Origami & Folding
what one sheet can do
Generator

Fold into 10, cut once

A generator in the who found it, and when library, called 11 times across 3 essays. Below: what it draws at its defaults and at the arguments the essays give it, what it checked while drawing, and everywhere it is used.

wedge-and-cut 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

Fold into 10, cut onceThe traditional method, which is not the theorem. The sheet is folded into equal wedges about a point, one straight cut is made, and the shape that falls out has the symmetry the folding imposed. It gives a regular star for nothing and it gives nothing at all for a shape without that symmetry.10 layers, one cutwhat the fold decides5 points, 10 cornerscut at 54° to the foldwaist 0.309 of the pointregular, and checkedequal radii to 1e-12equal turning to 1e-12the symmetry is the method — a shape without it is not reachable thisway, and that is what 1998 changed

points: 5, cutAngleDeg: 48

Fold into 10, cut onceThe traditional method, which is not the theorem. The sheet is folded into equal wedges about a point, one straight cut is made, and the shape that falls out has the symmetry the folding imposed. It gives a regular star for nothing and it gives nothing at all for a shape without that symmetry.10 layers, one cutwhat the fold decides5 points, 10 cornerscut at 48° to the foldwaist 0.209 of the pointregular, and checkedequal radii to 1e-12equal turning to 1e-12the symmetry is the method — a shape without it is not reachable thisway, and that is what 1998 changed

view: "layers"

How many wedges the paper allowsFor each order of folk star, the stack the scissors have to cut through — twice the number of points, in thicknesses of the sheet — and whether each paper allows it in one clean pass. The geometry of the construction says nothing about this: it will draw a star of any order. What stops the repertoire is the substrate, and where it stops is arithmetic on a measured thickness.the stack under a single cuta clean cut taken as 0.6 mm of stackstarnewsprintcopier paperkamiwashifoil-backed tissueunryu tissue3 points6 layers390 µm600 µm420 µm240 µm156 µm108 µm4 points8 layers520 µm800 µm560 µm320 µm208 µm144 µm5 points10 layers650 µm1.0 mm700 µm400 µm260 µm180 µm6 points12 layers780 µm1.2 mm840 µm480 µm312 µm216 µm8 points16 layers1.0 mm1.6 mm1.1 mm640 µm416 µm288 µm10 points20 layers1.3 mm2.0 mm1.4 mm800 µm520 µm360 µm12 points24 layers1.6 mm2.4 mm1.7 mm960 µm624 µm432 µm16 points32 layers2.1 mm3.2 mm2.2 mm1.3 mm832 µm576 µm20 points40 layers2.6 mm4.0 mm2.8 mm1.6 mm1.0 mm720 µma k-pointed star is folded into 2k wedges, so the scissors pass through 2k layers · a clean single cut is taken here as 0.6 mm of stack

view: "layers", points: [3, 4, 5, 6, 8, 12, 20, 30], shear: 900

How many wedges the paper allowsFor each order of folk star, the stack the scissors have to cut through — twice the number of points, in thicknesses of the sheet — and whether each paper allows it in one clean pass. The geometry of the construction says nothing about this: it will draw a star of any order. What stops the repertoire is the substrate, and where it stops is arithmetic on a measured thickness.the stack under a single cuta clean cut taken as 0.9 mm of stackstarnewsprintcopier paperkamiwashifoil-backed tissueunryu tissue3 points6 layers390 µm600 µm420 µm240 µm156 µm108 µm4 points8 layers520 µm800 µm560 µm320 µm208 µm144 µm5 points10 layers650 µm1.0 mm700 µm400 µm260 µm180 µm6 points12 layers780 µm1.2 mm840 µm480 µm312 µm216 µm8 points16 layers1.0 mm1.6 mm1.1 mm640 µm416 µm288 µm12 points24 layers1.6 mm2.4 mm1.7 mm960 µm624 µm432 µm20 points40 layers2.6 mm4.0 mm2.8 mm1.6 mm1.0 mm720 µm30 points60 layers3.9 mm6.0 mm4.2 mm2.4 mm1.6 mm1.1 mma k-pointed star is folded into 2k wedges, so the scissors pass through 2k layers · a clean single cut is taken here as 0.9 mm of stack

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.

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 tree cannot argue

A molecule fills a polygon with creases taken from its straight skeleton, and a straight skeleton is a tree. So a molecule's panels have almost no closed chains for its letters to contradict themselves round — one to three, against thirty-six on the smallest tessellation patch. Two hundred and eighty independent letterings across seven outlines, including an L and a five-pointed star, and not one of them disagrees with itself.

How many wedges the paper allows

A k-pointed folk star is folded into 2k equal wedges and cut once, so the scissors go through 2k thicknesses of the sheet. The geometry is indifferent to k and the paper is not: at a stack a pair of scissors will shear cleanly in one pass, ordinary copier paper takes a three-pointed star and nothing more, and the five-pointed one everybody knows needs washi or thinner.

The star that was cut before it was proved

Fold a sheet into ten wedges, make one straight cut, and a regular five-pointed star falls out. The trick is at least two centuries old and the theorem that any straight-line drawing can be released by one cut is of 1998 — because the traditional method is not the theorem, and works only on shapes with the symmetry the folding imposes.

Every generator · The who found it, and when field · The patterns a reader can fold