The corrugation that closes on itself

The Miura fold is the most-studied pattern in this subject and the one with the most hardware built on it. It is one vertex repeated, it has exactly one degree of freedom, and it collapses in both directions at once, which is why it packs solar arrays and folds maps.

It is also not symmetric between its two directions, and gluing it makes that visible in a way nothing else has.

The cell #

The Miura’s drawing repeats in a rectangle one column wide and two rows high — two rows, because the zigzag alternates its offset and returns to itself only after two.

Cut such a rectangle, place its corners so the edges miss every vertex, and identify opposite edges. What comes out has two interior vertices, four creases and two panels at one period; eight vertices, sixteen creases and eight panels at two.

One crease across and four along #

A loop running once round the cell in the horizontal direction crosses the vertical creases: one per period. A loop running once round in the vertical direction crosses the zigzag courses: four per period, because a period is two rows and each row contributes two.

That is the whole of the asymmetry and everything follows from it.

Crossing a crease exchanges which face of the paper is up. A loop that comes back having crossed an odd number of them comes back with the sheet turned over, and a sheet on which that happens has no consistent two-colouring and no flat folded state.

So the horizontal gluing has a parity: a cell of one period refuses, two folds, three refuses, four folds. The vertical gluing has none: four is even at every size, and the count is even at every size, and no cell of the Miura ever fails that way.

Two cylinders that are not the same object #

The two gluings produce sheets with the same Euler number, the same number of edges left, and the same name. They are different sheets and the counts say so.

At two periods, gluing across gives eighteen free letters over ten panels; gluing along gives twenty letters over twelve panels. At three periods it is thirty-nine over twenty-one against forty-two over twenty-four. The difference is not rounding and it is not a choice of drawing: it is that the two directions of the pattern cut different numbers of creases.

What one glued pair saves is not what the other saves, and the two savings add to the total. That is the arithmetic; the sheets are the reason it was worth checking.

The tube the pattern already wanted to be #

There is a reason to care about this beyond bookkeeping, and it is that the horizontal gluing is a real object.

A Miura tube — a corrugation rolled round and joined — is one of the standard forms in deployable structures, and it is exactly the cylinder above. Whether such a tube can be pressed flat is a question with an answer, and the answer is a parity: an even number of columns round the tube and it flattens, odd and it does not.

That is a designer’s constraint stated as a count, and it is the kind of statement this subject is good at producing and bad at circulating: the people who build folded tubes arrive at even column counts by trying, and the reason is one line of arithmetic.

The vertical gluing, which never fails #

The direction that cannot fail is worth a moment, because cannot fail is a strong statement and it has a structural reason rather than an empirical one.

Each row of the Miura contributes two crease-crossings to a vertical loop: one going into the zigzag and one coming out. A period is two rows. So the count is a multiple of four, hence even, at every cell size and every aspect ratio.

That is not luck. It is the same fact that makes the Miura a corrugation: its creases come in pairs, a mountain and a valley bounding each fold, and a path crossing a fold crosses both. A pattern whose pleats come in pairs has no parity problem; a pattern whose creases are singletons does.

What the search costs on each #

The three sheets that fold behave much as the rest of the family does, and the numbers are small enough to be worth quoting.

Cut out of the plane, a two-period Miura cell has twenty-two free letters over fifteen panels and settles in fifteen nodes — one per panel exactly, which is the reading every repeating pattern in this collection gives on a cut patch. Glued across, eighteen letters over ten panels, eleven nodes. Glued along, twenty letters over twelve panels, thirteen nodes. Glued both ways, sixteen letters over eight panels, ten nodes.

Why this had to wait for a general gluing #

The construction that produced these numbers could not glue a Miura until recently, and the reason is worth recording because it explains why the family was measured last rather than first.

The gluing machinery was written for twist tessellations, which are generated by a construction that fills the plane: given a tiling and a turn, it produces polygons and pleats everywhere, and cutting a rectangle out of that is straightforward.

