One vertex, repeated
A crease pattern with one vertex is a local question. A crease pattern with four hundred identical vertices is something else: the individual vertex stops mattering and the pattern acquires behaviour.
This is the best-known example, it is on several spacecraft, and its crease pattern fits on a postcard.
The geometry
Start with a grid of parallelograms rather than rectangles. Each row is offset from the one below by a fixed amount, so the vertical lines zig-zag rather than running straight.
At each interior vertex four creases meet: two horizontal, continuing the row in both directions, and two from the zig-zag above and below. The four sectors come in two equal pairs — two acute and two obtuse — with the two acute ones adjacent.
Kawasaki’s condition is satisfied: the alternating sums are each 180°, because opposite sectors are supplementary by construction. Developability holds, because the four angles sum to a full turn.
So the geometry folds flat at every vertex, and it does so for any offset. The parallelogram angle is a free parameter.
The assignment is the trick
The geometry is easy and the assignment is not, and getting it wrong is the standard mistake.
The obvious thing is to make every horizontal crease a mountain and every vertical crease a valley, or something similarly regular. Try it and every vertex has two of each — and Maekawa forbids that. The pattern will not fold.
What is needed is three of one and one of the other at every vertex. The two horizontal creases at a vertex are halves of the same zig-zag line and share an assignment, so if the two vertical creases also shared one the count would be two and two.
Therefore the vertical creases must change assignment every time they cross a row.
That single rule is what makes the Miura fold work, and it is not obvious from looking at a folded one. A vertical crease line running up the sheet is mountain, then valley, then mountain, alternating at every row — so it is not really one crease at all, but a sequence of them that happen to be collinear.
Checking it
This site’s Miura generator does not assert the assignment; it derives it and the result is verified.
The first version assigned every vertical crease the same way and every horizontal crease by row. The checker rejected it: four mountains and no valleys at every interior vertex, which satisfies Kawasaki and fails Maekawa. The picture had looked entirely convincing.
The corrected rule was then cross-checked a second way, by enumerating all assignments of a small Miura and confirming that the derived one is among those passing the local tests. Constructed one way, verified another.
That is the site’s method in miniature, and this is the pattern that most needed it.
What a tessellation is for
A single vertex has properties. A tiling of identical vertices has material properties, and that is the shift worth naming.
An ordinary sheet, pulled in one direction, gets narrower in the other. A Miura-folded sheet pulled in one direction gets wider in the other. That is a negative Poisson’s ratio, it is unusual in ordinary materials, and it arrives entirely from the fold pattern.
The paper is unchanged. What changed is the geometry it has been given, and the geometry now dominates the behaviour. That is the definition of a mechanical metamaterial, and origami tessellations are among the cleanest examples anybody has.
Why the vertices must be identical
A tessellation could in principle use different vertices in different places. In practice the useful ones do not, and the reason is kinematic.
A single degree-four vertex is a one-degree-of-freedom mechanism: fix one dihedral angle and the other three follow. Two adjacent vertices share a crease, so they share that crease’s fold angle — and if the two vertices are different, they generally demand different values for it.
With identical vertices arranged periodically, the demands agree everywhere by symmetry, and the whole sheet moves as one. With mixed vertices they conflict, the sheet locks, and the only way it moves is by the facets bending.
So periodicity is not an aesthetic choice. It is what makes the tessellation a mechanism rather than a sculpture.
The free parameter
The parallelogram angle is not fixed, and varying it changes what the pattern is good for.
Near 90° the parallelograms are almost rectangles, the zig-zag is shallow, and the folded sheet is nearly a plain accordion — it collapses in one direction and barely at all in the other.
At a strong angle the zig-zag is pronounced, both directions collapse strongly, and the packing ratio improves considerably.
Past a point the facets become long and thin, the layers pile up at the corners, and thickness starts to dominate.
