The folded strip lies at its own slant

A shrink is two numbers replaced the single figure for how much smaller a folded sheet gets with a pair: the flat sheet’s width over the folded state’s, and its height over the folded state’s. The plane the five points were in turned every family’s dial and traced the pair as a curve, and the direction that gets longer found one of the pair falling below one. All three read the folded state against the same frame: the axes the pattern was drawn on, with the first panel of the sheet left lying where it was drawn and every other panel carried to wherever the folds put it.

Neither half of that frame belongs to the folded object. The axes belong to the drawing. The panel that stays put is a choice, and the folded state is the same rigid object whichever panel is put on the table — it just lies differently.

On two of the families measured, the difference is the whole of the effect. A flat-folded Miura is a strip of fixed width, and it lies at exactly its own slant to the drawing’s axes. Its two factors are the shadow of that strip on those axes, which is why they change when columns are added that change nothing across the pattern, why on some sheets they change when a different panel is held, and why the dial the plane reported as turning round turns round in the wrong place.

A strip at the slant #

Fold a four-by-four Miura at a slant of 0.35 radians and look for the smallest rectangle its paper fits in, allowing the rectangle any orientation. The rectangle is found exactly rather than searched for: the smallest box round a set of points always has one side along an edge of their convex hull, so trying every hull edge finds it.

It is 0.939 wide and 2.436 long, and its long side lies 20.1 degrees off the drawing’s vertical. A slant of 0.35 radians is 20.05 degrees. At a slant of 0.80 the strip is 0.697 by 4.305 and lies 45.8 degrees off, which is 0.80 radians. The strip lies along the slanted creases of the panel that was held still, and it lies exactly along them.

That is not surprising once it is seen. A Miura’s folded state is a stack in which the zigzag creases have brought every column over its neighbours and every row onto the rows beside it, and the stack’s long direction is the direction the slanted creases run. What is surprising is how much of the box on the drawing’s axes the tilt accounts for. At a slant of 0.35 the drawing’s box round the folded state is 1.600 by 2.288, an area of 3.66, and the strip inside it has an area of 2.29. The box the factors are read from is 1.6 times the box the folded paper needs, and the difference is nothing but the angle.

The leaf corrugation does the same thing, and it had been described as the pattern that does not. Its folded state is a strip too, and at every zigzag angle from 0.2 radians to 1.1 its long side lies at that angle to the drawing’s vertical, to within half a degree at every setting. At the leaf’s usual zigzag of 0.42 radians the strip is 24.1 degrees off the axes; at the 0.78 where its cross factor dipped below one it is 44.7 degrees off.

So the two families on which the pair of factors did the most interesting things are the two whose folded states lie furthest off the axes the pair is read on. The accordion, the square twist and the Yoshimura lie along those axes exactly — tilts of zero, zero and ninety degrees — and the pair is a clean description of them. The Miura and the leaf lie at their own slants, and on them the pair is a description of a strip and an angle together.

A cross factor that depends on the columns #

The quickest way to see that the Miura’s pair is the strip’s shadow rather than the fold’s is to change something the fold does not care about. Add columns.

At a slant of 0.35 and four rows, the folded strip is 0.939 wide with two columns, with four, with six, with eight and with ten — the same to three figures at every width of sheet. Across the pattern nothing changes when a column is added: the rows are the same rows, stacked the same way, and the strip’s width is set by them.

Its length does change, by 0.343 a column, because each column the fold brings over its neighbour lands 0.343 further along the strip than the last. That is a genuine property of the fold — the along direction is where the columns go — and it is exactly what an along factor should measure.

But the strip lies at twenty degrees to the vertical, so lengthening it lengthens its shadow in both of the drawing’s directions. The cross factor falls from 2.433 on two columns to 1.748 on four, 1.364 on six, 1.118 on eight and 0.948 on ten, and at ten columns it is below one: a Miura whose folded state’s box is taller than the flat sheet, by the same reading that found the leaf’s window. No row was added and no fold across the pattern changed. The fall is the tilted strip’s shadow growing as the strip does.

