Origami & Folding
what one sheet can do
The evidence, in the hand

Patterns to fold

Every claim on this site is made twice — once as a picture and once as a sheet of paper. These are the patterns a reader can print at true scale, fold, and use to check the argument against the only authority that matters.

No other kind of illustration can be executed. A graph of a function is a picture of an argument and a reader can only look at it; a crease pattern is the argument, written in the only notation this subject has, and anybody with a sheet of paper can run it in ninety seconds.

There are 8 of them here. Each one is built by code in lib/patterns.js, checked against developability, Kawasaki, Maekawa and the big-little-big lemma before it is allowed onto a page, and printed at a stated physical size — so a printer set to fit to page can be caught out by measuring the sheet.

Each also downloads as FOLD, the interchange format the field uses, which is the same representation this repository computes in. A pattern here is therefore reusable outside this site rather than trapped in an illustration.

Nothing on this page is somebody else's design. The patterns are traditional, published as mathematics or engineering, or generated by this repository's own code — the distinction is stated on each one, and it is why the list is short.

at the centre8 creases, all sectors 45°3 mountain, 5 valleydifference 2 — Maekawa holdsfour and four would fail,which is what most people drawfold every line, then collapse — the four corners meetmountainvalleyraw edge

The preliminary base

Both diagonals and both midlines of a square: the base under the crane, the lily and half the traditional repertoire, and the one most folders letter wrongly first time.

3M · 5V · 150 mm sheet at every vertexthree of one, one of the other15 interior vertices, all identicalwhat the sheet gainsone degree of freedom, not manyit opens and closes in both directions at oncea negative Poisson's ratio22 mountain and 16 valley creases · 6.2 sheet-widths of foldingmountainvalleyraw edge

The Miura fold

A grid of identical parallelograms, and the pattern behind every deployable this subject reaches. Folded, it opens and closes in both directions at once.

22M · 16V · 170 mm sheet 4 corners, all alikesectors 90°, 90°, 90°, 90°two equal pairs, so no sectoris strictly the smallestthe assignment256 of 4096 fold6 mountain, 6 valleythe ring takes two lettersthe panels can be orderedwhat is checked4 interior verticesand not the tilinga twist of radius 0.17 sheet-widths12 creases, 4.70 sheet-widths of foldingmountainvalleyraw edge

The square twist

One twist unit rather than a tessellation, and the cheapest of the family to fold: nine panels, twelve creases, and a middle square that turns as the sheet closes.

6M · 6V · 150 mm sheet 6 corners, all alikesectors 120°, 120°, 60°, 60°two equal pairs, so no sectoris strictly the smallestthe assignment18 creases, searched9 mountain, 9 valleythe ring takes two lettersthe panels can be orderedwhat is checked6 interior verticesand not the tilinga twist of radius 0.17 sheet-widths18 creases, 6.11 sheet-widths of foldingmountainvalleyraw edge

The hexagon twist

Six pleats arranged round a hexagon, and harder on the hands than it looks: the ring closes only if all six are persuaded upward together.

9M · 9V · 150 mm sheet found rather than designedcrushed drink cans, tree bark,deployable boomsat every interior vertexsectors 60°, 60°, 60°, 60°, 60°, 60°two courses and four diagonalstwo of one letter, four of the otherwhy the height is not freea steeper diagonal makes the topsector strictly smallest, flankedby two of the same letter22 interior vertices · 26 mountain and 60 valley creasesmountainvalleyraw edge

The Yoshimura pattern

The diamond pattern a thin cylinder buckles into when it is crushed. Nobody designed it; it is what the shell does, folded flat and rolled back into a cylinder.

26M · 60V · 170 mm sheet the cut line3 straight edgesthe pattern3 skeleton arcs3 perpendiculars1 interior vertexassignments that fold30 of 646 creases in allarcs one way, perpendicularsthe other: fails Maekawaequidistance off by 1.9e-16mountainvalleyevery node sits the same distance from each edge that formed it,which is why one fold can carry several edges onto the line at once

Fold and cut — the triangle

Fold on every line, make one straight cut along the fold, and the triangle falls out whole. The heavy outline is the shape rather than a crease.

2M · 4V · 150 mm sheet 18 interior vertices26 mountains · 19 valleyscolumns taper 2.44 : 1packs to 11.2% of flatmountainvalleyraw edgethe taper is in the columns, because Kawasaki does not mention their widthtapering the rows instead puts the alternating sums at 186.4° and 173.6°

The tapered corrugation

The fold a corrugated leaf packs into, tapered in the column widths — because tapering across the folds instead, which is the obvious thing to try, will not close.

26M · 19V · 160 mm sheet two kinds of vertex, both forced16 of degree 490°, 90°, 90°, 90°9 of degree 690°, 45°, 45°, 90°, 45°, 45°40 mountain and 36 valley creases14.3 sheet-widths of foldingmountainvalleyraw edge

The waterbomb tessellation

Both diagonals of every cell, every row creased and none of the columns. That last omission is the whole pattern: it repeats the waterbomb base, not the preliminary one.

40M · 36V · 160 mm sheet

What “verified” means here, and what it does not

Each pattern passes every test this repository can apply at a vertex: the sectors close to a full turn, alternating sectors sum to a straight angle, mountains and valleys differ by two, and no strictly smallest sector is flanked by two creases of the same letter. Where the pattern's graph is connected, the faces are read off it as well and checked against Euler's formula and against area conservation.

None of that decides whether the whole sheet folds flat. That problem is NP-hard, and a page of printable patterns is exactly where it would be convenient to forget it. What the checks establish is that no local condition is violated; what folding one establishes is the rest.