The slant that stacks in one depth

A panel is not the unit of depth laid every panel of the printed Miura on the map of its folded pile and found each one lying over three or four depths, so no panel could be given one thickness. The uniform piles did the opposite — every panel of the Yoshimura, the waterbomb and the preliminary base lies over exactly one depth — and none of those three folds as a mechanism driven by one input. It ended on a question that would decide whether the Miura’s difficulty is an accident of one design or a law: is stagger the price of a single freedom?

It is not, and the reason is short. The Miura’s pile is set by one number, and the printed Miura’s value of it happens to be close to a bad one.

Rows stack, columns step #

The rows are free measured a folded Miura’s smallest box and found it growing by the same step for every column and not at all for its rows. The pile says the same thing from inside. Every row folds exactly onto the row before it, so a Miura four rows deep has exactly four times the layers of the same Miura one row deep, at every point: the two maps are the same map with its numbers multiplied. Measured on a Miura four columns wide, one, two, four and eight rows deep, the shares at each depth are identical to three decimals and the depths are one, two, four and eight times those of a single row.

So all the variety in a Miura’s pile belongs to its columns. Each column’s panels fold to a parallelogram image a unit long along the folded strip, and each column’s image lies one step further along than the last. The step is $\sin a\cos a$ for a slant $a$ — it is the same column walk that makes a folded Miura’s box grow with its columns — and a unit-long image stepping by that much is covered, on average, by

$$k(a) = \frac{1}{\sin a \cos a} = \frac{2}{\sin 2a}$$

column images. At the printed slant of 0.35 radians, 20.05 degrees, that is 3.10. At 45 degrees it is two, its least value. As the slant closes toward nothing it grows without bound, because the columns barely walk and nearly all of them pile onto one another.

The measured piles follow the curve. On forty columns, the mean depth per row is 2.95 at the printed slant against 3.10 predicted, 1.93 at 45 degrees against two, and 8.2 at 0.1 radians against 10.1; the shortfall in each is the two ends, which are shallower than the middle and matter more when the columns barely walk.

An even count is one depth, an odd count is three #

An average is not a pile. What decides whether a panel can be given one thickness is whether the count is the same everywhere, and the law says where to look: at slants where $2/\sin 2a$ is a whole number $k$, which are

$$a_k = \tfrac12 \arcsin \frac{2}{k},$$

and their complements. For $k = 2$ that is 45 degrees; for $k = 3$, 20.91; for $k = 4$, exactly 15, since $\arcsin\tfrac12$ is thirty degrees; for $k = 6$, 9.74.

The table splits cleanly by parity. At every even count, the share at the commonest depth climbs toward the whole footprint as columns are added: 94.4 per cent at two columns deep on eighty columns, 92.5 at four, 88.3 at six, 83.6 at eight, and in every case the commonest depth is exactly $k$. At every odd count it stays under half: 48.2 per cent at three, 45.5 at five, 43.6 at seven, with the rest split between $k - 1$ and $k + 1$. The complementary slants — 75 degrees for four, 69.09 for three — give the same pattern, since the step depends on the slant only through $\sin 2a$.

The odd counts have a shape of their own, and it is not a smear. On a hundred and twenty columns at 20.91 degrees, where the columns overlap three deep, the footprint is 25.8 per cent two deep, 48.7 three deep and 23.8 four deep, with 1.7 at the ends; at five deep it is 25.0, 47.0 and 22.9 at four, five and six. An odd count settles to half its footprint at the count and a quarter at each neighbour, which averages to the count exactly, so the law holds on odd counts too — it is only the uniformity that fails. A thick-panel designer facing an odd count is facing a pile of three depths in fixed proportions everywhere along the sheet, not a gradient toward one end.

The reason for the parity is in the first map. A column’s image is a parallelogram, not a rectangle: its ends cut across the strip at a slant, and neighbouring columns lean opposite ways. Every panel holds a frame found the same alternation in the frames a Miura’s panels impose, the middle columns holding the mirror of the outer ones. So the images repeat every two columns, not every one. Where the slanted ends of a leftward image and a rightward one fall on each other, they fill each other’s triangles; at an even count they always do, and at an odd count one end in every pair is left over, which is the alternating triangles of two, three and four that fill the printed Miura’s strip. That account fits both maps and all fourteen slants in the table; it has not been turned into a proof.