The four-by-four Miura’s 2.728 along and 1.748 across, which have been the Miura’s pair throughout these essays, are the pair for one width of sheet. They are correct numbers for that sheet on those axes. They are not the Miura’s, in the way that 8 along and 1 across is the accordion’s at eight folds: widen the accordion and its cross factor stays at exactly one, because its folded state lies along the axes and a longer pleat stack casts no shadow sideways.

The panel that stays on the table #

The second convention is the held panel, and it matters on more patterns than the tilt does.

A flat fold is computed by leaving one panel where it was drawn and carrying every other panel through the reflections along a path of creases to it. Which panel is left alone is arbitrary. Holding panel $k$ instead of the first moves the whole folded state by the inverse of panel $k$’s motion — a rigid motion, so nothing about the folded object changes, but its orientation against the drawing’s axes can.

Each placement’s motion is recovered from the held panel’s own corners and checked on every corner rather than only the three it is read from, so the comparison is between placements of one computed state and not between separate folds. Five of the ten patterns give one answer on every panel. The accordion’s pleats are all parallel, so every panel’s motion is a slide or a reflection in a line along the axes, and none of those changes a box. The square twist’s fourfold symmetry carries every placement onto every other. The Yoshimura and the four- and six-column Miuras have enough symmetry that every placement lands in a box of the same size.

The other five do not. The leaf gives 3.573 by 1.220 held on sixteen of its twenty-eight panels and 3.575 by 1.130 held on the other twelve. The Miura with three columns gives 2.103 by 2.433 on eight panels and 2.465 by 1.748 on four; with five columns, 2.923 by 1.748 on twelve and 3.353 by 1.364 on eight. The minority placements on the odd Miuras are the panels of the even-numbered columns — the middle column of three, the second and fourth of five — and their cross factors are, to every digit, the cross factor of the Miura one column wider, while the majority’s are those of the Miura one column narrower. That is what the strip reading predicts if holding one of those panels lays the strip across a slanted crease from where the first panel put it, though the mechanism is not traced further here.

The three- and six-sided twists are the cases where the convention had already done damage. Held on its first panel the triangle twist draws in 1.082 along and 1.168 across; held on four of its seven panels it draws in 1.448 and exactly 1.000. The hexagon twist gives 1.287 by 1.168 on five of thirteen panels and 1.417 by 1.732 on eight. The plane essay noticed that the triangle and the hexagon twists had the same cross factor to four figures and asked what in the pleats could make it so. Nothing in the pleats does. Both 1.168s are the placement the first panel happens to fall in; on each twist’s commoner placement the two share nothing.

Where the dial really stops paying #

The plane essay’s most quoted result was that the Miura’s dial turns round: steepen the slant and the folded sheet gets smaller, until past a point it stops getting smaller and starts getting larger again. On the drawing’s axes the product of the two factors is least at a slant of 1.05.

If the product is a tilted strip’s shadow, the turn could be the strip turning rather than the paper, and the test is to ask the question again with no axes in it. Two readings of how small a folded state is have no orientation in them. One is the paper over the smallest box it fits in — the container a given amount of paper needs, allowed to be turned. The other is the paper over the area it actually covers, which is the mean depth of the stack and which the conservation law makes an identity rather than a measurement.

Both are least at a slant of 0.80. The smallest container falls from 11.4 times smaller than the paper at a slant of 0.1 to 5.32 at 0.80 and rises again to 7.85 by 1.3; the depth of the stack falls from 12.3 to 6.40 and rises to 9.03. The product on the drawing’s axes goes on falling past 0.80 to its minimum of 1.773 at 1.05.

So the Miura’s dial does turn round — a steeper slant does stop buying a smaller folded sheet — and the plane had the fact right and the place wrong. Between 0.80 and 1.05, while the strip swings from 46 degrees off the vertical to 60, the paper is already getting less compact by both readings that have no axes in them, and the product on the axes goes on falling. The product’s turn is the conventions’; the paper’s turn is at 0.80.

Two conventions move the product, not one. Its numerator is the flat sheet’s box, and a Miura’s sheet, with alternate rows offset by the slant, fills less of its own box the steeper it is: 92 per cent at a slant of 0.35, 57 per cent at 1.25. Its denominator is the folded state’s box on the drawing’s axes, which is the tilted strip’s shadow. Neither is the paper, and the two readings beside them are.

