Every panel holds a frame

The folded strip lies at its own slant put ten folded patterns down on each of their panels in turn and read the two directional shrink factors off the box each placement needed. Five gave one pair whatever panel was held. Five gave two. The Miura gave one on four or six columns and two on three or five, which is not the kind of difference a pattern’s own look suggests, and the essay asked the question that would turn the table into a rule: which symmetry is needed?

The answer has a part that can be read off the crease pattern before anything is folded, and a part that belongs to the folded object — and the second is not quite a symmetry of the object. It is a symmetry of its widths, which is weaker, and the difference is exactly what the even Miura exploits.

Holding a panel turns the frame, not the paper #

A flat fold is computed by leaving one panel where it was drawn and carrying every other panel to its place by reflections. Hold panel $k$ instead and the whole folded state moves by the inverse of panel $k$’s motion. Nothing about the folded object changes. What changes is which way it faces the drawing’s axes, and the box the directional factors are read from is a box on those axes.

That box can be read the other way round. The box of the moved state on the axes has exactly the widths of the unmoved state measured along two other directions: the directions panel $k$’s motion carries the axes to. So a placement is not really a new position for the paper. It is a frame — a pair of perpendicular directions — in which the one folded state is measured, and each panel imposes its own.

A panel lies face up or face down. Face up, its motion turns it by some angle, and its frame is turned by that angle. Face down, its motion is a mirror in some line, and its frame is turned by twice that line’s angle, because a mirror at angle $\varphi$ sends the horizontal to the direction $2\varphi$. A width does not care which way along a direction it is measured, so a frame is an angle modulo a half-turn.

In the three-column Miura above, the eight panels in the outer columns hold the folded state in the drawing’s own frame and the four in the middle column hold it in a frame turned 139.9 degrees. The two boxes are the two answers the table recorded.

The frame is written in the creases #

The motion that puts a panel in place is a chain of reflections, one for each crease crossed on the way from the held panel, and two reflections in lines at angles $a$ and $b$ make a rotation by $2(b - a)$. So a panel reached across creases at angles $c_1, c_2, \dots, c_m$ holds the frame

$$\beta = 2\,(c_m - c_{m-1} + c_{m-2} - \cdots \pm c_1) \pmod{180^\circ},$$

twice the alternating sum of the crease directions crossed. The route does not matter, and the reason is Kawasaki’s condition. Two routes to one panel differ by loops round vertices, and going once round a flat-foldable vertex crosses its creases in order; their alternating sum is a multiple of a half-turn exactly when the alternate sectors sum to a half-turn each, which is the condition at a point that lets the vertex fold flat at all. The theorem that decides whether a pattern folds is the same one that makes its frames well defined.

So the frames are a property of the crease pattern, readable by walking its panels with nothing folded and no layers ordered. For the accordion every crease is vertical, twice ninety degrees is a half-turn, and every panel holds the one frame there is. The Miura’s creases run at 0, 69.9 and 110.1 degrees, and its panels hold two frames, 0 and 139.9 — twice its narrow sector. A five-sided twist’s creases run in five directions and its panels hold five frames.

The drawing makes a thing visible that the earlier table could only report. On the odd Miuras the minority placements were the panels of the even-numbered columns; here they are the panels whose motion includes the mirror in a zigzag crease, face up or face down — half of them lie face down in that mirror and half are turned by twice it. The columns alternate frames because crossing a zigzag adds twice its angle to the frame, and on an odd number of columns the two frames are held by unequal numbers of panels, eight against four on three columns and twelve against eight on five.

Twelve patterns, and how many frames each holds #

With the frames in hand, the factors each placement gives can be predicted from the folded state’s widths alone, without holding anything. Every prediction below was checked against holding a panel of that frame and reading its box, and agreed to six decimals.

The table sorts the shelf into three kinds.

Patterns with one frame give one answer, whatever they fold into. The accordion is the only one on this shelf, and it needs no symmetry of any kind: a lopsided accordion with pleats of every width would give one answer too, since its creases would all still be vertical. The rule is general. Any pattern whose creases all run along the drawing’s axes has one frame, because twice ninety degrees is a half-turn and twice zero is nothing. A map fold, a grid of square pleats, any box-pleated pattern without its diagonals: all of them are measured the same way whatever panel is put down.