The ends are the only departure #

On an even count the part of the footprint not at the commonest depth is the two ends, where the pile ramps up over $k - 1$ columns before it reaches its full depth. The ramps have a fixed length, so their share falls as the sheet gets longer.

At fifteen degrees the share at four deep is 46 per cent on eight columns, 68 on sixteen, 83 on thirty-two and 91 on sixty-four: each doubling roughly halves what is left, which is what two fixed ramps on a growing strip do. At 20.91 degrees and at 20.05 it barely moves, from 35 to 48 per cent, because the mixture is not at the ends but in every pair of columns along the strip. The even case can be counted exactly. The folded strip of $c$ columns is $1 + (c - 1)\,w$ long, where $w = \sin a\cos a$ is the step, and at an even count $k w = 1$. Each end ramps through depths one to $k - 1$, and each of those depths occupies exactly one step of length at each end, so the share of the footprint at the full depth is

$$\frac{c - k + 1}{c + k - 1}.$$

On sixteen columns at fifteen degrees that is 13/19, 68.4 per cent, and each ramp depth takes a step at both ends, 10.5 per cent apiece — the map’s own numbers to the decimal. On eighty columns it is 77/83, 92.8 per cent against the 92.5 measured. The odd case has no such count, because its mixture is not at the ends: on forty columns at 20.91 degrees the middle of the strip is about half at three deep and a quarter each at two and four.

The printed Miura’s graded pile is not a property of its size. A longer printed Miura is exactly as graded; a longer fifteen-degree Miura is more nearly uniform.

Every panel away from the ends lies over one depth #

The question the earlier essay asked was about panels, and the map of depths can be read panel by panel again.

At the printed slant none of the sixteen panels lies over a single depth; every one lies over three or four, as the earlier count found on the printed pattern. At fifteen degrees ten panels lie over exactly one depth, and they are precisely the panels more than three columns from either end. The six that do not are the ramps. On a Miura long enough to be worth building thick, that is nearly every panel, and each can be given the one thickness its depth asks for — which is the property the uniform piles had and the Miura was thought to lack.

It is worth being precise about what made the printed Miura look like a counterexample. The earlier essay measured one printed pattern at one slant, four columns wide, and a four-column Miura is too short to show a uniform pile at any slant: at fifteen degrees its strip is seven steps long and its two ramps would take six of them. The printed pattern was both at an uneven slant and too short to show an even one, and either alone would have produced a graded map.

So the answer to the earlier question is no. A Miura at fifteen degrees folds rigidly with one input, as every Miura does, and it stacks its panels in whole units of depth. The stagger belongs to the slant, and within the Miura family it is a choice.

What the choice costs #

A uniform pile is not free, and the price is written in the same number.

$2/\sin 2a$ is also how many layers of paper a folded Miura stacks over its footprint per row, on average — its compaction. A Miura that piles two columns deep at 45 degrees has compacted its sheet the least any Miura can, by a factor of two a row; the printed Miura compacts by 3.1 a row; a fifteen-degree Miura by four; one at 0.1 radians by ten. The uniform slants therefore form a scale of compaction — two, four, six, eight columns deep a row — and a designer who wants panels that are units of depth chooses a step on that scale rather than a slant.

The scale also says which step a use wants. A deployable panel that must pack small wants a high step — six or eight columns deep a row — and pays in a slant under ten degrees, where the zigzag is nearly flat and the vertices are nearly crossings. A thick-panel structure that must carry load through its layers wants a low step, where each point is under as few panels as possible, and the lowest is 45 degrees, two columns deep. Nothing between the steps is uniform, so a design specified by its compaction alone — “about three layers a row” — has specified a graded pile without saying so.

The scale has a top, and it is instructive. At no slant at all the columns do not walk, every column’s image lies on every other, and the pile is uniform at the full column count: eight columns two rows deep fold sixteen layers over the whole footprint. That is the map fold, and its vertices are right-angled crossings, which two mechanisms at one point found fold in two independent ways rather than one. So the square grid is the one uniform pile the family had before this measurement, and it is uniform precisely because it gave up the single freedom. The even steps are the uniform piles that keep it.

