The turn a column costs

A tessellation repeats. That is what makes it one, and the rectangle it repeats in is the unit every measurement about it is quoted in: so many panels per cell, so many creases, so many vertices.

Fold it flat and it repeats again — the folded state of a periodic pattern is periodic, since the same drawing folds the same way everywhere. The natural assumption is that the two periods are the same rectangle.

On one of the families here they are not, and the difference is a factor of three.

What the fold does to a column #

Take the equilateral Yoshimura: courses of horizontal creases with diagonals zigzagging between them, the standard drawing, the one that appears on every corrugated bellows and most collapsible lampshades.

Its drawing repeats every column. Shift it sideways by one column’s width and it lies on top of itself exactly.

Fold it flat and compare the panel one column over with the panel it came from. The two are related by a rigid motion, as any two panels of a folded state are. That motion is not a slide. It is a turn of two hundred and forty degrees.

Two columns therefore leave a hundred and twenty degrees, and three leave nothing. The folded period is three drawn columns, and it is six, and it is not one, two, four, five or seven.

Why the two periods can differ at all #

A folded state assigns each panel a rigid motion of the plane. For the state to be periodic under a translation of the drawing, the motions of corresponding panels have to differ by a translation — that is what periodic means for a folded object.

Nothing forces that. The motions of corresponding panels differ by some rigid motion, and a rigid motion can be a translation, a rotation, a reflection or a glide. If it happens to be a translation, the drawn period is a folded period; if it is a rotation, it is not.

Which it is comes out of composing the reflections in the creases crossed on the way from one cell to the next. Two reflections in lines at angles $\alpha$ and $\beta$ compose to a rotation by $2(\alpha - \beta)$. A Yoshimura column is crossed by two diagonals at sixty degrees to each other, giving a rotation of a hundred and twenty — and the horizontal courses contribute another, taking it to two hundred and forty.

Three of those is seven hundred and twenty, which is two full turns and therefore nothing.

The consequence for gluing #

A sheet made by identifying opposite edges of a cell is a sheet whose folded state, if it has one, has to be invariant under the identification. So a cell one column wide cannot be glued: the fold would have to be unchanged by a turn of two hundred and forty degrees, and it is not.

That is a refusal, and it is not the refusal the grid produces.

The grid’s failure is a parity: the paper returns the other way up, there is no two-colouring, and the sheet has no flat folded state at all. The Yoshimura’s is nothing of the kind — its paper returns the same way up, it two-colours perfectly, and every path across a column crosses an even number of creases.

What the Yoshimura lacks is not a colouring. It is a periodic folded state at that period.

One number was reporting two facts #

Before the two were separated, the collection had a single count: how many identified panel pairs had folded motions that disagreed. That number is nonzero in both cases and it means different things.

A pair whose motions differ in the sign of the determinant has been turned over. That is the parity, it kills the two-colouring, and the sheet has no flat folded state on any account.

A pair whose determinants agree and whose matrices differ has been rotated. The sheet may fold perfectly well; what fails is the assumption that it folds periodically at that period.

Lumping them together gave a count that refused both cases and could not say which, and on the Yoshimura it refused a family whose only offence is that its folded period is three times its drawn one.

Seeing it with paper #

The turn is easy to be sceptical about and easy to check, and checking it takes one strip.

Fold a Yoshimura from a strip of paper: two courses of horizontal creases with the diagonals zigzagging between them, six or eight columns across. Collapse it. The result is a compact bundle, and the thing to look at is which way the panels point.

Mark one panel with a pencil line running along it before collapsing. Collapse, and find the marked panel and the one two columns over. The two lines are not parallel; they are at a hundred and twenty degrees to each other. Find the panel three columns over and the line is parallel to the first again.

That is the whole result, in the hand, and it is more convincing than the arithmetic because it is very hard to believe until it is seen. A pattern that repeats every column produces a folded object that does not.

The reason the effect is not common knowledge among people who fold Yoshimuras is that the collapsed bundle is small and the panels are stacked, so the direction any particular one is pointing is not something a folder normally has occasion to notice.

Why the drawn period was always assumed to be enough #

There is a reasonable-sounding argument that the two periods must agree, and finding the hole in it is instructive.

The argument: the pattern repeats, so the folding rule repeats, so the folded state repeats. Every panel folds the way its counterpart folds, because they see identical surroundings.

