Origami & Folding
what one sheet can do
Generator

Thickness: chamfer

A generator in the rigid folding library, called 19 times across 5 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.

thickness-technique 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

Thickness: chamferA cross-section through one fold in a panel of real thickness, driven until the two panels touch. The contact is tested on the actual outlines rather than judged by eye, so the angle underneath is the travel the technique buys — and every technique buys it by giving something else up.0°61°108°travel before contact110.0°measured by contact testpanel thickness22% of the panel lengthwhat it gives upmaterial at the crease, sothe panel is thinnest whereit is worked hardesta zero-thickness pattern says the panels meet along a line; nothing that is built does

technique: "chamfer"

Thickness: chamferA cross-section through one fold in a panel of real thickness, driven until the two panels touch. The contact is tested on the actual outlines rather than judged by eye, so the angle underneath is the travel the technique buys — and every technique buys it by giving something else up.0°61°108°travel before contact110.0°measured by contact testpanel thickness22% of the panel lengthwhat it gives upmaterial at the crease, sothe panel is thinnest whereit is worked hardesta zero-thickness pattern says the panels meet along a line; nothing that is built does

technique: "surface-hinge"

Thickness: surface hingeA cross-section through one fold in a panel of real thickness, driven until the two panels touch. The contact is tested on the actual outlines rather than judged by eye, so the angle underneath is the travel the technique buys — and every technique buys it by giving something else up.0°99°177°travel before contact180.0°measured by contact testpanel thickness22% of the panel lengthwhat it gives upsymmetry: the hinge sits onthe side the fold goes, sothe assignment is builtinto the hardwarea zero-thickness pattern says the panels meet along a line; nothing that is built does

technique: "solid", t: 0.22, L: 1, beta: 35, gap: 0.3

Thickness: solidA cross-section through one fold in a panel of real thickness, driven until the two panels touch. The contact is tested on the actual outlines rather than judged by eye, so the angle underneath is the travel the technique buys — and every technique buys it by giving something else up.0°0°0°travel before contact0.0°the panels touch at oncepanel thickness22% of the panel lengthwhat it gives upnothing, and it does notmove: the control casea zero-thickness pattern says the panels meet along a line; nothing that is built does

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.

Getting thickness round a corner

There are half a dozen ways to build a fold in a panel that has depth, and the useful way to arrange them is not by what the cross-section looks like. It is by what each one gives away.

Nowhere to put the error

Paper takes a misplaced crease and spreads it along its whole length as a curvature nobody notices. A panel is flat by definition and cannot, so the error arrives at the hinge — and the room to receive it is a length that has to be drawn, is paid for in fold angle, and has to grow with the crease count.

Panels with somewhere to go

Every way of giving a folded panel real thickness costs something. Tachi's offset-panel technique costs the least interesting thing there is — it stops the panels being a surface, and leaves the hinges exactly where the zero-thickness pattern put them.

Paper through paper

Every test the subject has for rigid folding is a statement about a neighbourhood, and a neighbourhood cannot see the far side of the sheet. So a pattern can satisfy all of them while driving one panel straight through another, and the sharpest witness has no interior vertex in it at all.

Thickness round a closed loop

Real panels have thickness, and every technique for accommodating it works by shifting a hinge off the ideal crease by a small amount. On a flat sheet the shifts accumulate outward and end at the edge. On a closed sheet they accumulate round a loop and have to come back to where they started, which is a condition none of the techniques was designed to satisfy.

Every generator · The rigid folding field · The patterns a reader can fold