About

This is a growing collection of illustrated essays about the geometry of folding. Each one takes a single idea and draws it until the argument is visible — and where the argument is about a crease pattern, prints it at a size a reader can fold.

What this is not

It is not a set of instructions. There are no models to make here, no step-by-step diagrams for a crane or a dragon, and no gallery. The internet has a very large number of those and most of them are good. An essay exists here because there is a figure that explains something better than a paragraph can, and if there is no such figure there is no essay.

The crease patterns are checked, not drawn

Every illustration is generated as SVG at build time by code in this repository. That matters more here than in most subjects, because a crease pattern is not a picture of an argument — it is the argument, and a reader with a sheet of paper can find out whether it is true.

So the patterns are verified rather than asserted. Before one reaches a page it is put through the flat-folding theorems:

• Developability — the angles round every interior vertex sum to exactly 360°, or the paper was cut or stretched to make it.
• Kawasaki's theorem — alternating angles each sum to 180°, which is necessary and sufficient for a single vertex to fold flat.
• Maekawa's theorem — mountains and valleys differ by exactly two at every flat-foldable interior vertex.
• The big-little-big lemma — a strictly smallest sector is flanked by creases of opposite assignment, or the panels either side of it collide.

A generator that would emit a pattern failing any of those throws, and the build stops. That is not a formality. The preliminary base on this site had its assignment found by enumeration rather than stated, because the assignment most people would draw — both diagonals one way and both midlines the other — puts four mountains and four valleys at the centre and cannot fold flat.

What is deliberately absent is any claim to decide flat-foldability for a whole sheet. That problem is NP-hard, proved so by Bern and Hayes in 1996, and a figure implying otherwise would be dishonest in a more interesting way than a wrong drawing.

Several patterns are printable

Where a figure is a pattern somebody could fold, it carries a second copy of itself sized in millimetres. Print the page and that is what comes out: a sheet at the stated size, with mountains and valleys distinguished by dash as well as by colour, so a monochrome printer loses nothing. The dimensions are repeated on the sheet, because a printer set to fit-to-page will silently rescale it and a reader deserves a way to notice.

This is the site's way of showing its working. A claim that a pattern folds flat is checkable by anyone who owns paper, and making that easy is the strongest argument these essays can make.

Whose patterns these are

Origami models are creative works, and the folding community treats a designer's crease pattern as theirs. This site therefore restricts itself to three things: traditional bases, centuries old and unattributable; patterns published as mathematics rather than as models — the Miura fold, the Yoshimura pattern, Resch's tessellations; and patterns generated here from this repository's own code.

No living designer's crease pattern is reproduced. Where an essay discusses one, it describes and cites it and does not print it.

Where the model stops

The whole subject rests on a sheet with no thickness, no stretch, creases that are lines rather than radii, and perfect memory. All four are false. The engineering versions of this field are largely about the first one, wet-folding exists because of the second, and anyone who has folded a complex model has met the third and the fourth. Where a figure relies on an idealisation, the essay names it.

On being wrong

Corrections are welcome and will be made. A crease pattern that is beautiful and does not fold is worse than none, because somebody will spend an evening finding out.