The rows are free

The pattern cheapest to trust priced a one-shot deployable by the tests it needs. Demonstrating that a structure will open takes a number of consecutive successes proportional to its hinge count, and what the structure delivers is layers of compaction, so the pattern cheapest to trust is the one with fewest hinges per layer. Refined from two cells a side to eight, the waterbomb settled at 2.47 hinges a layer and the Yoshimura at 1.47, and the Miura — the pattern that actually flies — climbed at every step: 1.32, 2.20, 2.95, 4.33, 5.69.

The explanation offered was that the Miura’s folded footprint grows with the sheet while the other two fold onto a footprint about a few cells in size. That is true of a Miura refined equally in both directions, which is how the families figure refined it. It leaves out that a sheet has two directions and the Miura treats them completely differently.

The footprint does not grow in one of them at all.

Two directions, two behaviours #

A Miura is drawn with two families of crease. One family is straight: lines running across the sheet, with the zigzag’s vertices strung along them. The other is the zigzags themselves, crossing the straight lines and changing direction at each one by twice the slant. Between two neighbouring straight creases lies a band of the sheet, which these essays call a row; between two neighbouring zigzags lies a column.

Fold a Miura four columns wide with two rows and with six. The folded states are drawn above at one scale, and their smallest boxes are identical: 0.939 by 2.436 in both. Four more rows went into the second, and the folded paper did not move outward by a hair — the stack is three times deeper and exactly the same shape.

Fold one four rows deep with two columns and with six. Now the box goes from 0.939 by 1.750 to 0.939 by 3.122. It keeps its width and gets longer.

Why one direction stacks and the other walks #

A flat fold places every panel by reflecting it across each crease on a path from a panel held still, and two facts about reflections decide everything here.

Two reflections in parallel lines are a slide, square to the lines, by twice the distance between them. Two reflections in lines that meet are a turn. So what happens to a panel two bands away from a held one depends on whether the two creases between them are parallel, and how the slide they make compares with where the panel was drawn.

Across the rows, the creases are the straight ones, and every straight crease is parallel to every other. A panel two rows up is two cell heights away, square to the straight creases, and the two reflections slide it by twice the one-cell spacing, square to the same creases — exactly back onto the held panel. The slide and the offset are the same vector, so nothing is left over, and every second row lands where the first did. That is an accordion, and the rows of a Miura are one.

Across the columns, the creases are the zigzags, and within one row every zigzag’s segment is parallel to its neighbours’, slanted at $a$ to the vertical. A panel two columns over is two cell widths away along the row. The two reflections slide it square to the slanted segments, by twice their spacing, which is $2\cos a$. The offset and the slide are no longer the same vector: they differ by $2\sin a$, and the difference points along the slanted crease. So every second column lands $2\sin a$ further along the zigzag direction than the first — a step of $\sin a$ a column, which is the walk.

That is the whole mechanism, and it predicts both the direction and the size of what the figures measure. The folded stack runs along the zigzags, which is why the folded strip lies at its own slant — exactly at the slant angle to the drawing — and it lengthens by the sine of the slant for every column and by nothing for every row. A zigzag with no slant would be a straight crease, the step would vanish, and the Miura would be a map fold stacking both ways; the slant that gives the pattern its single freedom is the same slant that makes it walk.

The box, in closed form #

Because the rows stack exactly and the columns walk by a fixed step, the smallest box a flat-folded Miura fits in has a closed form, and every one of the figures here is checked against it on the computed folded state rather than drawn from it:

$$\cos a \;\times\; \left(\sec a + c\,\sin a\right)$$

for a sheet of unit cells at slant $a$ with $c$ columns — and no term in the number of rows. At a slant of 0.35 that is 0.9394 wide, and 1.0645 plus 0.3429 for every column long; at a slant of 0.80 it is 0.6967 wide and 1.4353 plus 0.7174 a column. The width is a single row’s height folded to the slant; the length is one panel’s slanted side plus a step of the sine of the slant for each column.

The formula has the two behaviours in it. Rows appear nowhere, so any number of them fold into the same box. Columns appear once, linearly, so the box grows in proportion to them.

Rows: hinges per layer that settle #

Take four columns and add rows. The box stays at 0.939 by 2.436. The paper grows by one row of four cells each time, and all of it lands in the same box, so the mean depth of the stack rises in proportion: 4.07 layers with two rows, 8.14 with four, 16.28 with eight, 24.43 with twelve. The hinges rise in proportion too, seven for every row added once the first is in.

