The band that needs an odd number

Every crease drawn in this collection carries a letter. A mountain is a crease whose fold takes the paper away from the reader; a valley brings it toward them. The letters are the first thing a folder learns, the thing a crease pattern is coloured by, and the thing Maekawa’s condition counts.

They are also, quietly, relative to a side. Turn a folded model over and every mountain becomes a valley. Nobody minds, because turning the model over is a thing the reader does rather than a thing the paper does, and the paper has a front.

A Möbius band does not have a front.

Following a face round #

The demonstration is older than any of this and takes twenty seconds. Join a strip into a band with a half twist, put a pencil on it and draw a line down the middle without lifting the pencil or crossing an edge. The line comes back to its own start on what looked, when the drawing began, like the other face. There was never another face.

That is the whole property, and everything below is a consequence of it. A sheet where a walker can return to the start having exchanged the two faces is called non-orientable, and it is the first such sheet this collection has been able to build, because everything it could build before — a square, a square with holes, a cell of a tessellation with its opposite edges joined — has two sides.

The consequence for the letters is immediate. Take a crease, look at it, call it a mountain. Walk once round the band and come back to the same crease. It is now a valley, and no fold was made and no paper moved: the walker is looking at the same physical crease from the other side, and there is no other side, so the two readings are the same reading.

What survives, and what does not #

The first reaction is that the subject’s whole vocabulary has been lost, and that is too strong. Precisely one thing is lost and quite a lot survives, and the line between them is worth drawing carefully.

The assignment does not survive. A function giving each crease a letter, defined on the whole sheet and consistent, does not exist. Any attempt to build one is a walk, and the walk contradicts itself on its way round.

The crease pattern survives untouched. Which creases there are, where they run, what angles they make with each other, which panels they separate: all of that is drawn on the paper and cares nothing about sides.

Maekawa’s condition survives, and this is the interesting one. The condition says that at an interior vertex the number of mountains and the number of valleys differ by exactly two. Swap every letter and the two counts exchange places; their difference is unchanged in size. So the condition is a statement the two readings agree about, and it can be asked of a vertex on a non-orientable sheet even though neither reading is preferred.

Kawasaki’s condition survives trivially, because it reads angles and angles have no sides.

The two-colouring survives, in the corrected form. It is not a colouring of panels into front-up and back-up any more, because there is no front. It is the statement that the signs multiply to one round every closed path, and the seam of a Möbius band contributes a sign of its own. That is why the parity inverts: the band needs an odd number of creases where a loop of paper needs an even one.

Two things that are easy to run together #

There is a distinction here that repays being slow about, because the two statements sound alike and only one of them is about the drawing.

The colouring is about panels and it exists on this sheet. It is not a two-colouring in the usual sense — the two colours are not front and back, since those are not defined — but the underlying object is a consistent assignment of two labels to panels with adjacent panels differing, and on a five-crease Möbius band it exists.

The assignment is about creases and it does not exist. There is no consistent way to say which creases are mountains.

Both are parity statements and they behave oppositely, which is the source of most of the confusion. The colouring survives because it is asked as a relative question — do these two panels differ — and a relative question is unaffected by the absence of an absolute reference. The assignment fails because it is asked as an absolute one.

The same distinction is familiar from elsewhere: on a surface with no preferred normal, the angle between two directions is perfectly well defined and the sense of rotation is not.

What a folded state on such a sheet is #

If the letters are gone, it is fair to ask what has been folded.

A flat folded state is a map from the sheet to the plane that is an isometry on every panel. That definition never mentions a letter. It says where each piece of paper goes, and it says so by giving each panel a rigid motion of the plane. Two panels sharing a crease have motions differing by the reflection in that crease, which is a statement about motions and not about sides.

So a flat folded state exists or does not exist, on any sheet, with or without a consistent notion of front — and on a five-crease Möbius band one exists. What is missing is only the summary of it in letters. The paper is folded; the diagram cannot be drawn in mountain and valley, because a diagram in mountain and valley presupposes a reader standing on one side of the whole sheet.

That is why the condition is worth stating as a composition rather than as a count. The count needs a letter to count; the composition needs only a crease.