Patterns with one answer and several frames need their folded widths to agree. The square twist and the waterbomb hold frames at 0 and 90 degrees, and the second swaps along and across — so both give one answer because each folds to a state as wide one way as the other, 1.515 by 1.515 and a square. The Yoshimura’s panels hold three frames, at 0, 60 and 120 degrees, twelve panels each, and its folded state is 4.000 by 4.000 in all three. The four- and six-column Miuras hold two frames each and give one answer.

Patterns with more answers than one fail to agree across some pair of frames. The odd Miuras and the leaf corrugation have two frames and two answers. The three-sided twist has three frames and two answers — its frames at 0 and 120 degrees agree, the one at 60 does not — and the six-sided twist has three frames too, whose frames at 0 and 60 agree and at 120 do not. The five-sided twist has five frames and five answers, 1.232 by 1.253 on two panels, 1.566 by 1.613 on six, and three more on one panel each: the frame count is the most answers a pattern can give, and the five-sided twist gives all of them.

A twist’s frames follow its polygon: its panels hold frames at multiples of $360/n$ degrees modulo a half-turn, which is $n$ frames when $n$ is odd and $n/2$ when it is even. The square twist’s two frames are reconciled by the sheet it is cut from, which shares its fourfold symmetry. The three- and six-sided twists are cut from square sheets as well, and a square has no threefold symmetry to lend them.

Where two frames agree #

For the patterns with two frames, the question is now a single comparison: are the folded state’s widths the same along the drawing’s axes as along the turned frame’s?

A folded state has a width in every direction, and a box on any frame reads two of them. The curve is the four-column Miura’s. Its width along the drawing’s horizontal is 1.600, and so is its width along the turned frame’s first direction; across, both read 2.288. Drag the slant and the turned frame moves — it is always twice the narrow sector, 139.9 degrees at a slant of 0.35 — and the curve changes shape, and the two pairs of dots stay level at every setting.

The rows play no part in any of this. A Miura’s straight creases run along the drawing’s horizontal, and crossing a horizontal crease adds twice nothing to a panel’s frame, so a Miura four rows deep and one twelve rows deep hold exactly the same frames on the same columns. That is the frame’s side of what the rows are free found in the folded box itself, which has no term in the rows at all: the rows neither move the strip nor turn the frame it is read in.

The leaf corrugation is the one pattern on the shelf where the two frames nearly agree and do not. Its zigzag is 65.9 degrees, its turned frame 131.9, and sixteen of its twenty-eight panels hold the drawing’s frame. Along the strip the two frames read factors of 3.573 and 3.575, the same to three figures; across they read 1.220 and 1.130, eight per cent apart. The direction that gets longer had quoted the leaf’s across factor from one frame, and its window of dial settings moved to 0.54–1.02 radians in the other; the frame reading says that the whole of that disagreement is across the strip, since along it the two frames agree.

The three-column Miura is a different curve with the same frames on it. Its widths along the axes are 1.600 and 1.644; in the turned frame they are 1.365 and 2.288. No slant makes them agree. The width curve has a single sharp minimum where the folded strip is measured square across its narrow way, and the two frames sit symmetrically about the strip’s long direction — 0 and 139.9 are each 69.9 degrees from it — so they agree exactly when the curve is symmetric about that direction.

A box cannot tell a shape from its half-turn #

The natural reading of symmetric about the strip’s direction is that the folded strip is its own mirror image across its long axis. It is not.

Neither Miura’s folded outline is its own mirror image across the strip. Both miss by 0.343, which is the step one column adds to the strip’s length. The four-column outline is instead its own mirror image across the line square to the strip — the same as a mirror across the strip followed by a half-turn — and a half-turn changes no width, since a width measured one way along a direction is the width measured the other way. So the four-column Miura’s widths are symmetric about its strip although its outline is not.

The three-column outline is a parallelogram, which is its own half-turn already, so turning its mirror image changes nothing: the mirror image is simply a different parallelogram, leaning the other way, and its widths differ.

That is the answer to which symmetry is needed. Not a symmetry of the crease pattern, and not a symmetry of the folded object: a symmetry of the folded object up to a half-turn, under the motions that relate its panels’ frames. For the Miura that is the mirror in the line square to the strip. On four and six columns the outline’s two ends are each other’s image in that line; on three and five the ends are parallel and the outline is a parallelogram, which that mirror sends to a parallelogram leaning the other way.

