The preliminary base
Fold it
The sheet is on the printed page, at the size it says.
Printing this page gives the page, and adds one more: the pattern alone, at 150 mm across, on a sheet of its own. Cut a square that size, transfer or trace the lines, and fold. The size is repeated in the corner of the sheet, because a printer set to fit to page rescales silently and there has to be some way to notice.
Mountain and valley are distinguished by dash as well as by colour, so the sheet survives the monochrome laser printer it will almost certainly come out of.
What it is
| Provenance | Traditional |
|---|---|
| Creases | 3 mountain, 5 valley |
| Interior vertices | 1, every one checked |
| Panels | 8, read off the pattern |
| Folding to do | about 724 mm of crease at this size |
| Printed sheet | 150 mm across |
What was checked
Four theorems at every interior vertex, and the faces two ways.
- Developability — the sectors around each of the 1 interior vertices sum to a full turn, so the sheet was flat before it was creased.
- Kawasaki — alternating sectors sum to a straight angle at each of them.
- Maekawa — mountains and valleys differ by exactly two.
- Big-little-big — no strictly smallest sector is flanked by two creases of the same letter.
- The faces — 8 of them, found by walking the planarised graph, and checked against Euler's formula and against the area they cover. A face walk that goes the wrong way round or merges two faces usually still satisfies Euler; it does not conserve area.
None of this decides whether the whole sheet folds flat, which is NP-hard in general. Every local condition holds. That is a different and weaker statement, and it is the one being made.
Take it away
The field's own interchange format, so the pattern is reusable outside this site.
preliminary-base.fold — 9 vertices, 16 edges, 8 faces, 1781 bytes. It opens in ORIPA, Rabbit Ear and the rest of the FOLD ecosystem.
The export is short because this repository never converts anything: FOLD's
vertices_coords, edges_vertices and edges_assignment
have been the in-memory representation of a pattern here since the site's first phase. What
the file adds is the metadata that makes it openable, and the faces where they can be read.
Coordinates are in sheet widths, and the file says what one unit measures on paper.
What is argued with it
Essays that call vertex-conditions — read off the figure index rather than listed by hand.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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 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.
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.
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.
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.
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 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.
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.
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.
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.
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.