The printed slant is not on a step of the scale, and it is nearest the odd count three. Moving it from 20.05 degrees to 15 costs the zigzag five degrees of lean and buys a pile four columns deep instead of a mixture of two, three and four. It also changes everything else the slant decides: how far the folded strip lies off the sheet’s axes, as the shrinkage essays measured, and the sector at every vertex, which where two bands part found the fineness of a Miura on a given paper to depend on. At fifteen degrees the narrow sector is 75 degrees, wider than the printed 69.9, so a fifteen-degree Miura is also slightly finer on any paper.

What the depth maps assume #

Panels have no thickness, which is the idealisation every pile count makes and the thing a thick-panel design then corrects. The count says how many layers lie at each point of an ideal folded state; it is the input to a thickness allowance, not an allowance.

The maps are sampled. Depth is read at the centres of a grid of cells over the footprint, two hundred to a side, and a panel’s depths are read only at cells clear of every panel edge, so the boundary between two depths is placed to within a cell. The shares in the tables are shares of those cells.

The parity account is an account. The average overlap is derived; that even counts give one depth and odd counts do not is measured at every count from two to eight, on sheets from eight to eighty columns, and the explanation offered for it — slanted ends that pair off every two columns — fits the maps without being proved.

And the slants are exact. A fifteen-degree Miura is uniform at fifteen degrees; at fourteen or sixteen the count is 4.3 or 3.8, and a sliver of another depth appears at every pair of columns. How much tolerance a real sheet’s slant has before its pile stops being usefully uniform is a question the tolerance essays have the tools for and have not asked.

A number that is a count #

The connection that was not expected is between a length and a count. The column walk is a length, measured in the rows are free as the growth of a box; its reciprocal turns out to be the number of layers a point is buried under, and whether that number is even decides whether a panel can be built as one piece. Nothing in the kinematics of the Miura — its single freedom, its one-parameter motion — refers to parity. The parity comes from the alternation of its columns, which is also why the Miura folds at all.

The count also explains a thing the printed maps showed without explaining. The earlier essay found the printed Miura’s deepest layers in three separate strips, and the tapered corrugation’s in four, and read the separation as a feature of graded piles. On a Miura it is the odd count at work: at 3.1 columns deep on average, the fourth layer appears only in the triangles where the slanted ends of neighbouring columns overlap, one between each pair of neighbours, pointing alternately across the strip. A four-column sheet has three such pairs, and three strips.

And it reframes the three kinds of pile. Three kinds of pile sorted the printed shelf into uniform, island and graded, and put the Miura among the graded. The Miura is all three kinds of pile at different slants: uniform on the even steps, graded between them, and — on a short sheet at an even step — a large uniform island with graded ends. A kind of pile is a property of a pattern at a setting, not of a pattern.

Still open: the tapered corrugation, and the rows #

The tapered corrugation is the next test. Its cells change size along the sheet, so its column step is not constant and no single slant puts every pair of columns on an even count. Whether a taper can be graded so that the local count stays on one step — a slant varying with the cell size — would say whether the uniform pile survives the one modification of the Miura that the printed shelf carries.

The rows could be put to work. Every row adds the same depth everywhere, so rows never grade a pile — but they never help one either. A pattern whose rows walked, as the columns do, would have a second count, and the pile would then depend on two numbers; whether any quadrilateral pattern with one freedom has walking rows and walking columns whose counts are both even is a search over the family the Miura belongs to, with the parity now known to be the target.

Sideways from here, a panel is not the unit of depth proposed a stepped panel, three or four levels, for the printed Miura. The count here says the printed Miura needs three levels on every panel along its whole length, and a fifteen-degree Miura needs one; a stepped panel is the answer to a slant that could have been changed.

The habit worth carrying is about averages that are whole numbers. When a quantity is a count on average, ask whether it is the same count everywhere, and at which settings the average is a whole number. A mean of 3.1 layers and a mean of 4 look like neighbours; one is a pile of three different depths and the other is a single one.