It also accounts for what the earlier essay measured and did not trace, that the minority placements on an odd Miura give the cross factor of the Miura one column wider and the majority that of the Miura one column narrower. The flat sheets have the same height whatever the column count, so equal cross factors mean equal widths across, and at every slant on the dial the three-column Miura’s turned frame reads the four-column Miura’s width across — 2.288 at a slant of 0.35 — while its drawing’s frame reads the two-column Miura’s. The five-column Miura’s frames read the six- and four-column widths the same way, 2.933 and 2.288. Holding a middle-column panel mirrors the strip across its own axis, as far as a box can tell, and the mirrored odd strip casts the shadow of the next even one. Why the shadows match column for column is measured here and not derived.

What the frame rule takes as given #

The folded state is flat and has no thickness, as in every shrink measurement on this shelf. A real folded Miura is a stack of some depth and its box has a third side; nothing here reaches it.

The widths are the paper’s corners, not its layers. A box depends only on the extreme points of the folded paper in each direction, so the frame rule needs no layer order and holds for any folded state the reflections produce. It is a statement about the placement convention of a shrink measured along both axes, not about how the layers are stacked.

Agreement is measured to four decimals. Two frames that agree to four figures are reported as one answer. On every pattern here the agreeing frames agree exactly, because the symmetry that makes them agree is exact; a pattern whose widths agreed by accident to that tolerance would be reported as agreeing too.

And the untested prediction is stated as one. A three- or six-sided twist cut from a sheet with its own symmetry — a hexagon rather than a square — should give one answer on every panel, as the square twist does from its square. The shelf has no such sheet and the prediction is not measured.

What Kawasaki does for a box #

The connection that was not expected is between the placement convention and the flat-folding condition. Kawasaki’s theorem is usually met as a statement about when a vertex folds flat; here it turns up as the reason a panel’s frame is well defined, since the frame is an alternating sum of crease directions and Kawasaki is exactly what makes alternating sums round a vertex vanish. Which polygons twist read a twist’s possibilities off its polygon’s symmetry; the frames read a twist’s measurement off the same polygon, through the same alternating sum.

The other connection is about what a box can see. A box on a frame reads two widths, and widths are blind to a half-turn, so a box sees any shape only through the part of it that a half-turn leaves unchanged. The symmetry a measurement needs is a symmetry of what the measurement can see, and that is a larger set than the symmetries of the object — large enough to take in the four-column Miura, which has no mirror of its own that does the job.

Still open: a frame the paper chooses #

The frames suggest the repair the earlier essay asked for. Its first open question was the right flat-sheet frame for a folded strip at an angle, and the table gives a candidate: report a pattern’s directional factors in every frame its panels hold, as a set, with the number of panels in each. For a one-frame pattern that is one pair and the familiar numbers; for the Miura it is two pairs, one of which is the drawing’s; and for the five-sided twist it is five. Whether some frame in the set is the natural one — the frame most panels hold, say, which on the three-column Miura is the drawing’s own and on the five-sided twist is the one at 144 degrees — is a question about what a shrink factor is for.

The twists cut from matching sheets are the direct test. A six-sided twist on a hexagonal sheet should close the table’s gap between its frames at 0 and 60 degrees and the one at 120, and a five-sided twist on a pentagon should collapse five answers to one. Both are twists that could be drawn and folded on those sheets, and neither has been.

Sideways from here, the turn a column costs found the Yoshimura carrying one column onto the next by a turn of 240 degrees. A turn of 240 is a frame of 60 modulo a half-turn, which is one of the three frames the Yoshimura’s panels hold, and it would be worth knowing whether every such column turn on the shelf is one of its pattern’s frames. And downward, what a corrugation costs measured covered area, which has no frame at all: it is the one shrink number on the shelf that every panel agrees on without any symmetry.

The habit worth carrying is about conventions that can be undone. When a measurement depends on an arbitrary choice, find what the choice actually changes before asking what symmetry would remove it. Holding a panel changes one thing, the frame, and the frame is a number the crease pattern already knows; the symmetry question then becomes a question about the one quantity the frame reads.