A grid glued

Box pleating is the practical compromise that won. It gives up some of the packing efficiency of a free circle arrangement in exchange for creases that land on grid lines, and most complex designers use it — because a grid is easy to draw, easy to fold accurately, and because the whole alphabet of a grid is a small vocabulary that composes.

Its selling point is reliability. A box-pleated pattern’s vertices are grid vertices, its angles are right angles and half-right angles, and the local conditions are satisfied by construction.

Close the sheet and half of them stop folding.

The object #

Take a box-pleated pattern on an $n$-by-$m$ grid, roll it so that the left edge meets the right, and join them.

That is a tube of grid: creases running round it and creases running along it, with $n$ of the latter — the vertical lines of the original pattern, plus the seam if the seam becomes a fold.

The condition #

A path running once round the tube crosses every crease that runs along it — $n$ of them. Crossing a crease exchanges which face of the paper is showing, so the path comes back with the paper the other way up when $n$ is odd.

A sheet on which that happens has no two-colouring and therefore no flat folded state at all.

So a box-pleated tube needs an even number of creases round it, and round it means along the tube’s length in the original flat pattern.

Why the local conditions do not see it #

Every vertex of a box-pleated grid is a grid vertex: four creases meeting at right angles, with three of one letter and one of the other. Developability holds, Kawasaki holds, Maekawa holds, and the big-little-big lemma has nothing to say because no sector is strictly smallest.

All of that is unchanged by the gluing, because the gluing changes no vertex. The cell’s edges are placed to miss them, so identifying the edges joins crease pieces at ordinary interior points and every vertex is exactly where it was, at exactly the same angles.

So a checker reading vertices certifies the odd tube without objection, and the tube cannot be pressed flat.

What a designer has to do differently #

One thing, and it costs an addition.

Count the creases that will run round the finished tube. If the count is even, the pattern has a flat state and everything the designer knows applies. If it is odd, it does not, and no adjustment to the crease assignment will help — the obstruction is a parity and no arrangement of letters changes it.

The remedy is to change the count: add a crease, remove one, or choose the seam’s position so that the seam itself becomes the extra crease.

That last option is worth noting because it is free. A tube’s seam is a line the flattened paper has to bend along whether or not anybody creased it, so it counts, and where the designer chooses to put it decides whether it lands on an existing grid line or between two.

Where the seam goes, and what it costs #

The seam’s position is the one free choice a designer has when closing a pattern, and it is worth thinking about properly, because it is doing three things at once.

It decides the parity. A seam that becomes a fold adds one to the count. So a pattern with an odd number of lengthwise creases can be rescued by placing the seam between two grid lines, where it will crease, and a pattern with an even number is ruined by the same choice.

It decides where the object is weakest. A seam is adhesive or stitching or a fold-over, and it is stiffer and thicker than the paper. Putting it in the middle of a wide panel is different from putting it along an existing crease, and both are different from putting it at a vertex.

It decides how the pattern is cut. A flat pattern for a tube has to be cut somewhere, and the cut is the seam. A designer choosing it is choosing where the pattern’s own repetition is interrupted.

Only the first is a folding question and it is the one nobody checks. The other two are manufacturing questions and they are the ones anybody making a tube thinks about, which is a fair description of why the parity gets missed.

A worked case #

It helps to do one, since the counting is easy to get wrong by one.

Take a box-pleated pattern on a sixteen-by-sixteen grid, intended for a tube. Roll it so that the sixteen vertical lines run round the tube — no, along it: the tube’s circumference is the pattern’s horizontal extent, so the creases that run along the finished tube are the vertical lines of the flat pattern.

There are fifteen of them strictly inside the sheet, plus the two edges, which become one seam. If the seam creases, the count is sixteen: even, and the tube flattens.

If the two edges are joined without creasing — overlapped and glued flat, say — the count is fifteen: odd, and it does not.

So the same pattern gives opposite answers depending on how the join is made, which is a manufacturing decision that nobody would expect to change whether the object folds.