The Miura is not generated that way. It is drawn as a finite grid of quadrilaterals, with a boundary, by a routine that takes a column count and a row count. A rectangle cut from that would glue the edge of a sample to the edge of a sheet, which is not a gluing of the pattern at all — and the tell would be exactly the kind of nonsense Euler’s number catches.

So what was needed was a way to say: here is a drawing that covers the plane, here is the rectangle it repeats in, and here is where the rectangle should be placed. Three facts, one of which — the period — is a property of the pattern that had never been written down anywhere. Once they existed the rest was the machinery that already worked on twists, unchanged.

That is the general shape of extending a measurement to a new family here. The hard part is stating what the family’s period is; the rest is arithmetic that was already running.

The counts, in full #

Since the argument turns on a handful of integers, they are worth having in one place.

At one period the Miura’s cell holds two interior vertices. Cut out of the plane it has seven free letters and six panels; glued across, five letters and three panels; glued along, six letters and four panels; glued both ways, four letters and two panels. Euler’s number reads one, nought, nought, nought.

At two periods: eight vertices; twenty-two letters and fifteen panels cut out; eighteen and ten across; twenty and twelve along; sixteen and eight both ways.

At three periods: eighteen vertices; forty-five letters and twenty-eight panels cut out; thirty-nine and twenty-one across; forty-two and twenty-four along; thirty-six and eighteen both ways.

Read the letter columns and the savings are four and two at one period, four and two at two, six and three at three — the across gluing saving twice what the along gluing does at every size, because the across direction cuts twice as many creases per unit of edge as the along direction does on this drawing.

Read the vertex column and nothing moves, which is the control.

The asymmetry, and where else it shows #

An anisotropic drawing produces asymmetric counts, and the Miura is not the extreme case in this collection.

On the elongated triangular tiling’s twist tessellation, gluing one pair of edges saves ten letters per cell and gluing the other saves two — a factor of five, on one rectangle. On the square twist, the two directions are alike and the savings are equal. The Miura sits between them at a factor of two.

The consequence for the collection’s earlier language is small and worth correcting anyway. Phrases of the form four letters per cell of rim were derived on isotropic drawings, where the two directions genuinely are alike, and read as though the boundary were a single quantity with a single price. It is two quantities with two prices, and their sum is the only number that belongs to the cell rather than to a direction.

What this does not change about the Miura #

Everything the pattern is known for survives, and it is worth saying so plainly because a result of this shape can read as an attack on the object.

The Miura is still rigid-foldable, still has one degree of freedom, and still collapses in both directions at once. Those are statements about the pattern on a disc of paper, which is what every application of it uses, and nothing above touches them.

What the gluing adds is a statement about a sheet the pattern can be put on, and the sheet is one nobody was using. The parity condition is invisible on a rectangle of Miura because a rectangle has no loop that cannot be shrunk; it appears the moment the rectangle’s edges are joined, and it appears in exactly one of the two ways they can be.

Making one #

A Miura tube is easy to make badly and instructive to make carefully, and doing it settles what the parity is about.

Fold an ordinary Miura from a rectangle: a zigzag of parallel courses, each offset from the last, so that the sheet collapses to a small parallelogram when pushed from two corners. Six columns across is a comfortable size. Then bring the two short ends together and tape them, so the corrugation runs round a tube.

Push the tube’s ends towards each other and it collapses, into a flat ring of paper. Six columns is even.

Now make the same thing with five columns. The corrugation folds exactly as before while the sheet is flat, and the tube does not collapse: pushing the ends together forces the paper to buckle somewhere, and the buckle is a sixth crease appearing where the count needs one.

What the second attempt makes clear is that the failure is not local. Every vertex of the five-column tube is a perfectly ordinary Miura vertex, satisfying every condition this subject checks, folding exactly as it does on a flat sheet. The refusal is a property of going all the way round.

Where the tube’s parity is already known #

Folded tubes are a going concern in engineering, and it is worth saying what is and is not new here.

That a corrugated tube needs an even number of facets round its circumference is known to the people who build them, in the way that practical constraints are known: it comes out of trying, it is passed on as a rule of thumb, and it is not usually connected to anything.