The cross factor alone is least at 0.80 as well, which the plane essay recorded as a separate fact. Whether that coincidence with the paper’s turn has a reason is not settled by anything measured here: the cross factor is the flat sheet’s height over the strip’s vertical shadow, and the shadow’s largest value landing on the paper’s least compaction may be a property of the Miura’s geometry or an accident of four rows.

The leaf repeats the pattern exactly. Its product on the drawing’s axes is least at a zigzag of 1.05; its smallest container and its stack depth are both least at 0.80, at 4.72 and 6.98. Two families with different cells, different outlines and different column profiles agree to the setting on where the paper stops compacting, and both disagree with the box by the same quarter radian. The agreement is a coincidence of the two families’ particular numbers rather than a law. The disagreement is not: it is what reading a tilted strip on fixed axes does.

What the leaf’s window was #

The direction that gets longer found the leaf’s cross factor below one between zigzags of 0.70 and 0.88, with a minimum of 0.9856 at 0.78, and explained it by the folded extent across the pattern being a constant of the cell. It set aside the possibility that the window was an artefact of the axes on the grounds that the leaf does not turn.

The leaf does turn, in the only sense that matters for a box: its folded strip lies at its zigzag to the drawing’s axes, 44.7 degrees at 0.78. And the window depends on the panel held. Held on the first panel it is 0.70 to 0.88; held on any of the twelve panels of the other placement it runs from about 0.54 to 1.02, with a minimum of 0.9067 at the same 0.78. On every placement there is a range of zigzags over which the folded state’s box is taller across than the sheet, so the essay’s refutation stands — no directional factor is bounded below by one — but where the window is and how deep it goes are properties of a placement rather than of the leaf.

The Yoshimura, which that essay called the cleaner witness, is exactly that. Drawn four columns across at a row height of one, with two, four, six and eight rows, it gives one answer on every panel of every sheet, and its cross factor of half the row count is a property of the Yoshimura rather than of a placement.

Six patterns, with and without the axes #

Put the six patterns shrinkage has been measured on here side by side at their usual settings and the readings sort cleanly by whether the folded state lies along the axes.

The accordion reads 8 on all three: its folded state is a rectangle lying along the axes, filling its box. The square twist reads 2.296 on the axes and in its smallest box, because it lies square to them, and 3.011 by covered area, because its folded outline fills three quarters of its box. The Yoshimura reads 16 on the axes and in its smallest box and 32 by covered area, because its folded state is a right triangle lying along the axes and fills exactly half.

The waterbomb is the surprise of the six. Its folded state lies at 45 degrees — a square turned onto its corner — so its box on the axes is twice its smallest box, and it reads 16 on the axes and 32 in its smallest box and by covered area alike. The Yoshimura and the waterbomb read the same on the drawing’s axes and the same by area, and the census that compared them told them apart only by their creasing. They differ in shape: one is a triangle that needs a box twice its size whichever way it is turned, the other a square that needs one only when it is turned the wrong way.

The Miura and the leaf read 4.768 and 4.361 on the axes, 6.992 and 5.867 in their smallest boxes, 8.137 and 8.311 by area. On the axes the Miura is the more compact; in any box it could actually be packed in, it is by a fifth. By area the leaf is.

What a box on the drawing’s axes is for #

None of this makes the directional factors wrong. They answer a precise question: if the folded object is left lying where the fold put it, with that panel where it was drawn, how much of the drawing’s width and height does it take up? That is the right question for a folded sheet that is going back into the frame it came out of — a panel folded on a table and slid into a slot the same shape as the unfolded sheet was.

It is the wrong question for the use the pair was introduced for. A shrink is two numbers proposed the pair as the way to separate a pattern that leaves one direction alone from one that draws in equally from one that does neither, and to say which folded state fits which container. A container can be turned. A folded Miura packed into a canister — the use Koryo Miura designed the pattern for, deploying large membranes in space — goes in along the strip and not along the drawing’s axes, and the number that says whether it fits is the smallest box’s, 6.99 times smaller than its paper, rather than the 4.37 its box on the drawing’s axes gives. The Miura folds two ways is a reminder that even the folded state is a choice among several; the tilt is the same on every one measured here.

