Origami & Folding
what one sheet can do
Generator

From a stick figure to a crease pattern

A generator in the designing a base library, called 11 times across 7 essays. Below: what it draws with nothing specified and with the settings the essays give it, what it checked while drawing, and everywhere it is used.

tree-to-base is one function. Everything below was drawn by it when the site was last published, at settings 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.

With nothing specified

From a stick figure to a crease patternThe tree method in three steps. A subject is reduced to a skeleton with measured limbs; each limb becomes a circle; the packing that results dictates where the creases go. Robert Lang's TreeMaker automates the middle step, which is the one that is genuinely hard.the subjecta stick figure with limb lengthsthe packingone circle per limb, no overlapthe basea flap for every circlethe lengths in the skeleton become the radii, and the radii become the flaps

r: 0.22

From a stick figure to a crease patternThe tree method in three steps. A subject is reduced to a skeleton with measured limbs; each limb becomes a circle; the packing that results dictates where the creases go. Robert Lang's TreeMaker automates the middle step, which is the one that is genuinely hard.the subjecta stick figure with limb lengthsthe packingone circle per limb, no overlapthe basea flap for every circlethe lengths in the skeleton become the radii, and the radii become the flaps

r: 0.18

From a stick figure to a crease patternThe tree method in three steps. A subject is reduced to a skeleton with measured limbs; each limb becomes a circle; the packing that results dictates where the creases go. Robert Lang's TreeMaker automates the middle step, which is the one that is genuinely hard.the subjecta stick figure with limb lengthsthe packingone circle per limb, no overlapthe basea flap for every circlethe lengths in the skeleton become the radii, and the radii become the flaps

r: 0.24

From a stick figure to a crease patternThe tree method in three steps. A subject is reduced to a skeleton with measured limbs; each limb becomes a circle; the packing that results dictates where the creases go. Robert Lang's TreeMaker automates the middle step, which is the one that is genuinely hard.the subjecta stick figure with limb lengthsthe packingone circle per limb, no overlapthe basea flap for every circlethe lengths in the skeleton become the radii, and the radii become the flaps

What it checked while it drew

Collected while it drew, not written by hand. The count is how many separate times that claim was put to the test.

Where it is called

Every figure listed here is drawn by the same rule, so a mistake in the rule would be in all of them at once, and so would a correction.

A flap costs a circle

A flap of a given length uses up every point of the sheet within that distance of it. Two flaps whose circles overlap are asking for the same paper twice — and that one observation turned origami design from an art into an algorithm.

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.

Spelling a tree on a grid

Box pleating asks every limb of a design to be a whole number of grid squares, which sounds like rounding and is not. Rounding each limb to its own nearest whole number is one way to choose the numbers, and at most resolutions it is not the best way — the best whole-number version of a subject is often a coarser one, with fewer squares and a shape twice as close.

The last free parameter

Once the packing is fixed, one number is left in the whole design: how far a leftover polygon can be shrunk before it stops being a polygon. Everything else about the crease pattern has already been decided.

The second term

A feature grafted into a finished design costs exactly the paper slid in for it, and the bill has one term. Add a second feature across the first and it has two: the rectangle where the strips cross is paper both features are charged for and neither uses. At a strip a fifth of the sheet wide it is nine per cent of the bill; at four fifths it is nearly a third.

The skeleton changes its mind

The universal molecule fills any convex polygon, always, which is what makes it the part of the tree method with no special cases. It does not fill it continuously. Slide one corner along its edge and the number of creases in the molecule sits at six, jumps, and sits at seven — so two designs a hairsbreadth apart have crease patterns that are not small variations on one another.

Two packings, one radius

A packing search reports a number, and the number is not the design. What a crease pattern is built from is the graph of which discs touch which — and at five and six flaps, runs of the same search that agree about the best radius to four decimal places come back with contact graphs that are provably not the same graph. The answer an optimiser gives has not determined the pattern it is supposed to have found.

All the figures · The designing a base field · The patterns a reader can fold