The Miura fold
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 170 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 | Miura, 1970 — published as engineering |
|---|---|
| Creases | 22 mountain, 16 valley |
| Interior vertices | 15, every one checked |
| Panels | 24, read off the pattern |
| Folding to do | about 6679 mm of crease at this size |
| Printed sheet | 170 mm across |
What was checked
Four theorems at every interior vertex, and the faces two ways.
- Developability — the sectors around each of the 15 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 — 24 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.
miura-fold.fold — 35 vertices, 58 edges, 24 faces, 4613 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 miura-pattern — read off the figure index rather than listed by hand.
One vertex, repeated
Take a single flat-foldable vertex and tile the plane with it. The sheet stops being a sheet and becomes a material — with a stiffness, a packing behaviour and a Poisson's ratio that the paper never had.
A square that turns
A twist is a small polygon that rotates as the sheet closes around it. The geometry is forced rather than designed — Kawasaki fixes one sector, the big-little-big lemma forbids a strictly smallest one, and what is left is the pattern Ron Resch was drawing in the 1960s.
A state no motion reaches
Flat-foldability asks whether a folded state exists. Rigid-foldability asks whether there is a path to it. The two sets are different, and the difference can be counted on a single vertex.
The name is not the date
Kawasaki's theorem is in Husimi's book ten years before Kawasaki's paper. Maekawa's is Justin's too. The mean gap between a result in this field and the name it is known by is twenty-two years, and it runs in one direction.
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 only pattern that moves
A rigid motion is not a generic property of a folded pattern. Move one vertex of a Miura by a thousandth of a panel and the sheet has no isometric folded position of that kind at all — and the amount by which it fails is first order in the displacement, so no move is small enough to be free.
Where an error goes
A misplaced crease in a folded sheet has to be paid for somewhere, and this subject has two answers already — the error is folded too, and the hinge is where it ends up. There is a third. In a quadrilateral mesh a mistake in one row has no consequence in that row at all: it is felt by the columns, which is to say by every other row on the sheet.
The corrugation that curves
A Miura is a flat sheet that becomes a flat slab. Open its straight creases into a fan and the same construction gives a corrugation that wraps a cone — exactly a cone, with every straight crease passing through one point to fifteen decimal places, at every moment of the fold, with the apex travelling as the sheet closes.
One crease decides the sheet
Fix one crease of a flat-folding problem, propagate every condition the subject has, and three creases out of a hundred and fifty-eight follow. Fix one fold angle of a rigid one and every crease on the sheet follows, with a single consistent answer. The same experiment, two questions, opposite answers — and it is why a self-folding sheet needs one biased vertex rather than one per vertex.
The patterns nobody owns
This site prints crease patterns at true scale and prints no designer's work, and that has always been stated as a rule applied at the end. Read the printed shelf as a documentary record instead and the rule turns out to be a property of the record: every pattern that carries a date was published as mathematics, every undated one belongs to nobody, and the two silences are one silence.
How much line is on the paper
A crease pattern is described by its creases: how many, at what angles, in what arrangement. What a folder spends is length. The Yoshimura this site prints has eighty-six creases and 2,380 millimetres of folding on a sheet seventeen centimetres across; the Miura has thirty-eight creases and 1,049, and the fold-and-cut triangle has six and 258 — and the two counts do not rank the eight printed patterns the same way.
The paper a pattern asks for
A Miura of c columns and r rows at a slant α wants a sheet whose proportion is (c + tan α) / r — one equation tying the two counts, the angle and the shape of the paper. A square is the case where it comes to one, which needs the tangent of the slant to be a whole number: 45° for a pattern one row taller than it is wide, 63.43° for two, and nothing at all for the slants anybody draws.
A corrugation agrees with itself
A Miura fold of forty-eight panels and a twist tessellation patch of forty-nine have almost exactly the same number of independent closed chains for their letters to contradict themselves round — thirty-five against thirty-six. Sixty-four per cent of the Miura's drawn letterings are consistent and thirteen per cent of the patch's. A Yoshimura at thirty-three chains manages ninety-three. The room to fail sets the scale; the construction decides where in it a pattern lands.
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.