does any closed loop of cuts. A cut pattern separates the sheet when its cuts contain a curve that either closes on itself or runs from one edge of the paper to another, and a covered line is only the simplest way to have one.

So the honest general statement is topological rather than arithmetic, and it is the same statement a hole makes about a sheet: what matters is whether the cuts enclose anything, not how long any of them is.

What changes between patterns is how many lines there are and how much slack each carries. The offset array has the least slack of any pattern that opens: its cuts are as long as they can be while every line keeps a ligament at every period, which is what makes it the standard shape rather than an arbitrary one.

The one straight cut theorem is the extreme case of the same arithmetic, and it goes the other way: there the cut is meant to separate, the whole line is cut through, and the interesting question is what shape the separated pieces are. A kirigami sheet is the same object with the covering condition deliberately failed at every line.

Which is why the dumbest method is the general one #

That correction earns the flood fill its place, and it changes what the second opinion is for.

The interval argument is fast and exact and it is a shortcut that knows about this pattern: it assumes the cuts lie on parallel lines spanning the sheet, and under that assumption a covered line is the only way to separate anything. The flood fill assumes nothing. It joins neighbouring cells wherever the segment between them crosses no cut, and it finds a separated region however that region came to be enclosed — by a covered line, by a loop of cuts, by four short cuts meeting at their ends.

So the two are not two implementations of one test. One is the general question and the other is a special case of it, and they agree on the slit array because the slit array is the special case. Hand both of them a pattern with a closed loop of cuts in it and only one returns the right answer.

That is the reverse of how the pair reads at first. The flood fill is described above as the dumbest possible method, kept as a check on the clever one — and it is the clever one that carries an assumption, which the dumb one does not. Where a cheap method and an expensive one disagree, the usual diagnosis is that the expensive one is wrong; here the cheap method is wrong exactly where it stops applying, and the expensive one is the definition.

And what the resolution failure really shows #

That also puts the flood fill’s own failure in its proper place. Run on a raster coarser than the ligament, it reports a sheet in pieces — and the essay reads that as a measurement failing to see what holds the sheet together, which it is.

What it is not is evidence against the method. A raster fine enough to resolve the narrowest ligament gives the right answer on every pattern, including the ones the interval shortcut cannot express; the requirement is a resolution rather than an assumption, and a resolution can be met by counting cells while an assumption can only be met by being true.

So the pair has the shape a checking arrangement ought to have. The general method needs one number chosen well — a cell size below the narrowest ligament — and is otherwise unconditional. The fast method needs no numbers and one structural fact about the pattern. Where both apply they agree, and where they disagree the disagreement identifies which condition has failed: a coarse raster, or a cut pattern that is no longer a set of intervals on parallel lines.

Where the model stops #

Nothing here is a claim about strength. The ligament is a length and every number above is geometric. How much load a ligament carries, when it yields and how a sheet fails are mechanics; what a sheet’s material does is a thing this site names and does not derive.

The cuts are straight, equal and periodic. A real cut pattern has a purpose, and cut patterns designed to produce a particular shape have cuts of varying lengths in varying directions. The step behaviour survives — a sheet is one piece until some line is cut through — and the location of the step does not: with unequal cuts it depends on which line runs out first.

The sheet has no thickness and the cuts have no width. A knife removes material. A cut of finite width shortens the ligament at both ends by half the blade, so a real sheet reaches the step earlier than the geometry says, and by an amount that depends on the tool rather than on the pattern.

This is a cut and the site’s rule is one sheet, no cuts. The rule bends here as it has bent twice before, and the terms are the same: what is claimed is local — the material at one place along one line — and nothing above is an argument about a region, a total or a surface. The oldest book cuts the paper and one straight cut both say why the rule is worth having and why it is not absolute.

What a folded sheet does instead #

The comparison that makes the threshold legible is with the folded sheet that reaches the same behaviour without cutting.

A Miura-folded sheet has a negative Poisson’s ratio too, and it gets there by a completely different route: the panels stay whole and the creases do the accommodating. There is nothing in it that narrows toward disconnection, because a crease is a line the sheet bends along rather than a place material has been removed. A folded sheet can be worked as hard as its material allows and it is still one sheet.

That is the sharpest way to say what a cut costs. It is not the holes, which the previous rung priced at exactly half the sheet, and it is not the strength, which is somebody else’s subject. It is that a cut sheet has a threshold and a folded sheet has none — one of them can stop being a sheet and the other cannot.

Why the threshold is where the interest is #

There is a reason cut-sheet designs sit close to the step rather than comfortably away from it.

The auxetic behaviour — the tiles rotating, the sheet opening in both directions at once — needs the tiles to be nearly free to turn, and they are free in proportion to how little material is joining them. A generous ligament is a stiff hinge, and a sheet with stiff hinges does not open; it just bends. So every property the cut sheet is used for improves as the ligament narrows, and every one of them is bounded by the fact that at zero ligament the sheet is confetti.

That is an unusual shape of design problem and it is worth naming. Most structures are designed away from their failure mode with a margin. A cut sheet is designed toward its failure mode, because the failure mode and the function are the same geometry: what makes it fall apart is exactly what makes it work. The ligament is not a safety margin that happens to be small; it is the design variable, and the whole of the sheet’s behaviour is a function of how close to nothing it has been taken.

Who noticed, and where #

The threshold is not a discovery. Anybody who has cut a paper snowflake has taken a sheet to the edge of falling apart and occasionally past it, and the failure is memorable precisely because it is sudden — the sheet is intact right up to the moment two cuts meet, and then there is a piece on the table.

What is worth having is the separation of the two quantities, because it is the thing a picture cannot show. A photograph of a cut sheet at 97% and one at 100% look almost identical, and one of them is an object and the other is six. Every property that makes cut sheets useful improves toward that boundary, so every real design sits near it, and the only way to know which side is to compute the covering rather than to look.

Where the ladder goes next #

The obvious continuation is the cut pattern designed for a target rather than a lattice. A field of cuts whose lengths and directions vary can be made to open into a prescribed shape, and the ligaments then vary too — so there is a weakest ligament, and the sheet’s whole behaviour is set by wherever that is. Finding it is a question about a cut pattern that this machinery can already ask.

The other direction is the one the previous rung opened and did not close. The mechanism’s Poisson’s ratio is exactly −1 and a real sheet’s is not, and the difference is the ligament: a hinge with width resists, so the tiles do not rotate freely and the two directions do not open by the same factor. How much of the −1 survives a ligament of a given width is a measurement, and it is the number that would tell a designer what the step is actually worth staying away from.