The slant belongs to the line

The slant that stacks in one depth found that a folded Miura’s pile is set by one number. The rows fold exactly onto each other; each column’s folded image steps along the strip from the last; and the columns overlap $2/\sin 2a$ deep on average for a slant $a$. When that is an even whole number — at 45, 15 and 9.74 degrees — the pile is one depth everywhere but the ends, and every panel away from the ends lies over a single depth, so a thick panel of one thickness can follow it.

It ended on the pattern that could not have that. The tapered corrugation — the fold a plicate leaf packs into, broad in the middle and narrow at both ends — changes the width of its columns along the sheet, so its step is not constant and no single slant puts every pair of columns on an even count. The question it left was whether a taper could be graded so that the count stays on one step: a slant varying with the cell size.

It can, and the reason it can was stated long ago and read too narrowly. The slant of a corrugation is not a property of the pattern. It is a property of each zigzag line.

A column’s pile in two lengths and an angle #

The count the earlier essay used was drawn for columns and rows one unit each. Let both go, and the count becomes $2h/(w\sin 2a)$ for a column $w$ wide in rows $h$ high: the column’s folded image runs $h/\cos a$ along the strip, and the next column’s image starts $w\sin a$ further on, so the images overlap their ratio deep.

The folded depth maps agree at every setting where the count is whole: a column one unit wide in a row one unit high at fifteen degrees piles four deep; half as wide, eight; half as high, two; half as wide and half as high, four again; twice as wide at forty-five degrees, one. Width and height enter as a ratio, and the slant through the sine of twice itself. For a taper this is the whole problem. A column half the width of its neighbour piles twice as deep at the same slant, and a leaf’s corrugation runs from its broadest column to one a third as wide.

Where the count comes from #

The count is worth deriving rather than quoting, because the derivation shows which parts of the pattern it reads. Fold a corrugation flat and hold one panel still. Every other panel reaches its folded place by reflections across the creases between it and the held one. The row creases are straight lines, and reflecting across them brings every row onto the one below: that is why the rows stack exactly and every depth is a whole number of rows. Within a row the zigzag segments of one row all lean the same way, so the reflections across them are reflections across lines that are parallel — and two reflections across parallel lines are a slide, perpendicular to the lines, by twice the distance between them.

So a row’s panels fold into a strip that runs along the zigzags. Each panel keeps its position along that direction, since a reflection across a line does not move anything along the line; measured along it, a panel of height $h$ spans $h/\cos a$, and the panel one column over starts $w\sin a$ further along, because that is how far along the zigzag direction its corner sits. The images overlap $h/(w\sin a\cos a)$ deep, which is $2h/(w\sin 2a)$, and every term in it belongs to one column and one row: its own width, its own height, and the lean of the zigzags on either side of it.

That last phrase is the one the taper needs. The count of a column reads the zigzags at its two edges and nothing else, so if those zigzags may lean differently from the rest of the pattern, each column’s count can be set on its own.

Why the slant is the line’s #

A leaf packs by corrugating found which way a corrugation may taper. At an interior vertex the two row creases are collinear, and the zigzag’s two segments leave at the same lean above and below; Kawasaki’s alternating sums come to a half-turn exactly when the rows above and below have the same height. So a corrugation may taper along its fold lines and may not taper across them: vary the row heights and the alternating sums go to 186.4 and 173.6 degrees, and the pattern does not fold flat. The column widths never enter the condition, and every one is free.

The same reading says more than that, and it was not drawn out. The angles at a vertex are set by the zigzag line through it and by nothing else.

A zigzag leaning $a$ from upright meets the straight row crease at $90° - a$ on one side and $90° + a$ on the other, above and below alike, so the four sectors are $90 - a$, $90 + a$, $90 + a$ and $90 - a$ in turn and the alternate sums are $180°$ whatever $a$ is. Each zigzag line is checked against its own lean only, and two zigzags a column apart need not lean alike. A corrugation with a different slant on every zigzag line passes Kawasaki and Maekawa at every vertex; the one with rows of two heights fails. The freedom the taper has in its columns is matched by a freedom in its zigzags, and it is the second one the pile needs.