Every step of that is true and the conclusion does not follow. The folded state does repeat — the panel one column over is folded exactly as its counterpart is, in the sense that its shape, its layers and its relations to its neighbours are the same. What differs is where it is in the plane, and the map taking one to the other is a rigid motion of some kind.

The argument establishes that the folded structure is invariant under the drawing’s translation. It says nothing about the motion being a translation, and periodic for a folded object means the second thing rather than the first.

That is a distinction the collection had no reason to draw while every family it could glue happened to slide.

Where this matters and where it does not #

The turn changes nothing about the Yoshimura as an object anybody uses, and it changes several things about how it can be measured.

It does not affect folding a Yoshimura. A rectangle of the pattern collapses exactly as it always did, its rigid-foldability is untouched, and every bellows made from it works.

It does not affect the vertex conditions. Those read angles at a point and no point moved.

It does affect any gluing at a period that is not a multiple of three, which is refused.

It affects any claim about the folded object stated per drawn cell. The shrink factor, the folded footprint, the layer structure: each is a statement about the folded state and each has to be quoted per folded period, or per drawn period with a note that consecutive ones are rotated.

The last of those is where the correction actually bites, and it bites gently: the collection’s existing measurements of Yoshimura shrinkage are made on whole patches rather than per cell, so nothing published has to move.

The two periods, in general #

The general situation is easy to state and it is worth having, because the Yoshimura is not going to be the last case.

A repeating pattern has a lattice of translations under which its drawing is invariant. Folded flat, it has a lattice of translations under which its folded state is invariant, and the second is a sublattice of the first: every folded period is a drawn period, since a folded state that repeats certainly has a drawing that does.

The index of that sublattice is the number this essay is about. It is one when the fold slides, three for the equilateral Yoshimura across its columns, and one for the same pattern along its courses.

Computing it is a small search: try one drawn period, two, three, and see which is the first whose identified panels are related by a translation. What makes the search terminate is that the turn per period is a rotation by a fixed angle, and a rotation by a fixed angle either has finite order or has none — and if it has none, the pattern has no finite folded period in that direction and no cell of it can ever be glued.

None of the three families here is in that case, and the twist tessellations, whose turn is thirty-six point six degrees, very likely are.

Which families turn and which do not #

Of the three that can be glued, only one turns, and the reason is visible in the drawing.

The grid crosses one crease per period in each direction. One reflection is not a rotation, so consecutive cells are related by a reflection — which is the parity — and never by a nontrivial rotation.

The Miura crosses one across and four along. The four compose to a rotation by twice their alternating sum, and the Miura’s alternating sum is nought by the symmetry of its zigzag, so the rotation is nothing and the cells slide.

The Yoshimura crosses two per column, at sixty degrees to each other, and the composition is a genuine turn.

The turn is the shrink, seen from the side #

There is a connection here to a measurement this collection has already made and it is worth drawing.

Folding a twist tessellation flat is a similarity: the lattice comes back scaled by a fixed factor and turned by a fixed angle, and the angle is thirty-six point six degrees on every tiling tried. The scaling is the shrink and the turn is what makes the folded pattern sit at an angle to the drawn one.

The Yoshimura’s two hundred and forty degrees is the same kind of quantity measured on a different family — the turn part of the similarity between the drawing’s lattice and the folded state’s. Where the twists’ turn is an awkward angle that never closes, the Yoshimura’s is exactly two thirds of a full turn, which is why it closes at three.

That is the whole difference between a family whose folded period is finite and one whose folded lattice is a scaled and rotated copy that never lines up again.

Why the number is 240 and not something else #

The angle is not fitted and it is not approximate: it comes out at two hundred and forty degrees to fourteen figures, and it is worth saying where the exactness comes from.

The equilateral Yoshimura’s sectors are all sixty degrees — that is what equilateral means for this pattern, and it is why the big-little-big lemma says nothing at any of its vertices. A composition of reflections in lines whose angles are all multiples of sixty degrees has a linear part that is a rotation by a multiple of a hundred and twenty, and there are only three such rotations.

So the turn per column is nought, a hundred and twenty or two hundred and forty, and it is measured to be the last of the three. Being one of three exact values rather than a number near one of them is what makes the folded period exactly three rather than approximately so.

