A wedge out, and the cone that closes
kirigami is one function. Everything below came out of it during this
build, at arguments taken from the essays rather than invented for this page — so a figure
here is the same figure a reader meets in an essay, and when the generator changes, this
page changes with it.
At its defaults
view: "wedge", wedgeDeg: 60
view: "slits", cols: 4, rows: 4
view: "wedge", wedgeDeg: 45
What it checked while it drew
Collected by running this generator with a listener on the assertions, not written here. The count is how many separate times this build put that claim to the test.
- a wedge of 60° leaves 83.3% of the turn, and the sheet closes into a cone of half-angle 56.44° — the sine of that angle is what is left of the turn, checked against the closed form ×3
- at 25° the tiles cover 56.6% of the sheet, measured off the polygons drawn rather than off the rule that placed them, and tile plus hole account for the whole cell ×2
- all 92 interior vertices of the pattern library keep a full turn of paper, the worst of them short by 1.4e-16 of a turn — folding moves paper and never changes how much surrounds a point ×1
- and not one of the 6 cut vertices does: the nearest is 30.0° away from a full turn, which is the whole of what a cut buys ×1
- and the 2 wedges that add paper have no half-angle at all: there is no surface of revolution with more than a full turn at a point ×1
- both numbers are finite differences of two measured dimensions, so the flat line is a measurement and not a definition ×1
- every node of the straight skeleton is equidistant from each edge that defined it, so one fold serves them all — 1 checked ×1
- every point of the profile lies on one straight line from the apex, so the surface is flat everywhere except at the single point the cut was made ×1
- opening the array from 0° to 45° takes it from 100% solid to 50%, with the tiles never changing size — every square millimetre of the growth is hole ×1
- the cone's half-angle satisfies sin θ = 1 − δ/2π at every wedge sampled, to machine zero — a 60° wedge gives 56.44° and a straight-angle wedge exactly 30° ×1
- the corners meant to be one hinge coincide across the array, so nothing has been drawn that a sheet of paper could not be cut into ×1
- the cut sheet's Poisson's ratio is minus one to machine zero at every opening sampled, while the folded sheet's starts at -0.12, is never the same number twice, and is past -49.1 by the time its rows close ×1
- the sheet is one piece at every cut length short of the pitch — 6 of 7 lengths — and falls into 6 at it ×1
Where it is called
Changing this generator changes every figure on this list, which is what makes the list worth publishing rather than keeping in a check script.
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.
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.
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.
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.
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.
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.
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.
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.
Every generator · The curves and material field · The patterns a reader can fold