The census, and its symmetry #

Setting the two sheets against each other across a range of crease counts shows the inversion cleanly, and shows also that the inversion is exact rather than approximate: every count at which one sheet folds is a count at which the other does not.

Two computations produce the table. One multiplies signs round the panel cycle. The other composes reflections and compares the result with the map the gluing demands. They agree at every entry, which is the check that matters, because a claim that a rule is exactly reversed on a second object is precisely the shape of claim a dropped sign would produce.

How far the refusals differ #

A bit says which bands fail. It does not say how badly, and the two kinds of failure here are not the same size.

The distinction is not decoration. A band that misses by a translation is a band that would close if the strip were a different length, and that is a solvable problem — the positions of the creases are unknowns and the closure is a linear condition on them. A band that misses in its linear part cannot be repaired by moving anything, because no arrangement of a fixed number of reflections changes whether the product turns the paper over. The first refusal is about a drawing and the second is about a count.

The cylinder, for contrast #

None of the strangeness is caused by gluing as such. A cylinder is glued too, and it behaves in the way the subject expects.

A cylinder has two sides. A walker round it comes back the way they went, the letters are consistent, the assignment exists, and the parity is the familiar one. Everything the subject says about letters continues to hold, and it holds because the seam happens to contribute nothing.

That is the point the pair of sheets is here to make. The subject’s vocabulary is not built on paper but on two-sided paper, and the two are the same thing for every sheet anybody normally folds. The distinction becomes visible only when a sheet is available that separates them, and until one is, the hypothesis reads as a fact about paper.

Where the letters do live #

The assignment is not lost so much as displaced, and saying where it goes makes the situation less mysterious than there are no letters suggests.

Go round the band twice instead of once. A walk of two circuits returns the paper the way it started, because two exchanges of the faces cancel, so a strip twice as long joined without a twist covers the band exactly: every point of the band corresponds to two points of the longer strip, and the longer strip is an ordinary two-sided cylinder.

On that cylinder the letters are perfectly well defined, and they have a property that says everything. The two points covering one crease of the band get opposite letters — one is a mountain and the other is a valley — because the two of them are the same crease seen from the two sides. A five-crease Möbius band lifts to a ten-crease cylinder, ten is even, and the cylinder folds. The parity works out on both sheets and the arithmetic is the same arithmetic.

So the honest description is not that the band has no assignment. It is that the band’s assignment lives one level up, on a sheet that covers it twice, and is anti-invariant under the deck exchange rather than invariant. A quantity that comes back negated after a circuit is a familiar object elsewhere and has no established name in this subject, which is part of why the situation reads as a paradox rather than as a bookkeeping fact.

This is also the practical way to compute anything about such a band. Every existing piece of machinery here assumes a sheet with sides; handing it the double cover and remembering that the answers come in pairs is cheaper and safer than teaching it about sheets that have none.

In the hand, and what a folder actually does #

The band that folds is worth making, because it is a pleasant object and because doing it settles a suspicion that all of this is bookkeeping.

Take a strip about four centimetres wide and a little over seven long — the reason for that proportion is a whole essay of its own and for now it is a recipe. Join the ends with a half twist and a piece of tape. Then crease it three times, not square across but at sixty degrees to the edges, alternating the direction of the slant. Press.

It goes flat into an equilateral triangle, with three layers everywhere and each of the three creases forming one of the sides. Turn it over and it looks the same, which it would, since the two apparent sides of the flattened object are the one side of the paper.

What a folder notices immediately is that the letters cannot be written on it. Each of the three creases is a mountain from where one is standing and a valley from the other end of the same piece of paper, and there is no vantage point that makes them agree. A folding diagram for this object cannot be drawn in the usual notation, and that is not a limitation of the notation’s authors: the object has no property for the notation to record.

What the folder can record is the geometry — the strip’s proportion, where the creases go, what angle they make — and that is complete. The paper is completely specified by it, and anybody with the strip can reproduce the fold exactly.

The order the layers come in #

Two-colouring and letters are the first two global questions this subject asks. The third is the layer ordering: given that the paper folds, which sheet lies over which.

