A vertex that folds flat
vertex-conditions 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
a: 60, b: 90, valid: true
a: 60, b: 90
a: 70, b: 110
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.
- developability holds at every interior vertex — its sectors close to 360° — 1 checked ×1
- eight creases meet at the centre in equal 45° sectors, so Kawasaki holds for any assignment at all, and the one drawn is 3 mountain to 5 valley — the difference of two Maekawa requires and the four-and-four most people draw would break ×1
- Kawasaki holds at every interior vertex — the alternating sums of the sectors agree — 1 checked ×1
- Maekawa holds at every interior vertex — mountains and valleys differ by exactly two — 1 checked ×1
- the big-little-big lemma holds at every interior vertex — no strictly smallest sector is flanked by two creases of one assignment — 1 checked ×1
- the four sectors close to 360°, the opposite pairs each sum to 180°, and the assignment is 3 to 1 — so the vertex drawn satisfies every condition the figure names ×1
- the preliminary base is put past all four theorems before it is drawn ×1
- the Preliminary base is put past all four theorems before it is drawn ×1
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 near miss is nearly as rare
Flat-foldability is a coincidence of measure zero, which is usually where the argument stops. Measure how far a random vertex is from folding rather than whether it does, and the answer is thirty-one degrees a sector — so the tolerance real paper has does not buy back anything at all, and a pattern that nearly folds had to start near one that did.
A theorem with an unstated hypothesis
Maekawa's and Kawasaki's conditions are quoted everywhere without saying which sheet they are about, and they do not need to be — they are conditions at a point and every point is the same. The two-colouring is quoted the same way and it is not a condition at a point, and the omission there is not harmless.
Almost every pattern fails
Kawasaki's condition is one equation for each interior vertex, and a drawing satisfies an equation with probability zero. Every pattern on this site folds because it was constructed to, and the fraction that would fold by accident can be measured.
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.
Four things that are not true
Zero thickness, no stretch, creases that are lines, and perfect memory. Every theorem on this site rests on all four, every one of them is false, and the interesting engineering is exactly where each fails.
Half the recipe is decoration
Every account of the Miura fold gives its letters as two instructions: the rows go one way, and the columns change letter every time they cross a row. Enumerate all sixty-four repeating rules and the second instruction is the whole of the condition — all four ways of writing the rows appear among the sixteen that fold, in every combination. The first instruction has never constrained anything.
Nothing meets at three
Every interior vertex of a flat-foldable pattern carries an even number of creases and at least four. So a crease cannot stop in the middle of the sheet, three creases cannot meet anywhere, and every crease pattern in the subject ends up looking the same way — all crossings and no stars.
One marking, many objects
A crease pattern with every mountain and valley written on it is spoken of as though it named a folded model. At four creases it does. At six it need not, and at the eight-crease vertex in the middle of the first base anybody folds, a single marking can be folded into four genuinely different objects — same creases, same letters, four answers.
Patterns nobody designed
Crush a thin cylinder and it folds into a diamond lattice. Nobody chose the pattern — it is the buckling mode with the lowest energy, and it satisfies the flat-folding theorems because it just folded.
Sixty-four rules, sixteen fold
The Miura fold's letters are usually given as a recipe: rows one way, columns changing at every row. Write down every rule of that shape — the letter on a crease depending only on which row and which column it is in — and there are sixty-four. Sixteen fold flat. They are exactly the ones whose columns change at every row, the row letters do not matter at all, and every one of the forty-eight refusals is the counting theorem's alone.
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 creases a sheet gives itself
A crease pattern drawn at random satisfies the flat-folding condition at essentially none of its vertices. A sheet crumpled at random satisfies it at every single one, on every seed, at every size — and the reason is a tautology that is very easy to miss.
The family the Miura belongs to
Move one vertex of a Miura and the sheet has no rigid folded position at all — which leaves the obvious question unanswered. What else moves? A row of paper reflected in each of a fan of lines is flat-foldable for nothing at all, and whether it also folds rigidly turns out to be a condition on a table of cosines: it has to be a column of numbers times a row of numbers.
The lengths are free
Kawasaki reads angles, Maekawa counts letters, and the big-little-big lemma compares one sector with its neighbours. Not one condition in the subject mentions how long a crease is — so a single vertex is not a pattern but a whole family of them, every member folding, no two folding into the same shape.
The loop is in the rule
Of the forty-eight repeating rules that do not fold a grid corrugation, thirty-eight send four panels round in a circle and ten merely fail the count. Which is which can be read off three of the rule's six bits, without building the pattern, folding it or walking a single arrow — and the closed form agrees with the arrows on all sixty-four rules of both grid families.
The outline is mostly crease
The edge of a folded model is what a reader looks at, and almost none of it is the edge of the paper. Measured across five patterns, the sheet's own boundary accounts for between nothing and a third of the exposed edge; the rest is fold, and on a waterbomb tessellation the raw edge does not reach the outside at all.
The sheet has two sides
Read a crease pattern as a set of panels rather than a set of lines and a condition appears that no vertex theorem states: the panels take two colours, no crease has the same colour on both sides, and the colour is which face of the paper each panel ends up showing.
The sheet remembers
Perfect memory is the fourth idealisation, and the least examined of the four. It is usually read as a complaint that paper will not lie flat again; the large half is the opposite. A sheet folded once is no longer blank, so folding a second model into it is folding the union of two patterns — and a union folds flat only where every new crease meets every old one at a right angle.
The whole alphabet of a grid
Box pleating is defended as a trade — give up packing efficiency, buy creases that land where they should. There is a third thing it buys and it is much stronger than either: on a forty-five degree grid there are exactly six kinds of interior vertex a flat-foldable design can contain, ever. On a thirty degree grid there are thirty.
Two conditions at a point
Whether a single vertex folds flat is decided completely by two tests — one on the angles, one on the assignment. They are independent, they are easy to check, and together they settle the case entirely.
Why the difference is two
Maekawa's theorem says mountains and valleys differ by exactly two at every flat-foldable vertex. The constant is not empirical — it is a full turn, and the theorem is about winding rather than about paper.
Every generator · The flat-folding field · The patterns a reader can fold