Engineering uses of the pattern tune this parameter deliberately: a solar array wants maximum packing, a shock absorber wants a particular force-displacement curve, and both are the same crease pattern at different angles.
Reading the pattern
A useful skill, and the Miura is the best pattern to learn it on.
The horizontal lines are the major folds — they run the full width and they alternate mountain and valley by row, which is an ordinary accordion pleat. Fold those first and the sheet becomes a concertina.
The zig-zag lines are the minor folds, and they are what turns the concertina into a Miura. They run across the accordion at an angle, and they alternate along their length.
That is also the practical folding order: accordion first, then the zig-zag, then collapse. Anyone who has folded one knows the collapse is the satisfying part, and it works because every vertex is doing the same thing at the same moment.
The pattern is older than its uses
Koryo Miura published the fold in 1970 as a way of packing large membranes, and it reached the Space Flyer Unit’s solar array in 1995. That is the canonical story and it is incomplete.
The pattern occurs in nature and had been noticed. It is close to the buckling pattern of a thin cylinder under axial load, it appears in the folded wings of some insects, and something very like it turns up in the epidermis of certain leaves as they emerge from a bud.
It also appears in traditional folding under other names, and the “Miura map fold” — a road map that opens with one pull on opposite corners — was patented and marketed long after folders had been making the pattern for fun.
So the attribution is to the person who characterised it and put it into orbit, which is the usual and reasonable convention, and it is worth knowing the pattern was not invented so much as recognised.
What repeating does not fix
A tessellation inherits every problem a single vertex has and adds some.
Layers still stack. A Miura sheet folded flat has as many layers as it has rows, and that is a real limit. A 20-row Miura in paper is a centimetre thick when packed.
The local conditions are still local. Every vertex satisfying Kawasaki and Maekawa does not mean the sheet folds — it is still the global problem — and for a tessellation the layer ordering is where any failure would appear.
Edges are different. The vertices at the boundary of the sheet are not interior vertices, the theorems do not constrain them, and a tessellation’s edge behaves differently from its middle. For an engineered panel that edge is where the mounting goes and where the problems are.
Folding one
The Miura is the pattern most worth folding by hand, and the sequence explains the structure better than the crease pattern does.
Start by accordion-pleating the sheet in one direction — mountain, valley, mountain, valley — into even strips. That produces every horizontal crease at once, and it is the easy part.
Now fold the accordion in half across the pleats, at a slight angle rather than square, and crease. Open it, refold the other way at the mirrored angle, and continue. Those diagonal creases are the zig-zag, and folding them through the accordion is what makes each vertical line alternate assignment as it crosses each row.
That last point is the one worth noticing. The alternation is not something the folder imposes crease by crease; it happens automatically, because folding a pleated stack reverses the sense of the fold on alternate layers. The rule that looked arbitrary in the crease pattern is what the accordion does on its own.
Then collapse. Every vertex closes at the same moment, and the sheet contracts in both directions together, which is the property the pattern exists for.
Why the pattern is printable
This is one of the figures on the site that carries a millimetre-sized sheet for printing, and the reason is that the argument is checkable in about ten minutes.
The claim is that a particular assignment folds and the obvious one does not. That is not a claim anybody should take on trust when the counter-example costs a sheet of A4 and a bone folder.
Print it, fold every line, and collapse. Then try it with the obvious assignment — every vertical crease the same way — and find that it will not go. The theorem is Maekawa’s and the demonstration is physical.
Where the model stops
Zero thickness. As always, and it bites hardest here because a tessellation multiplies the layer count by its row count.
Rigid facets. The kinematic argument assumes the parallelograms do not bend. Paper bends slightly and metal panels do not, which is why rigid-foldability is a stricter requirement than flat-foldability and why the Miura satisfies both.
Periodic and infinite. The material properties are properties of the infinite tiling. A small patch behaves differently, and most of the difference is at the edges.