Leaning each zigzag for its columns #

With a slant per line, the count can be put on a chosen number $k$ column by column. A zigzag between two columns leans $a = \tfrac12\arcsin\!\bigl(2h/(k\bar w)\bigr)$, where $\bar w$ is the mean width of the two columns beside it, so that both columns pile close to $k$ a row.

The test taper has twenty-four columns running from a third of a unit wide at the ends to one unit in the middle, in two rows one unit high. At one slant its columns pile anywhere from 3.8 to 10.9 deep a row, broad columns near four and the narrowest near eleven. Leaning each zigzag for its columns brings every column more than a little wider than half a unit to within eight per cent of four, with the zigzags running from fifteen degrees in the broad middle to forty-five at the narrow ends.

Four columns do not get there, and cannot. A column narrower than $2h/k$ piles deeper than $k$ at every slant, because $\sin 2a$ is at most one and a zigzag leaning forty-five degrees already gives the shallowest pile it can: $2h/w$. The four end columns of this taper, a third and two fifths of a unit wide, pile 5.8 and 4.6 at best. For a target of six deep a row every column of the taper is wide enough and every one reaches it; the dial runs from four to ten, and from six up the narrow ends stop being a limit.

The narrowest column sets the target #

The four end columns that miss four deep are not a failure of the rule; they are a fact about the taper, and it fixes how deep the graded pile has to be. A target $k$ is reachable by every column exactly when $k \ge 2h/w_{\min}$, since the narrowest column piles at least $2h/w_{\min}$ at any slant. For the test taper, with its narrowest column a third of a unit wide in rows one unit high, that is 5.8, and six is the shallowest even pile every column can make. At six deep the rule reaches every column, and the zigzags run from about ten degrees in the middle to 37 at the ends.

The printed tapered corrugation, drawn for a leaf, has columns from 0.09 to 0.22 of its sheet in rows 0.17 high. Its narrowest column piles at least $0.34/0.09$, about 3.8, so four deep a row is reachable by every one of its columns — and at the printed single slant of 24 degrees its columns pile from about two to five a row. A designer grading a taper chooses the target first, from the narrowest column, and the leans follow from it: a deeper target lets every zigzag stand more upright, which costs less at every vertex and piles deeper everywhere.

What the pile looks like #

Folded and read off the depth map, the difference is not in how much of the footprint reaches the target. It is in how much goes past it.

At four deep a row, equal columns put 78 per cent of their footprint there, the taper at its best single slant 55, and the graded taper 62. The single slant is sixteen degrees, chosen by trying every whole degree and keeping the one with most of the footprint four deep. The gain from grading, seven points, is modest on that measure. What grading changes is the tail: at the single slant 22 per cent of the footprint piles deeper than four, five and six and seven deep over the narrow columns; graded, 3 per cent does, and the deepest pile falls from seven to six. At six deep a row the shares are 66, 41 and 51, and the part piled past six falls from 25 per cent to 2.5.

The part below the target grows instead — a third of the graded taper’s footprint is shallower than four, against a fifth at the single slant — and that part is the ends. Two zigzags leaning differently are not parallel, so the folded strip fans at the narrow ends, where neighbouring leans differ most, and the end columns’ images presumably spread out there rather than overlapping fully. Grading trades depth past the target for shallowness at the ends, and for anything that stacks thick panels the trade is the right way round: a pile shallower than a panel’s design depth is a gap a spacer fills, and a pile deeper is a collision.

The rows stay out of it #

The earlier essay’s second open question was whether the rows could be put to work — a pattern whose rows walked as its columns do would have a second count, and the pile would depend on two numbers. The vertex check answers half of that for corrugations. A corrugation’s rows cannot differ in height, so they cannot be graded, and they fold exactly onto one another whatever the zigzags do: every depth on the graded tapers is even, two rows folding onto two. A row only multiplies the pile. Grading lives entirely in the columns and their zigzags, which is why a taper along the fold lines can be evened and a taper across them cannot even be folded.

