Parallel creases forgive every lean

The slant belongs to the line found that a corrugation’s slant is not one number for the whole pattern. At every vertex the straight crease runs through and the zigzag leans the same amount either side of upright, so Kawasaki’s condition reads each zigzag line against its own lean and nothing else. That freedom was what a tapered corrugation needed: lean each zigzag for the width of the columns beside it, and a leaf taper whose narrow columns piled eleven deep a row at one slant piles close to four almost everywhere.

It left one question, and it called it the one that decides whether the graded pattern is useful. A plicate leaf opens by corrugating and a deployable panel wants one input to open it, so a graded corrugation that folds flat but does not fold rigidly would be a leaf that packs evenly and opens badly. The essay pointed to the family the Miura belongs to, which reduced rigid folding in meshes of this kind to a condition on a table of cosines, and said the graded corrugation belonged to that family only if its fold angles still agreed round every panel.

They do, at every stage of the motion. The reason is short, and it shows that one practical rule in the family essay is stated more broadly than it holds.

The table that decides it #

The family essay’s argument, applied here, goes as follows. At a vertex where a straight crease runs through and the other crease reflects in it, the two fold angles are tied by a closed form,

$$\tan\frac{\rho_{\text{straight}}}{2} = \cos\beta \cdot \tan\frac{\rho_{\text{zigzag}}}{2},$$

where $\beta$ is the angle between the two creases. A straight crease is one crease along its whole length, so it has one fold angle; a zigzag line keeps one fold angle too, because its two segments at every vertex meet the straight crease at equal and opposite angles. So the table of $\cos\beta$ — one entry for each vertex, laid out by zigzag and by straight crease — must be a column of numbers times a row of numbers. That is the rank-one condition, and it is a condition on the flat pattern rather than on the fold.

Now read the table for a corrugation. The straight creases are parallel, and a zigzag line leaning $a$ meets each of them at $90° - a$ or $90° + a$, alternately. Every entry on the row belonging to that zigzag is $\pm\sin a$, the same number all the way along. A table whose rows are each a constant is a column of numbers times a row of ones. It is rank one whatever the constants are: whatever each zigzag’s lean, however unequal the leans, and whichever order they come in.

So the graded corrugation cannot fail the condition, and neither can any corrugation built the same way. The freedom the flat pile needed — a lean for every zigzag — and the freedom the rigid motion leaves are the same freedom. Every straight crease folds at one shared angle $\rho_s$, and each zigzag folds at the angle its own lean sets:

$$\tan\frac{\rho_z}{2} = \frac{\tan(\rho_s/2)}{\sin a}.$$

The relation says what the motion looks like before any panel is placed. A zigzag leaning 45 degrees, at the taper’s narrow ends, folds to 109 degrees when the straight creases are at 90; one leaning 15 degrees, in the broad middle, has already folded to 151. The upright zigzags lead and the steep ones lag, and every zigzag reaches flat exactly when the straight creases do, because the right-hand side goes to infinity with $\tan(\rho_s/2)$ whatever $\sin a$ is. The pattern starts flat, folds on one input and arrives flat, as the Miura does.

Measured rather than argued #

An argument from a closed form is worth checking against something the closed form cannot promise, and the family essay’s own method provides one. Every panel is placed in space by turning it about the creases between it and a fixed panel, through the fold angles the relation gives; each interior corner is then computed once for every panel that owns it, and the largest disagreement between two panels about where a shared corner went is a length a reader can see. The relation supplies the angles, but it cannot make the panels meet: on a pattern whose table is not rank one the angles disagree along some crease, and the panels open up by exactly that disagreement.

On the leaf taper at one slant of sixteen degrees, and graded for four and for six deep a row, every table is rank one to within $10^{-14}$ and every panel meets its neighbours to $3 \times 10^{-14}$ of a row’s height at straight-crease angles of 20, 60, 100, 140 and 170 degrees. The graded tapers lean their zigzags from 15 to 45 degrees and from 9.8 to 37.3.

The same test can refuse, and it is shown refusing. Slide one interior vertex of the graded taper a tenth of the way further along the direction its neighbour’s flat-folding condition forces, which keeps every vertex flat-foldable, and the loop round a face that contains it no longer closes: it misses by 0.22 radians. A test that passes only the patterns it should is a test, and this one passes the graded corrugation for the reason given above and no other.

