Kirigami — the series
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What one cut buys
A fold moves paper about and cannot change how much of it surrounds a point. A cut can, and that one difference is the whole of what this site's founding rule is worth. Take a wedge out and the sheet closes into a cone; let one in and it has more paper than the plane will accept.
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Bought with holes
A Miura-folded sheet gets wider as it is pulled, and by how much depends on its panels and on how far it happens to be folded. A sheet cut into squares joined at their corners does the same thing and holds the value at exactly minus one, everywhere in its motion — the same property, bought with different geometry, and paid for in holes.
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One cut short of falling apart
Everything a cut sheet can do is bought out of the material between the end of one cut and the start of the next. That material shrinks to nothing in a straight line as the cuts grow, and the sheet stays in one piece the whole way down — until the instant it does not, and then it is in six.
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A cut that removes no paper
Cuts in this subject are graded. Take a wedge out and the angle at a point falls by exactly the wedge; take twice as much and it falls twice as far. A hole is not like that. Its effect on what the sheet can do is the same whether it is a tenth of the paper or a ten-thousandth, and it is the same because it is not a quantity at all.
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A cut is a licence
What a cut buys is usually described in words — freedom, release, a shape a fold cannot reach. It can be counted, and the unit is vertices. Cutting one crease of a square twist turns two interior vertices into vertices no theorem applies to, and the share of letterings the pattern admits goes up by a factor of two for each vertex released: exactly, on every cut tried.
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A cut is not local
Cutting one crease of a square twist takes its letterings from sixteen mutually unreachable pieces to two. The cut crease is one of the four that were settled when the pattern was drawn — and it takes two others with it, because the vertices it releases were the far ends of those. Even a cut along a crease that was never settled quarters the count.
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A cut that reaches the edge
A ring of paper with three creases running from its hole to its rim satisfies every condition the subject has — vacuously, because it has no interior vertex at all — and cannot be folded: its panels take no two colours and the two routes to one of them end up 1.75 sheet widths apart. One cut from the hole to the edge, crossing no crease and changing no letter, and it folds exactly. The cut removes an adjacency, which is the one thing neither a fold nor an edge can do.
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One cut removes one arc
A crease pattern whose letters contradict themselves has, in principle, an obvious smallest repair: cut one crease and the statement it was making goes away. Cut every crease of four tessellation patches in turn — four hundred and seventy-four cuts — and sixteen of them leave a sheet whose panels still land anywhere at all. A cut gives the paper a freedom, and a sheet with a freedom in it has no folded state to order.
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A cut is surgery
Two cuts that look identical on the paper do completely different things to the sheet. A slit run inward from the rim changes nothing at all; a closed cut in the middle removes a disc and leaves a sheet carrying a condition it did not have before. What separates them is not the length of the cut or how much paper it removes.
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Two holes are two conditions
One hole in a sheet of paper gives one loop that cannot be shrunk and one parity to satisfy. Two holes give two, and they are independent: an arrangement of creases can satisfy the condition round one hole and fail the condition round the other, and the sheet refuses on the strength of the one it failed.
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The cut that changes nothing
A slit goes right through the material and leaves the sheet exactly the object it was. A closed cut removes almost no paper and produces a different sheet with a condition it did not have. Kirigami is made almost entirely of the first kind, which is why every result about it survives the distinction untouched.
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A slit tip keeps one condition
A cut was priced at a factor of two in the letterings a pattern admits for every interior vertex it releases — exactly, no more and not less — and the price was measured on the square twist. A released vertex is not free. It is a strip a full turn long, and a strip has its own big-little-big lemma: a middle sector smaller than both its neighbours forces the creases either side of it to differ. Across the printed shelf, 81 of the 298 vertices a single slit releases still refuse some of their letterings. On a Miura every slit along a straight crease multiplies the share by three where the ledger says four, and every slit along a zigzag by four. The square twist, on which the price was set, is one of the two patterns where the strip never bites.
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An array is priced by its vertices
One slit along a Miura's zigzag crease multiplies the share of letterings that pass every local test by four, and one along a straight crease by three. An array of slits is not the product of those factors. Two slits that meet at a vertex release it once, not twice, and on a small Miura the slits' own factors price correctly exactly the 22 of its 128 arrays in which no two slits touch. What prices every array is the product of each vertex's own share — the ring conditions where nothing was cut, the strip rule wedge by wedge where something was. So the cheapest way to release every vertex is a matching: zigzag slits laid in pairs that never meet, half as many slits as vertices. And the count can be made before anything is cut only on a pattern whose vertices decide independently, which the Yoshimura's do not.
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What a loop of creases carries
Multiplying each vertex's own share of letterings gives the exact share for a whole Miura, both twists and a tapered corrugation, and three quarters of a per cent too much for the Yoshimura. The difference has an exact form. The count is the product times a sum over loops in the graph of creases, each loop weighted by the correlations its vertices put on the pairs of creases it uses. A Miura vertex correlates no pair at all, so no loop can carry anything. An equilateral Yoshimura vertex correlates every pair by −1⁄15, and its 27 small triangles take the product down by 0.8 per cent before the larger loops pay a little back. Draw the rows taller than √3 and the big-little-big lemma locks three pairs and frees the rest, no loop closes, and the product is exact.