The walking rows the question imagined would need a pattern whose straight creases are not straight — a quadrilateral mesh outside the corrugations altogether, where the row creases bend at every vertex. Whether such a mesh folds with one freedom and walks in both directions is still the search over the Miura’s family that the earlier essay proposed, now with the parity known for one direction and the count known for both.

What it costs the paper #

The leaning zigzags are not free. The floor at degree four prices a corrugation’s vertex exactly as a function of its lean: $\tfrac14(4\sec a - \tan a + a)$ band widths squared creased twice, 1.03 at fifteen degrees, 1.06 at the printed Miura’s twenty, and 1.36 at forty-five. The graded taper’s narrow-end vertices lean forty-five degrees, so each doubles about a third more paper than a vertex of the taper at sixteen degrees does. The even pile is paid for in crease density, at the vertices on the narrowest columns — which are the columns with least paper between their creases to spare.

The other cost is that the pattern is no longer a Miura in the strict sense. Two zigzags leaning differently are not parallel, so the columns between them are trapezoids rather than parallelograms, and the folded strip is not straight. Whether such a pattern still folds rigidly, with the single freedom the Miura is prized for, is not something the depth maps can say; the family the Miura belongs to is where that would be settled, and a graded corrugation is a member of it only if its vertices’ fold angles still agree round every panel.

What the depth maps assume #

The folded state is the flat one, read as geometry. Every panel is placed by reflecting it across the creases between it and a fixed panel, and the depth at a point counts the panels over it. That places the panels without ordering them, and for the graded corrugations no layer order was searched: even a graded corrugation of seven columns has fourteen panels, past what the ordering search finishes. That every vertex passes Kawasaki and Maekawa is checked; that the whole folds flat without the layers passing through one another is not.

Panels have no thickness. The purpose of evening the pile is to let thick panels follow it, and a panel is not the unit of depth found how much a thickness technique depends on each panel lying over one depth. The maps here are for zero-thickness panels; with real panels a graded taper’s fanned ends would also shift, and by an amount the thickness sets.

And the taper is a model leaf. Its widths follow half a sine wave from a third of a unit to one, over twenty-four columns so that the ends are a small part of the strip. The printed tapered corrugation has seven columns, on which the ends are nearly everything; its pile is graded, and a proud hundredth buys almost nothing found its deepest depth a sliver of 0.78 per cent of its footprint — presumably over its narrowest columns, which are the ones a graded slant would lean hardest.

How the corrugations were read #

Each corrugation is built as two rows of columns with a slant for every zigzag line, checked by the local conditions at every vertex, and folded flat; its depth map is sampled on a grid of 220 by 220 points and every depth is required to be even, since the two rows fold exactly onto each other, then halved. The law $2h/(w\sin 2a)$ is required to match the commonest depth on equal columns at every setting where it is whole. The best single slant is found by trying every whole degree from six to forty-five. The graded taper is required to put more of its footprint at the target than the best single slant, and no more than equal columns do.

Still open: the smallest lean that evens the pile #

The rule used here leans each zigzag for the mean of its two columns, which is a guess, and a better one would even the pile further. Each zigzag’s lean enters two columns’ counts, and choosing twenty-five leans to put twenty-four columns on one count is a small least-squares problem; solving it, and weighting the narrow columns’ vertex cost against their pile, would give the graded taper that a designer who cares about both would draw.

Whether the graded corrugation keeps one freedom is the question that decides whether it is useful. A plicate leaf opens by corrugating, and a deployable panel wants one input to open it; a graded corrugation that folds flat but not rigidly would be a leaf that packs evenly and opens badly. The rigid-folding test on trapezoidal quadrilateral meshes is the one the rows are free looked for in a different form, and it applies here directly.

Sideways from here, three kinds of pile sorted the printed patterns into uniform piles, islands and graded piles. A graded pile was a property of the pattern there; here it is a property of a choice — one slant — that the pattern did not force.

The habit worth carrying is about parameters that look global. When one number seems to govern a whole pattern, find the smallest piece of the pattern the condition on it actually reads. The slant was read off the pattern because every zigzag of the printed ones leaned alike; Kawasaki reads it one line at a time, and a taper needed exactly that freedom.