So hinges per layer settle along the rows: 2.46, 2.78, 2.95, 3.11, 3.19, 3.24, 3.28 as the rows go from two to twelve, closing on a limit a little above 3.4 that is fixed by the columns. It is exactly the behaviour the waterbomb and the Yoshimura showed, and for exactly their reason: the footprint stops growing, so each new row of hinges buys a new layer.

A four-column Miura with many rows is therefore as cheap to trust, per layer, as a waterbomb. It is not a special Miura. It is the ordinary pattern laid out long in the direction its footprint does not care about.

Columns: hinges per layer that climb #

Take four rows and add columns instead.

The box lengthens by 0.3429 each time, the sine of 0.35. The paper grows by four cells a column, but the footprint grows too, so the depth of the stack rises ever more slowly: 6.04 layers with two columns, 8.14 with four, 9.83 with eight, 10.55 with twelve. The depth has a ceiling. With many columns each one adds four cells of paper and about $\sin a \cos a$ of footprint, so the mean depth closes on a limit near $2r/\sin 2a$ — about 12.4 layers for four rows at this slant — and never passes it.

The hinges have no ceiling, since every column adds seven. So the hinges per layer climb at every added column: 1.66, 2.33, 2.95, 4.12, 5.29, 6.43, 7.58. Past a few columns every column is almost pure cost — seven more hinges to test, a fraction of a layer more compaction.

That is the whole of the climb the families figure found. Refining a Miura equally in both directions adds columns as fast as rows, and the columns’ cost dominates, since a row’s cost is bounded and a column’s is not.

A ceiling on depth, and what it means for a panel #

The ceiling on depth along the columns is worth stating as a design number, because it is the number a folder of a real panel runs into. A Miura $r$ rows deep can never fold to more than about $2r/\sin 2a$ layers, however many columns it has. At a slant of 0.35 that is 3.1 layers for every row; at 45 degrees, where the walk adds most footprint, it is exactly two a row. Adding columns past a few brings the stack ever closer to the ceiling and never through it.

Along the rows there is no ceiling at all. Each row adds its own layer over the whole footprint, so depth rises in proportion to the rows without limit, which is why a long-in-rows Miura can be made as compact as wanted by lengthening it, and a long-in-columns one cannot.

The practical reading is that a Miura’s compaction is chosen in its rows and its footprint in its columns, almost independently. A designer who needs a panel to fold into a slot of a given length chooses the columns from the closed form above; a designer who needs a given number of layers chooses the rows. The two decisions interact only through the slant.

The same cells, five proportions #

The design consequence is immediate, and it is large.

Take sixty-four cells and arrange them five ways. Two columns by thirty-two rows need 1.94 hinges a layer; eight by eight need 5.69; thirty-two columns by two rows need 16.14. The hinge counts are 94, 108, 112, 108 and 94 — almost the same bill — and the layers they buy run from 48.3 down to 5.8. The two-column layout folds its whole sheet into a box 0.94 by 1.75; the thirty-two-column one needs a box 0.94 by 12.04.

In the terms the one-shot analysis priced it in, the difference is a factor of eight in the tests needed for each layer of compaction delivered, between two layouts of one pattern with nearly the same number of hinges. The crease count is a reliability budget found the probability of a full deployment falling as a power of the hinge count; that budget is spent almost identically by all five arrangements, and what it buys differs eightfold.

The long-in-rows Miura is a better deployable than any square one, on this measure, and it is not a new pattern. A panel meant to fold to a small footprint should be laid out with its straight creases many and its zigzags few — long across the straight creases, short across the zigzags.

What a column costs, slant by slant #

The step each column adds is the sine of the slant, and the strip it adds it to is the cosine wide, so the footprint a column adds is their product, half of $\sin 2a$. That predicts how the cost of a column depends on the slant: largest at 45 degrees, falling toward both a gentle slant and a steep one.

Measured between four columns and eight at nine slants, the extra hinges per layer that one column costs are 0.285 at a slant of 0.15, 0.585 at 0.35, 0.899 at 0.785 — which is 45 degrees — and back down to 0.598 at 1.2. Divided by $\sin 2a$ the nine numbers lie between 0.89 and 0.96, close to the $(2r - 1)/2r = 0.875$ the counting predicts for four rows once the walk is all that is left.

