Origami & Folding
what one sheet can do
Generator

3 creases on a Möbius band

A generator in the flat-folding library, called 67 times across 11 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.

glued-band 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

3 creases on a Möbius bandA rectangular strip of paper with 3 creases across it and its two ends glued into a Möbius band. The arrows on the left and right edges show which way round the gluing carries one onto the other; on a Möbius band they point in opposite directions, which is the whole of the difference between the two sheets. The strip has 3 panels and no interior vertex at all, so every vertex condition in the subject is satisfied here without deciding anything.3 creases on a Möbius bandthe panels take two coloursseamthe same seam123the right edge onto the left, turned over3 creases, 3 panelsinterior vertices: 0two-coloursoff by 4.000 of a widthand turn the paper the right waymountainvalleyraw edge

view: "sheet", kind: "cylinder", count: 4

4 creases on a cylinderA rectangular strip of paper with 4 creases across it and its two ends glued into a cylinder. The arrows on the left and right edges show which way round the gluing carries one onto the other; on a Möbius band they point in opposite directions, which is the whole of the difference between the two sheets. The strip has 4 panels and no interior vertex at all, so every vertex condition in the subject is satisfied here without deciding anything.4 creases on a cylinderthe panels take two coloursseamthe same seam1234the right edge onto the left, the same way up4 creases, 4 panelsinterior vertices: 0two-coloursthe reflections closeand turn the paper the right waymountainvalleyraw edge

view: "vacuum"

Four conditions with nowhere to holdThe four vertex conditions this subject checks, evaluated on a band of paper. Each reads the creases meeting at one interior point, and a band has no interior point where creases meet, so all four hold on every band at every crease count — including the 6 of 12 measured here that have no flat folded state at all.four conditions with nowhere to holddevelopabilityholds, at 0 verticesKawasakiholds, at 0 verticesMaekawaholds, at 0 verticesbig-little-bigholds, at 0 vertices6 of 12 of these bands have no flat folded stateand only the panel colouring can see it

view: "census"

Which bands fold, on each of the two sheetsFor a cylinder and for a Möbius band, whether a strip with that many creases across it has a flat folded state. A cylinder needs an even number and a Möbius band an odd one, and the reason is that a composition of reflections turns the paper over when there are an odd number of them while the two gluing maps differ in exactly that.which bands foldcreases across the strip123456nofoldsnofoldsnofoldsfoldsnofoldsnofoldsnocylinderMöbius bandthe gluing map of a cylinder is a slide and of a Möbius band a slide with a flipand a composition of k reflections turns the paper over exactly when k is odd

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.

A crease with no vertex to belong to

Crease density is measured as length of line per area of paper, and everything else about a crease is measured at the vertex it runs into. A band of paper has creases that run from one edge to the other and meet nothing, so it has density and no vertices at all — and it still refuses to fold.

A proof in no nodes at all

A parity refuses a sheet before any search begins. It costs one addition, it is certain, and it says nothing about why — while a search that exhausts on the same sheet costs thousands of nodes and produces a proof of the same fact. Two proofs of one thing, and the cheap one is available only where somebody has noticed the invariant.

An alternating sum of angles

Kawasaki's condition says the sectors round a vertex alternate to a straight angle. A glued band has no vertices and obeys a condition of exactly the same shape: the crease angles have to alternate to a multiple of a straight angle. Two different quantities, two different sheets, one arithmetic — and in both cases what is being said is that a product of reflections came back the right way.

Closure is not the identity

Walk a folded state from panel to panel, composing a reflection at every crease, and come back to where the walk started: the composition has to be the identity. That is the rule everybody states, and it is a special case. On a sheet whose edges are glued the walk does not come back to where it started, and what the composition has to equal is the gluing map.

Found by people not folding paper

The shortest strip that makes a Möbius band has a literature, and it is in differential geometry rather than in origami. The two subjects have the same number, they reached it by completely different routes, and neither of them cites the other — which is the fourth time this collection has found that shape.

How rare a band that folds is

Almost every crease pattern fails to fold flat, and the usual way of saying so is a count over discrete choices. A glued band fails for a reason that no count can reach: its crease angles have to satisfy an equation, and a set defined by an equation has no volume in the space it sits in.

Parity is not enough

A Möbius band needs an odd number of creases round it. Give it three, square across the strip, and it does not fold — nor does five, nor seven, nor any odd number at all. The counting argument is necessary and it is not close to sufficient, and the thing it cannot see is which way the creases point.

The band that needs an odd number

A Möbius band is the first sheet in this collection with one side, and the consequence is sharper than a reversed parity. Mountain and valley are defined relative to a side, so on a sheet with no consistent side a crease has no letter — and Maekawa's condition survives the loss while the assignment it is about does not.

The proportion a band asks for

√2 is a shape: a rectangle either has it or does not, and what it buys is that halving returns the same shape. √3 is what a Möbius band needs, and it is a different kind of number — a minimum rather than a shape, with every longer strip working and no shorter one.

The seam carries a sign

A loop of paper folds flat when it has an even number of creases round it. A Möbius band folds flat when it has an odd number. The drawing is the same in both cases, the creases are the same creases, and what changed is a factor of minus one contributed by the sheet rather than by anything drawn on it.

The triangle a strip becomes

A Möbius band of paper folds flat into an equilateral triangle, and the shortest strip that will do it is √3 times its own width. The number is not put in: the crease angles come out of a condition on their alternating sum, the positions come out of two linear equations, and the length is where the drawing stops fitting.

Every generator · The flat-folding field · The patterns a reader can fold