The Möbius band inherits the sharper version of a difficulty a sheet with no edge already has. On a disc, the enumeration of stackings starts from the bottom panel — the one with nothing below it — and works up. That panel exists because the paper has an edge, and the edge is where the bottom of a stack turns out to live.

The flattened triangle has one edge: the strip’s own long sides, which survive the gluing and become the triangle’s boundary. So there is a bottom layer, and the enumeration has somewhere to start. What it does not have is a consistent statement of what below means, because below is a direction relative to the reader and the reader is standing on a sheet with one side. Walking round the band, the panel that was below arrives above.

The resolution is the same one the letters take. Below is not a property of a panel; it is a relation between two panels, and relations survive where properties do not. The order of the three layers of the triangle is perfectly definite as a cyclic arrangement and has no top and no bottom that the whole sheet agrees about.

Why the case was never met #

The Möbius band is a hundred and sixty years old and completely standard, and origami has been mathematical for forty. It is fair to ask why the interaction was not noticed long ago, and the answer is not that it is deep.

It is that nobody folds one. The subject’s objects are squares and rectangles, occasionally with holes; its applications are sheets that have to pack into a volume or deploy out of one, and none of them is non-orientable. A designer has no reason to build such a sheet and a manufacturer has no way to.

So the hypothesis went unstated for the ordinary reason a hypothesis goes unstated, which is that no counter-example was in the room. The theorems are correctly proved for the sheets they are proved about; what is missing is a sentence naming those sheets, and a sentence naming a hypothesis is only ever written by somebody holding a case that violates it.

The counter-example arrives here as a by-product rather than as a discovery. Gluing a rectangle’s opposite edges was built to ask what a boundary costs, and gluing with a flip is one extra line of the same construction. Having built it, the parity result is a two-line consequence, and the letters result is one line after that.

One more thing the swap leaves alone #

There is a short list of quantities that survive a global exchange of the letters, and it is worth having, because it is the list of things that can still be said about a crease pattern on such a sheet.

The number of creases survives, trivially. So does the crease length, the panel count, and every angle at every vertex. So does the difference between the two letter counts at a vertex, in size if not in sign, which is Maekawa. So does the big-little-big lemma, which says that the two creases bounding a strictly smallest sector cannot both take the same letter — a statement about two creases agreeing, and agreement is preserved by swapping both.

What does not survive is anything that names one letter rather than the other. This crease is a mountain does not survive. Neither does the outermost fold is a valley, nor any instruction in a folding sequence, nor a count of mountains on the sheet.

The pattern is clear once it is written down: relations survive and properties do not. Every condition this subject checks turns out to be a relation, which is why all four of them can be asked of a sheet with one side and answered.

That is not a coincidence and it is not luck. The conditions are about whether the paper can be in one place at one time, and being in one place at one time is not a fact about which way a reader happens to be looking.

What the letters are for elsewhere #

Since the essay is about losing them, it is worth recalling what they do on an ordinary sheet.

A lettering is what a search for a flat folded state searches over, and how many of them fold is one of the subject’s basic counts. The order the layers come in is decided by them, and a folding diagram is a sequence of instructions in them.

All four of those presuppose a sheet with two sides, and all four are unavailable on a band with one — which is a substantial loss and is confined to an object nobody folds.

The thing this does not license #

It would be easy to over-read the result into a claim that mountain and valley are somehow arbitrary. They are not.

On a two-sided sheet the assignment is defined up to one global swap and no more. Choosing a side fixes every letter at once, and the choice is a convention in exactly the way that choosing which end of a ruler is zero is a convention: it changes every reading by the same operation and no measurement depends on it. Every folding diagram ever printed makes that choice silently and correctly.

What the Möbius band shows is that the choice is available because of a property of the sheet, and that the property has a name and can fail. That is a smaller claim than the letters are conventional and a more useful one, because it says exactly which sheets the convention is available on.

And it leaves the harder question open. Having the right parity is necessary and is a long way from sufficient — a Möbius band creased square across the strip has the right parity at every odd count and folds at none of them — so the letters are not the obstacle they might appear to be. The obstacle is geometric, and it takes a different instrument to see.