So a gentle slant walks slowly, and a Miura at 0.15 radians can be given many more columns before they cost much. That is not a free choice: a sheet with one freedom is the Miura’s reason to exist, and as the slant goes to nothing the pattern degenerates toward a plain grid, which folds flat by a sequence of separate motions rather than one. How gentle a slant can be before the single freedom stops being useful is a rigid-folding question, and this measurement cannot answer it.

A variant that was tried #

The essay this one continues suggested that a Miura variant whose rows stacked onto one footprint might settle. The rows turn out to stack already; what walks is the columns, and the obvious variant to cancel the walk is to mirror every other zigzag, so that one column’s step forward is followed by the next column’s step back.

That was built and folded. Mirroring alternate zigzags turns the parallelogram panels into trapezoids and keeps every vertex flat-foldable; the pattern folds flat. It walks less and does not stop walking: at four rows deep, its hinges per layer are 4.69 at eight columns against the plain Miura’s 5.29, and its folded box grows in width as well as length, since the mirrored columns no longer step along a single line. It is one variant of many, and it was not tested for whether it folds with a single freedom, which is the property that would make it worth having.

What these numbers rest on #

Every fold is to the flat, zero-thickness state. A real deployable is packed by folding until something stops it, and a real stack of thirty-two rows is thirty-two sheets thick; the pile, not the panel is where that cost is measured, and it grows with the rows, which is exactly the direction this essay recommends. A long-in-rows Miura is cheap to trust per layer and thick per layer, and a design has to weigh the two.

A hinge is an interior crease segment, counted as the one-shot analysis counted it: every segment between two vertices is a place the deployment can fail, whether it lies on a straight crease or a zigzag. Weighting the two families differently — a straight crease might be more reliable than a zigzag, being part of one long line — would change the numbers but not which direction walks.

Layers are the mean depth over the area the folded paper covers, sampled; the smallest box is computed exactly and agrees with the closed form to a millionth on every figure. The layer means are good to about a per cent, and the settling along the rows is measured to twelve rows and extrapolated beyond.

And a test is a success of the whole deployment. The one-shot analysis took the tests needed to be proportional to the hinges; that is the assumption this essay inherits, and it is conservative for a pattern with one freedom, whose hinges are not independent.

Why an accordion was always inside the Miura #

The unexpected thing is how exactly the Miura contains an accordion. Along its straight creases it is one — reflections in parallel lines, a stack that deepens and never spreads — and everything the Miura adds to an accordion is in the other direction, where the zigzag’s reflections in slanted lines compose into a slide. The pattern’s celebrated property, one freedom driving the whole sheet, comes from the zigzags; its cost, a footprint that spreads, comes from the same place.

That is also a clean reading of what makes the Miura a Miura rather than a pleat. The pair of shrink factors the plane of shrinkage reads was taken to be the Miura’s signature; what distinguishes it from an accordion is not a ratio of two numbers but the fact that one of its two directions walks. Measured on the smallest box, the Miura is an accordion in one direction and a slide in the other.

Still open: whether the walk can be cancelled with one freedom kept #

The walk is a slide, and a slide can be cancelled by an equal slide the other way. A quadrilateral mesh whose columns alternate their slide direction, and which still folds with a single freedom, would stack onto one footprint in both directions and settle in both. The mirrored variant is the simplest candidate and does not do it, because mirroring changes the panels’ shapes and fans the columns. Whether any member of the family the Miura belongs to does is a search over that family with one number to minimise — the growth of the smallest box with the columns — and the closed form above is the target it would have to beat.

The other open thing is practical. A long, narrow Miura is cheap to trust per layer and deep per layer, and splitting a sheet found reliability improving when a sheet is divided into separately driven modules. A sheet divided along its zigzags into narrow long-in-rows Miuras would get both benefits at once — few columns each, so no walk worth the name, and independent modules, so a stuck hinge costs one strip — and whether the joins between modules cost more than the walk they save is a measurement nobody has made.

The habit worth carrying is about refinement. When a cost grows as a pattern is refined, refine it one direction at a time before believing it grows in both. Refining equally is the natural way to make a pattern finer, and it averages two behaviours that have nothing in common.