Where the rule about copies holds #

The family essay drew its sheets by a single rule — a row that reflects in each of a fan of straight lines — and ended with a practical rule for designers: in such a mesh the rows have to be copies of one another, scaled about the fan’s centre if the columns meet, translated if they are parallel.

The first half is right. When the straight creases fan, a row reflecting in them changes its angle at every crease it crosses, so the entries along a row are not constant, and rank one then forces every row to change in step with the first — a scaled copy. The second half does not follow. When the straight creases are parallel, a reflecting row keeps its angle and its table row is constant, so any rows at all satisfy rank one, copies or not. The graded corrugation is exactly such a mesh, with rows that are not translates of one another, and it folds.

That was checked on the family’s own construction as well as on the corrugation. The parallel member built by the family essay’s rule, with one row’s direction changed so that it is no longer a translate of the others, has a rank-one table to $4 \times 10^{-16}$ and folds with its panels meeting to $9 \times 10^{-16}$. The same change on a fan leaves the table with the wrong pattern of signs to be rank one at all. In the parallel member the rows are free; it is the fan that ties them.

The correction is small in words and large in what it lets a designer do. Every corrugation with straight parallel creases and zigzags of any leans is a rigid mechanism with the Miura’s single input — including the tapered, graded and irregular ones a leaf or a panel array might want — and none of them has to be checked for rigidity once its straight creases are known to be parallel and its rows known to be equal in height.

Why the Miura looked so special #

The only pattern that moves made the case that a rigid motion is not generic: move one vertex of a Miura by a thousandth of a panel and the sheet has no isometric folded position of that kind, and the failure is first order in the displacement. The refusal in the closure table is the same experiment on the graded taper, and it fails the same way. So the graded corrugation is not generic either, and the question is what it shares with the Miura that a displaced vertex does not.

The answer is two properties, and the flat pattern shows both at a glance. Every straight crease is a straight line, running through every vertex it meets, and every pair of them is parallel. The first makes the straight creases single creases with one fold angle each; the second makes a reflecting zigzag keep its angle to all of them. A displaced vertex breaks the first, since the straight crease through it now bends there; a fan breaks the second. The lean of each zigzag was never part of what made the Miura move, and the Miura’s equal leans were a choice about the pile, not a condition of the motion.

That reframes what the essays on thick panels and piles have been doing. Three kinds of pile sorted the printed patterns by the shape of their folded depth, and the slant essays found how one number, and then one number per zigzag, sets a corrugation’s pile. Every one of those choices was free for the motion as well, so a corrugation can be graded for its pile without any accounting on the kinematic side, and the only kinematic consequence of grading is the uneven rate at which its zigzags close.

What grading costs the motion #

The motion has a price and it is paid in shape rather than in rigidity. In a Miura every zigzag folds at the same angle, so panels of one parity along a row stay parallel to one another at every stage: the strip compacts and does not bend. In the graded taper the middle closes ahead of the ends, and the strip has to accommodate that somehow.

It bends out of its own plane. Measured as the angle between each column’s panel and the first column’s, the graded taper turns by up to 18.8 degrees when the straight creases are at 20 degrees, 24.3 at 40, 18.7 at 80, 10.6 at 120 and 3.6 at 160. The same leaf taper at a single slant of sixteen degrees turns by nothing at any stage — its panels of one parity stay parallel exactly — so the bending belongs to the grading and not to the taper.

The profile along the strip says where. The turn grows from the narrow end inward as the leans change fastest there, peaks through the broad middle where the leans are nearly equal, and falls again toward the far end without quite returning to zero, because the turns compose in three dimensions rather than adding and cancelling. Early in the fold, when the middle zigzags have already closed a long way and the end ones have hardly started, the strip is most curved; as every zigzag approaches flat together the difference between them vanishes and so does the bending. The pattern arrives at a straight, evenly piled stack by way of a curved one, which matters for anything that deploys against a guide or inside a frame.

The deeper the target pile, the more the zigzags differ and the more the strip bends. Graded for six deep a row, where the middle zigzags stand at under ten degrees from upright and the end ones at 37, the peak turn is 35 degrees, reached at about 35 degrees of straight-crease fold. A designer choosing the target pile from the narrowest column, as the slant belongs to the line advised, is also choosing how far the strip will swing on its way there.

The leaf opens from its ends #

Run the motion backwards, from flat, and the relation between the fold angles decides the order in which a graded leaf opens.