So the pair should be read on the folded state’s own axes where it has them, and the smallest box supplies them: its long side is the along direction and its short side the across. On the accordion, the twist and the Yoshimura that changes nothing. On the Miura it changes everything the pair was used to say, because at the four-by-four size quoted the box it was read from is 1.6 times the box the folded paper needs.

What these readings rest on #

Every folded state is flat and of zero thickness, so the strip’s width is set by where the panels’ corners land and not by the paper piled between them. A real folded Miura is thicker in the middle of its stack than at its edges, as a panel is not the unit of depth measured, and its smallest container is a box in three dimensions; nothing here decides whether the tilt survives thickness, though nothing obvious would remove it.

The smallest box is a rectangle. A folded Miura strip is closer to a long parallelogram, and a container shaped like one would hold it more tightly still; the rectangle is the fairest container that is not tailored to the pattern.

The covered area is sampled on a grid 240 cells across the folded state’s longer side, and the paper under it is recovered to within a per cent; both minima sit at 0.80 on a sweep taken in steps of 0.05, so the claim is that both turn inside the same step, not that they turn at the same number to three figures.

And the held panel is any one of the sheet’s. A physical folder does not choose a panel to hold; they pick the folded object up. The readings with no axes in them are the ones that describe what is picked up, which is the argument for them, and the readings on the drawing’s axes describe it only for the patterns whose every placement agrees.

The picture the axes drew #

The surprising thing is not that a folded state can lie at an angle. Folding it flat is one similarity measured a twist tessellation’s folded state turning through a definite angle, 36.62 degrees, and said so. It is that the two corrugations — the patterns whose whole purpose is to bring pleats together along one direction — turn out to be the ones lying off the axes by exactly their own design angle, while the twist, the pattern named for turning, lies square.

The reason is the same fact seen from two sides. A twist’s pleats turn the paper, and its folded state is symmetric enough that every placement lies square to the drawing. A corrugation’s pleats do not turn the paper at all — every crease is a reflection — but its reflecting lines are slanted, and a stack of reflections in slanted lines is laid out along the slant. A corrugation lies along its creases; the drawing lies along its sheet, and on a Miura those are different directions by the whole of the slant.

The habit worth carrying from this is a check on any measurement that divides one extent by another: ask what the extent is measured along, and whether the thing being measured was put there or merely left there. A folded state is left wherever the fold computation puts it, and its box on the drawing’s axes is a fact about that placement.

Still open: the pair in the strip’s own frame #

The natural repair is to read both factors in the frame the folded state supplies: its smallest box’s long side as the along direction, the short side as the across. For that to be a pair of shrink factors the flat sheet needs a matching frame, and a Miura’s flat sheet has an obvious one only when its slant is small — the drawing’s axes — while its folded strip’s frame turns with the slant. What the right flat-sheet frame is for a folded strip at an angle is a question with a definite answer for each pattern, probably the direction the folded strip’s long side came from on the sheet, and it is not yet computed.

The other open thing is the placement count. Five of ten patterns give one answer on every panel and five do not, and the five that do are the ones with enough symmetry for the tilts to cancel. Which symmetry is needed — a mirror across a pleat, a half turn, the fourfold turn the square twist has — would turn the table of placements into a rule that could be read off a crease pattern before it is folded, and which polygons twist already classifies the twists by the symmetry their polygons carry.

Sideways from here, the waterbomb’s 45 degrees deserves its own reading. It is the one pattern on the shelf whose box ratio and whose area ratio disagreed for a reason that has nothing to do with empty space — its folded square fills its smallest box completely — and the turn a column costs, which found the Yoshimura’s folded state carrying one column onto the next by a turn of 240 degrees, is the place such a turn would be explained from the pattern’s own motions.

Downward, what a corrugation costs measured its whole census by covered area, which needs no axes; it is the one table on shrinkage that the tilt cannot have touched, and it is the one to trust where the two disagree.