What the count supplies is the connection. The rule is the same rule that says a loop of paper with three creases cannot be pressed flat and the same rule that refuses an odd cell of the plain grid, and all three are one statement: a closed path on a sheet that folds flat crosses an even number of creases, because crossing one exchanges the two faces of the paper.

Stated that way it also says what happens on a sheet with one side, where the rule inverts — which is not a situation any tube is in, and is the reason the general form is worth having anyway.

The four sheets, and which of them exist #

Of the four ways a Miura cell’s edges can be joined, two are objects a person can hold and two are not.

The cut rectangle is an ordinary piece of paper.

The cylinder glued across is a Miura tube, which is manufactured.

The cylinder glued along is a corrugation joined end to end in the direction of its courses — a ring of zigzag rather than a tube of it. That is also physically realisable, and it is the shape of a folded bellows closed into a loop.

The torus is not. Joining both pairs of a rectangle’s edges without stretching does not fit in three-dimensional space, and the object here is a drawing with a rule attached rather than something to fold.

That last one is worth being honest about, because three of the four objects in every table above are paper and one is not. The torus is included because the comparison needs it: it is the sheet with no rim at all, and the rim is what the whole measurement is about.

Why the cell is two rows and not one #

A small point of construction that decides everything above.

The Miura’s drawing looks as though it repeats every row: each row is a zigzag, the next row is the same zigzag, and a person sketching one repeats a single motif. But consecutive rows are offset from each other by half a column, and the offset alternates. So a shift by one row does not carry the drawing onto itself; a shift by two does.

Getting that wrong produces a cell whose edges do not match up when identified, and the symptom is immediate and loud: crease pieces on one edge with no partner on the other, panels that will not close, and Euler’s number coming out at something that is not nought.

The general lesson is that a pattern’s period is a fact to be established rather than read off a picture. The Miura’s is one column by two rows. The grid’s is one by one. The Yoshimura’s drawing repeats every column and its folded state does not, which is a third case again, and the one that shows the drawn period and the folded period are different quantities.

What a tube’s parity is not #

Three things it might be mistaken for, and none of them is right.

It is not about the fold angle. A Miura tube can be folded to any degree between flat-open and flat-closed, and the parity governs whether the fully flat state exists. A five-column tube collapses part of the way perfectly happily and stops.

It is not about rigidity. The Miura is rigid-foldable, meaning its panels can move as rigid plates hinged along the creases, and that property is local to the vertices. A tube with an odd column count is still made of rigid-foldable vertices; what it lacks is a flat state to arrive at, not a motion.

It is not about self-intersection. The tube’s failure is not that the paper would have to pass through itself, which is a separate and much harder question. It is that no assignment of the two faces to the panels is consistent, which is decided before any layer ordering is considered.

Keeping those apart matters because all three are real constraints on folded tubes and a designer meeting a tube that will not close has to know which of the four they have hit.

What is measured and what is inferred #

The parity claim is measured at every size the figures use, by two computations that share no code: a count of creases crossed by a swept family of paths on the flat drawing, and a comparison of the folded motions of two panels the gluing says are one panel.

They agree on every Miura cell built — one, two and three periods, glued each way and both ways. Odd cells across come back turned over by both readings; even cells do not; and no cell glued along ever does.

What is inferred rather than measured is the general statement about pleats coming in pairs. That is an argument about the drawing, it is short, and it is not checked by anything: a family whose creases paired in a way the argument did not anticipate would be a counter-example, and none of the three families here is one.

The general rule the Miura illustrates #

Stated once, because the Miura is the clearest instance and the rule is not about the Miura.

For a repeating pattern and a rectangular cell, each direction of gluing has its own crossing count and its own parity. The counts are properties of the drawing and the cell together; doubling the cell doubles the count and therefore repairs any parity that was odd; and a pattern whose creases come in pairs across a direction has an even count in that direction at every size.

So every repeating pattern has a torus that folds and — unless both directions pair up — half of its cells do not. The Miura pairs in one direction and not the other, which puts it exactly between the grid, which pairs in neither, and the twist tessellations, which pair in both.