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 refuses to draw. 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.

Those four are conditions at a point, and passing them is not folding. Since August 2026 every pattern printed here is also asked whether its panels can be put in an order: a crease decides which of its two panels is higher, a panel may not lie between the two a crease joins, and two creases in one place may not interleave. Where the pattern is small enough the orderings are searched exhaustively, and where it is not, a loop in the order the letters force is a proof of failure that costs one pass. The check refused two patterns that had been printed here for years, and both were redrawn.

That cheap proof turns out to be closed off at a single vertex, and the theorem that closes it is Maekawa's. Sending the panels round one point in a circle needs the letters to alternate strictly, an alternation has equal counts of mountain and valley, and a flat-foldable vertex has two more of one than the other — so of the hundred and fifty labellings a vertex of degree four, six or eight admits, not one puts its panels in a loop. A pattern's letters can only contradict themselves round a circuit that encloses more than one vertex, which is why the patterns printed here are consistent and a tessellation patch cut from the same construction usually is not.

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.

The patterns worth folding are collected on one page, each with the sheet at true scale, the list of what was checked, and a download in FOLD — the interchange format the field uses, which is also the representation this site computes in, so the export is the pattern rather than a conversion of it. Every exported file is parsed back and put through the theorems again before a build is allowed to finish: an artefact that was never checked as an artefact has not been checked.

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.