One vertex type. Everything here concerns tessellations of identical vertices. Mixed and aperiodic tessellations exist, are interesting, and do not move.
The figure draws a small patch. Six columns by four rows is enough to see the pattern and far too few for the material behaviour to be representative.
Other vertices, other tessellations
The Miura is one tiling of one vertex, and stepping sideways shows what the space looks like.
The waterbomb tessellation tiles a degree-six vertex. It has more freedom than the Miura, it can form domes and tubes rather than only flat sheets, and it is correspondingly harder to control.
The square twist tiles a vertex arrangement with a rotating centre. It is bistable — it snaps between open and closed rather than moving smoothly — which makes it useless as a deployable and interesting as a switch.
Resch’s patterns tile triangular arrangements and produce a stiff, deep, textured sheet. Ron Resch developed them in the 1960s, largely for architecture, and they are the most visually striking things in the field.
Hexagonal twists and their relatives fill out a large family of decorative tessellations that fold flat and have no engineering application at all.
What separates the useful few from the rest is the same short list every time: identical vertices, a single degree of freedom, rigid-foldability, and a packing ratio worth the trouble. Most tessellations fail at least one.
The pattern as a material
The framing worth ending on, because it is what the field has converged on.
A Miura-folded sheet is not really a folded sheet any more. It is a mechanical metamaterial: an object whose useful properties come from its internal geometry rather than from its composition, and which can be made from almost anything.
The same pattern in paper, in Mylar, in steel and in silicon has the same Poisson’s ratio, the same packing ratio and the same single degree of freedom. Those properties belong to the tiling. What changes with the material is the stiffness, the thickness penalty and how many times it can be cycled.
That is a genuinely different way of thinking about a crease pattern from the origami tradition, where the pattern is instructions for making an object. Here the pattern is the object, and the material is an implementation detail — which is why the engineering literature talks about deployables and metamaterials and hardly ever about paper.
What repeating gives up
A tessellation gains material properties and loses something worth naming.
A designed model has features — a head, a leg, a wing — and every part of the crease pattern is doing a specific job for a specific part of the subject. Circle packing exists to allocate paper to those jobs.
A tessellation has no features. Every region is doing the same thing as every other, and the pattern has nothing to say about any particular place on the sheet. That is exactly what makes it a material and exactly what makes it useless for representing anything.
The two branches of the subject barely interact for this reason. Representational design is about allocating paper unevenly and tessellation is about allocating it identically everywhere, and a technique from either is nearly useless in the other.
Where they do meet is at the boundary: a model that uses a tessellated region for texture — scales, feathers, a corrugated surface — inside an otherwise conventional design. That is a real and growing practice, and it is the only place the two vocabularies share a sheet.
What to notice when folding one
A short list, for anybody who prints the pattern.
The accordion pleats go first and they must be even, because everything else references them. Uneven pleats give a sheet that collapses crookedly and will not lie flat.
The zig-zag creases are made through the folded accordion, which is why the assignment alternates automatically — folding a stack reverses the sense on alternate layers, and that is the rule that looked arbitrary in the pattern.
The collapse happens all at once. There is a moment where the sheet resists and then goes, and the going is the several hundred vertices reaching their folded state simultaneously because they share their fold angles.
And the finished sheet opens with one pull on opposite corners. That is the single degree of freedom made tangible, and it is the most convincing thirty seconds available in this subject.
The ladder from here
Later rungs: the Miura’s kinematics derived. Negative Poisson’s ratio, computed. The parallelogram angle as a design parameter. Other tessellations — the square twist, the waterbomb tessellation, Resch’s patterns. Tessellations as metamaterials. Bistability and snap-through. Edge effects. Aperiodic tessellations. The Miura in nature. And the manufacturing question, which is where thickness stops being a footnote.
The Miura fold’s crease pattern is a grid of identical parallelograms with an assignment rule that fits in one sentence, and it is the reason a solar array the size of a tennis court fits inside a rocket.