How it was found #

The result arrived as a contradiction between two computations rather than as a discovery, which is the usual way here.

A count of identified panel pairs whose folded motions disagreed was being used as the test for whether a gluing was sound. On the Yoshimura it fired at one column and at two, and not at three — which looked like a construction that worked at some sizes and not others, and construction bugs of that shape are common.

The check that separated the cases was an independent count: how many creases a path across the cell crosses. That count is even on the Yoshimura at every size, so the paper comes back the same way up, so whatever was failing was not a parity.

Two computations disagreeing about which sheets are sound is exactly the situation the collection builds pairs of computations for, and the resolution was that they were not disagreeing — they were answering different questions, and one number had been standing in for both.

Splitting it took one line: compare the determinants of the two motions, and if they have the same sign but the matrices differ, the failure is a turn rather than a flip. The Yoshimura’s turns came out at two hundred and forty, one hundred and twenty and nought, in that order, and the number was recognisable on sight.

The other family, checked #

The twist tessellations are the collection’s main tessellation work and it is fair to ask what their folded period is, since folding one flat is a similarity with a turn of thirty-six point six degrees and thirty-six point six degrees is not a rational part of a full turn.

They slide, at every size and on every tiling. Measured on the square, triangular, hexagonal and elongated twist tessellations at one and two periods, no identified panel pair has a rotation between its two motions: the folded period is one drawn period, and every cell of every one of them glues.

The apparent contradiction dissolves because the two turns are different quantities.

The Yoshimura’s two hundred and forty degrees is a rotation between the motions of two panels that the gluing says are one panel. If that is not nought, the folded state is not invariant under the identification and the sheet is refused.

The twists’ thirty-six point six degrees is the angle between the drawn lattice vector and the folded one. The paper repeats every cell in the drawing and every cell in the fold, and the two lattices sit at an angle to each other — which is what makes a folded twist tessellation look rotated relative to the pattern it came from, and is a completely different fact from any panel being turned.

A twist cell’s folded lattice vector at one period comes out at about eleven hundredths of a sheet across and eight hundredths down, which is a vector at thirty-six point six degrees below the drawing’s own axis. That is the shrink’s turn, measured on the glued object, and it is present while the panels slide.

So one family has a rotation between panels and a lattice that lines up; the other has panels that line up and a lattice at an angle. Both are turns and neither implies the other.

Two turns, kept apart #

Since the two quantities are both angles and both come from folding, it is worth putting them side by side once.

The turn between panels. Take two panels the gluing identifies. Their folded motions differ by a rigid motion; if that motion’s linear part is a rotation, the folded state is not invariant under the identification. This is a property of the cell size: it is two hundred and forty degrees for a one-column Yoshimura, a hundred and twenty for two columns, nought for three.

The turn of the lattice. Take the drawing’s own translation vector and the vector by which the folded state repeats. They are two vectors in the plane and the angle between them is the shrink’s turn. This is a property of the pattern: thirty-six point six degrees on all five twist tilings, at every cell size, and unchanged by how much of the pattern is being looked at.

The first can be nought while the second is not, which is every twist tessellation. The second is nought while the first is not on the Yoshimura’s short cells — its folded lattice runs along the drawing’s own axis and its panels do not.

Confusing them is easy because both are reported as the fold turns things, and the sentence is true of both and means different things.

What else the Yoshimura has been used for here #

The pattern turns up repeatedly in this collection and it is worth checking which of those uses are affected.

Its vertices are degree six and its cost is linear in its panel count — a statement about a patch, unaffected.

At exactly √3 its sectors are all sixty degrees and the big-little-big lemma says nothing — a statement about a vertex, unaffected.

It is a pattern nobody designed, found rather than invented — history, unaffected.

So the folded period touches none of the existing Yoshimura work, which is luck rather than care and is worth stating as such.

What has to be recorded now #

The practical consequence is a column that no pattern in this collection has ever carried.

Every pattern’s drawn period is recorded, because the drawing cannot be generated without it. Its folded period is recorded nowhere, and on two of the three families measured it is the same number, and on the third it is three times larger.

That is a gap of exactly the kind this collection has found before: a quantity that has been equal to another quantity on every case anybody looked at, and was therefore never given a name of its own. It is taken up separately, because the fix is a measurement rather than an argument.