With the straight creases opened ten degrees from flat, the steep zigzags at the narrow ends have opened 7.1 degrees and the upright ones in the middle 2.6; at thirty degrees, 21.5 and 7.9. A graded leaf opens from its ends and its broad middle follows, which is the reverse of what a designer might guess from where the paper is: the broad middle is most of the sheet and the last part to move.

That has a use and a hazard. The use is that the narrow ends, which pile deepest and cost most paper at their steep vertices — the floor at degree four prices a vertex at forty-five degrees a third higher than one at sixteen — are the ones that clear their stack first. The hazard is that a mechanism driven from the middle, where the actuator would naturally sit, is driving the part of the pattern that moves least in the first stage of opening. The deciding set does not move found where an actuator on a Miura decides the folded state; on a graded pattern, the same question now has the fold-angle relation to answer it.

What the picture cannot show #

Whether the bending collides. The panels are placed without thickness and nothing checks whether one passes through another on the way; the turn is measured, and whether a strip that swings 24 degrees out of its plane meets itself or a neighbouring layer is not. A panel is not the unit of depth found how badly thick-panel techniques fare on the Miura’s mixed piles; the graded pile is more uniform flat and less uniform in motion.

What the flat stack is. The rigid motion ends at the flat-folded state that the depth maps of the earlier essays describe, but nothing here orders its layers or checks that the flat state is reached without the layers passing through one another in the last few degrees. The closing gaps are measured only for the rigid placement of the panels.

Whether the rows can differ. The rank-one argument needs every straight crease to be straight and parallel, and every row of panels to be equally high: rows of different heights fail Kawasaki before rigidity is asked, as a leaf packs by corrugating found. Nothing here reaches a pattern whose straight creases converge, which is the fan the family essay’s rule does govern.

The idealisations underneath #

Panels are rigid and have no thickness, and creases are perfect hinges. The motion is the one a sheet of rigid plates joined by ideal hinges would follow, and the fold angle of the straight creases is its one input; nothing is said about the forces needed or about paper bending instead.

The taper is the model leaf of the earlier essay: twenty-four columns whose widths follow half a sine wave from 0.3 of a unit at the ends to one in the middle, four rows one unit high, each zigzag leaning for the mean width of the columns either side so that a column piles close to the target when flat. The turn is measured between panels of one parity in the first row, because panels of the other parity differ by the corrugation’s own zigzag, which is not the shape of the strip.

How the claims were checked #

Two readings of one condition. The rank of the table of cosines is read from the flat pattern, as the classical two-way interaction, which is exactly zero for a rank-one table; the gaps are read from panels placed in space by composing rotations about the flat creases, where a table that failed would show as panels that do not meet. Both must vanish, to $10^{-12}$ and $10^{-9}$, on all three corrugations at every stage drawn.

The same loop test must refuse a pattern outside the family: a vertex slid along its flat-folding ray, which keeps every vertex’s own condition exact, must leave a face whose loop misses by more than a thousandth of a radian. It misses by 0.22.

The single-slant taper must not bend at all — its panels of one parity parallel to within a millionth of a degree at every stage — so that the bending measured on the graded taper cannot be an artefact of the measure.

And the order of opening must hold near flat: the end zigzags opened more than twice as far as the middle ones at both straight-crease angles drawn, which is the relation’s prediction and is read off the fold angles directly.

Still open: grading for the motion as well as the pile #

The leans used here were chosen for the flat pile alone, each zigzag for the mean width of its two columns. A lean schedule could be chosen for the motion as well — to keep the strip’s bending under some limit while the pile stays within a step of its target — and the fold-angle relation makes that a calculation rather than a search, since the bending is set by how fast the leans change along the strip. The least-squares schedule the earlier essay proposed, with twenty-five leans fitted to put twenty-four columns on one count, would be the natural starting point, with the turn added as a second term.

The other lead is the fan. With converging straight creases the rows must be scaled copies, so a tapered corrugation whose straight creases fan — a pattern that curves in the flat — cannot be graded freely at all, and whether some grading survives with the rows constrained is a question the family’s rank-one table can answer directly. The rows are free found how a Miura’s hinges per layer settle as rows are added; a graded corrugation adds the same rows at no cost to its rigidity, and its hinge budget is the next thing to price.

The habit worth carrying is about practical rules drawn from a family. Before stating what a family requires, check whether the requirement comes from the condition or from the family’s most general member. The rule that rows must be copies was true of the fan and was stated for the parallel case too, where the condition it came from is satisfied by every row there is.