That is the practical heart of it. The parity is not merely a property of the drawing; it is a property of the drawing plus the join, and the join is made at the last step by somebody who is not thinking about folding.

Counting the seam, and when not to #

Since the worked case turns on it, the rule for the seam deserves a clean statement.

The seam counts as a crease when the flattened tube bends along it. Pressing a tube flat produces two fold lines running its length; if the seam is at one of them, it is a crease and it counts.

The seam does not count when the flattened tube has the seam in the middle of a face, lying flat. Then it is a line of adhesive on an unbent panel and it contributes nothing.

Which of those happens is decided by the folding rather than by the designer, since the fold chooses where its own edges are. So the honest procedure is to count the interior creases, check the parity, and know that a seam falling on a fold flips the answer.

For a designer that is an argument for making the seam land on an existing crease deliberately, so that its status is decided rather than discovered.

What happens to an odd tube #

Worth describing, since no flat folded state is abstract and the physical behaviour is specific.

Push an odd tube’s ends together and it collapses partway, then resists. Push harder and the paper buckles, and the buckle is a new crease appearing somewhere — usually near the middle of a panel, running along the tube, in the place that makes the count even.

The tube then flattens, with one more crease than it was drawn with.

So the failure is not that nothing happens. It is that the material solves the problem itself, by adding what the arithmetic requires, and the result is a tube with an unplanned crease in it — which for paper is untidy and for a manufactured structure is a failure, since an unplanned fold line is where the thing will crack after a few cycles.

That is the practical cost of the condition, and it is why the empirical rule exists among people who make these objects.

Why box pleating is the case where it matters #

The condition applies to any pattern rolled into a tube. Box pleating is where it is most likely to catch somebody, for three reasons.

The counts are large. A box-pleated design uses a fine grid — sixteen, thirty-two, sixty-four divisions — and a designer keeping track of a great many creases is not counting their parity.

The patterns are designed flat and rolled afterwards. The uniaxial method produces a flat pattern; anybody wanting a tube makes one and joins it, at which point the condition applies to a pattern that was never checked against it.

The reliability is the selling point. Box pleating is chosen precisely because it does not surprise people, so a designer using it is not looking for a condition they have never met.

What was measured #

The claim is about box pleating and the measurements are on a plain grid, which is worth being straight about.

The objects built are cells of the unit grid — creases along every line of a square lattice — at one, two, three, four and six periods, glued across, along and both ways. Each was asked two independent questions: how many creases a loop crosses on the flat drawing, and whether the folded motions of two identified panels differ in orientation. The two agree at every size.

A box-pleated design is a grid with a subset of its lines creased and some diagonals added. The parity argument does not care which lines are creased or what the letters are; it counts the creases a loop crosses, and a loop round a tube crosses whatever runs along it.

So the measurement is on the simplest member of the family and the argument covers the family, because the argument is a count rather than a property of the particular pattern.

What has not been measured is a real box-pleated design rolled into a tube. That would be a worthwhile check and it needs a design, which this collection does not produce.

The other direction, which is free #

There is a second gluing available and it behaves differently, which is worth knowing.

Rolling the pattern the other way — joining the top edge to the bottom rather than the left to the right — gives a tube whose creases run round it in the other sense, and the count that matters is the number of horizontal lines.

On a square grid the two directions are alike, so the condition is the same condition. On a box-pleated design they are not: a design has different numbers of creases in the two directions, and one gluing can be even while the other is odd.

So a designer with a pattern that will not close one way should try the other, and the answer may differ. That is a free thing to try and it is not obvious that it would help, since the pattern is the same pattern is the natural thought.

The general version is that the two cylinders of a rectangle are different sheets, and on an anisotropic drawing they differ in every count including this one.

What the manufactured objects do #

The condition is already satisfied in practice, by people who arrived at it empirically.

Folded tubes are made — deployable booms, packing structures, energy absorbers — and the ones that collapse have even facet counts round them. That is known as a rule of thumb and it is passed on as one.

