Origami & Folding
what one sheet can do
Series

Kirigami — the series

11 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. the sheeta wedge of 60° marked for removal60°56.4°what it closes intoa cone of half-angle 56.44°83.3% of the turn is left, and the sine of the half-angle is that same fractionthe circles of latitude are the disc's own, arriving shorter than a flat sheet would needno fold can do this: folding moves paper about and never alters how much of it surrounds a point

    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.

    part 1 · material
  2. 010203040-3-2-10how far open — degrees for the cut sheet, the same fraction of the motion for the foldPoisson's ratiocut into squares−1 everywhere, exactlya fold, slant 0.35-0.12 at the startand without limit at the endthe two cross onceand agree nowhere elsethe flat line is a finite difference of two measured widths, taken the same way as the curve beside ita material made of matter cannot change its Poisson's ratio as it deforms; a material made of geometry can

    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.

    part 2 · material
  3. cuts at 85% of the pitch0.15 of material between one cut and the nextcut length, as a fraction of the pitchligament6 piecesone piece at every length up to 97%and the count is worked out from the cuts, not measured off a picture

    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.

    part 3 · material
  4. a disc, with a vertexa ring, with noneone interior vertex, 3 creases at itodd degree, so they do notno interior vertices at alland the panels still do notboth refuse: two routes round the sheet leave a panel 1.87 sheet-widths apart

    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.

    part 4 · material
  5. uncut: 4 vertices inside the paper, 6.3% of letterings admittedcut a crease with an interior vertex at each end25.0% admitted2 vertexes released · 4× the share · 4 such creases, all alikecut a crease that already reaches the edge12.5% admitted1 vertex released · 2× the share · 8 such creases, all alikea released vertex is one the four conditions no longer reach, and each is worth a factor of two

    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.

    part 5 · material
  6. The square twist, one crease at a timethe bar is the number of pieces, and a cut anywhere reduces itbefore any cut16 pieces · 0 vertices releasedcutting a crease that reaches the edge4 pieces · 1 vertices releasedcutting a buried crease2 pieces · 2 vertices released

    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.

    part 6 · material
  7. the bar is how far apart two routes to one panel end up, before the cut3 creases1.751.75 apart · cut open, 0e+05 creases1.591.59 apart · cut open, 0e+07 creases1.431.43 apart · cut open, 0e+09 creases1.321.32 apart · cut open, 0e+011 creases1.241.24 apart · cut open, 0e+0after one cut from the hole to the rim, every one of them places to rounding — with no crease changed

    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.

    part 7 · material
  8. the bar is the cuts after which the panels still place at allnone of them clears the contradiction, and no cut of a buried crease leaves a sheet that placessquare0 of 8484 cut, one at a time · 0 still place · 60 are buried and none of those doeselongated0 of 106106 cut, one at a time · 0 still place · 74 are buried and none of those doeshexagonal14 of 142142 cut, one at a time · 14 still place · 100 are buried and none of those doestriangular2 of 142142 cut, one at a time · 2 still place · 100 are buried and none of those doesa cut along a crease removes no paper — the two panels are still there and are no longer joined

    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.

    part 8 · material
  9. what each sheet's shape costs in conditionsa square0 loopsχ = 1 · no loop that cannot be shrunka slit from the rim0 loopsχ = 1 · the same paper, topologicallyone hole1 loopχ = 0 · one parity conditiona cylinder1 loopχ = 0 · the same sheet as one holetwo holes2 loopsχ = -1 · two independent conditionsa torus2 loopsχ = 0 · two conditions, no rim at alla slit inward from the rim changes nothing, and a closed cut changes everything

    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.

    part 9 · material
  10. two holes, two conditionsone loop is odd — the sheet refuses2 out to the left, 1 to the right, 1 betweenround the left hole: 3 creases, oddround the right hole: 2, evenround both: 3, oddno two-colouring exists0 interior verticesa loop round one hole says nothing about a loop round the other

    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.

    part 10 · material
  11. what each sheet's shape costs in conditionsa square0 loopsχ = 1 · no loop that cannot be shrunka slit from the rim0 loopsχ = 1 · the same paper, topologicallyone hole1 loopχ = 0 · one parity conditiona cylinder1 loopχ = 0 · the same sheet as one holetwo holes2 loopsχ = -1 · two independent conditionsa torus2 loopsχ = 0 · two conditions, no rim at alla slit inward from the rim changes nothing, and a closed cut changes everything

    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.

    part 11 · material

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