What the argument adds is that the rule is exact rather than approximate, that it has the same cause as a loop of paper with three creases refusing, and that it can be checked before anything is cut.

The grid’s own reliability, in perspective #

Box pleating is chosen because it does not surprise people, and it is worth asking what that reliability actually consists of, since the surprise here is real.

A box-pleated pattern’s vertices are all of one or two kinds, they satisfy the vertex conditions by construction, and the modules compose without interfering. That is a genuine and substantial reliability and it is entirely local.

What it does not include is anything global. A pattern all of whose vertices pass can still fail to fold, and box pleating’s guarantee is a guarantee about vertices.

On a flat sheet that turns out not to matter much, because a flat sheet has no loops that cannot be shrunk and the global condition is implied by the local ones. So box pleating’s local guarantee is, on a disc, effectively a global one — by an accident of the sheet rather than by anything about the method.

Close the sheet and the accident stops holding. The local guarantee is exactly as strong as it always was and it stops being sufficient, which is a clean illustration of what local has been buying all along.

Why the grid is the extreme case #

Among the patterns this collection can glue, the grid is the one most exposed to the condition, and the reason is structural.

A path round a cell of the grid crosses one crease per period. So the count is the number of periods, and half of all sizes are odd.

A path round a cell of the Miura crosses one per period in one direction and four in the other, so one of its directions has a parity and the other cannot fail at any size.

A path round a twist tessellation’s cell crosses an even number in both directions at every size, because its pleats come in pairs, so it never fails at all.

The general rule is that a pattern whose creases cross a loop in pairs has no parity problem and one whose creases are singletons does. The grid’s creases are singletons — one line per unit, in each direction — which is exactly what makes it the simplest pattern and the most exposed.

Box pleating inherits that, since box pleating is a grid with some of its lines creased. Which is a slightly uncomfortable observation: the property that makes the method easy to draw is the property that makes it fail half the time when the sheet closes.

Where the condition sits among the others #

Box pleating already has constraints and it is worth placing this one among them, because a designer meeting a tube that will not close has several candidate explanations.

The pattern does not fold flat at all. Check the vertices. Box pleating makes this unlikely by construction.

The layers collide. Two panels have to pass through each other, which the vertex conditions never see and which is the hard half of the subject. Common in dense designs.

The paper is too thick. Thickness accumulates and a design with many layers stops closing physically long before it stops closing mathematically.

The tube’s crease count is odd. New, exact, cheap to check, and the only one of the four that can be settled before anything is cut.

A designer meeting a stubborn tube should check the last one first, precisely because it is the cheapest and because none of the other three explanations would ever lead them to it.

Three sentences for a designer #

Count. How many creases will run along the finished tube, including the seam if it will fold. Even is fine; odd has no flat state.

Choose the seam. It is the one free variable and it flips the answer. Landing it on an existing crease makes its status decided rather than discovered.

Try the other roll. The two ways of closing a rectangle are different sheets with different counts, and a pattern that fails one may pass the other.

None of that changes how a pattern is designed. All of it happens after the design and before the tape.

What the condition does not touch #

Two things, and both are the reasons box pleating is used.

The vocabulary still works. Every box-pleating module — the pleat, the offset, the level shifter, the sink — is a local arrangement, and local arrangements are unaffected by the sheet.

The grid still divides. Binary division on a grid is a construction on the flat sheet, performed before anything is joined, and it is untouched.

So the correction is a single check at the end rather than a change to how anything is designed, which is the right size for a condition that only exists on a sheet the design method never mentions.

The general form #

For any pattern and any tube: count the creases a loop round the tube crosses, including the seam if the seam becomes a fold, and require the count to be even.

The count is the same for every loop, at every height, so it is a property of the tube rather than of the path. That is checked here by sweeping the path over every height that misses a vertex, and by an independent computation on the folded motions that agrees with it on every sheet measured.

An addition, and the answer is a proof.