An alternating sum of angles

Two conditions in this subject are alternating sums of angles, and they are about completely different objects.

The first is Kawasaki’s. Round an interior vertex the sectors between consecutive creases, taken alternately positive and negative, have to add to nought — equivalently, the odd sectors and the even sectors each add to a straight angle. It is a statement about one point of the paper, it is the oldest condition the subject has, and every folder meets it in the first week.

The second is about a sheet with no interior vertices at all. On a strip of paper glued into a band, the crease angles, taken alternately positive and negative, have to add to a multiple of a straight angle. It is a statement about the whole sheet, it constrains the drawing before a single crease is placed anywhere, and it has no vertex in it.

That two conditions of the same shape govern two such different situations is not a coincidence, and the common cause is worth having: both are saying that a product of reflections came back as the right kind of motion.

What the condition is #

A band’s creases run from one edge of the strip to the other. Each has an angle to the strip’s long direction — ninety degrees for a crease square across, sixty for one slanting forwards, a hundred and twenty for one slanting back.

The closure condition says the reflections in those creases, composed in order, have to equal the map that glues the band’s two ends together. That equation has a linear part and a translation part, and they behave differently.

The linear part is a two-by-two matrix and it depends on the crease angles and nothing else. Reflection in a line at angle $\phi$ has a linear part determined by $\phi$; where the line sits enters only the translation. So composing $k$ reflections gives a linear part determined by

$$\phi_1 - \phi_2 + \phi_3 - \cdots \pm \phi_k$$

and by nothing else about the drawing.

For a Möbius band the gluing map’s linear part is a reflection in the band’s own axis, at angle nought. So the condition is that the alternating sum is a multiple of a straight angle.

Decided before anything is placed #

The practical force of the condition is when it applies.

Every other constraint on a band involves where the creases are: whether they fit inside the strip, whether they are in order, whether they cross each other, whether the translation comes out right. All of those need positions.

The alternating sum needs none. Given three angles, the linear part is already determined, and if it does not match, no arrangement of those creases at those angles on a strip of any length will ever close. The drawing can be abandoned before it is drawn.

That is an unusual shape for a condition in this subject. Almost everything else here — the letters, the layer order, the packing — is a search over arrangements. This one refuses whole families of arrangements without looking at one.

The same arithmetic at a vertex #

Kawasaki’s condition is the same statement about a different loop.

Round an interior vertex, the panels form a cycle. Walking round it and composing the reflection in each crease returns a rotation, and the rotation is by twice the alternating sum of the sector angles between consecutive creases. The paper has to come back where it started, so the rotation has to be nothing, so the alternating sum has to be nought — which is Kawasaki, written as a composition.

So the two conditions differ in three ways and agree in the fourth.

They differ in which loop: a vertex’s loop is a circle round one point; a band’s is a circuit of the whole sheet.

They differ in which angles: a vertex’s condition reads the sectors between the creases; a band’s reads the directions of the creases.

They differ in what the composition has to equal: at a vertex, the identity, because the walk returns to the same paper; on a band, the gluing map, because it does not.

And they agree in why: both are the statement that a composed product of reflections is the motion the sheet demands, and in both cases the linear part of that product depends on an alternating sum and the positions do not enter it.

Why the sum alternates #

The alternation is not a convention, and it is worth seeing where it comes from, because it is the same in both cases.

Composing two reflections in lines at angles $\alpha$ and $\beta$ gives a rotation by $2(\alpha - \beta)$. The minus sign is not chosen: reflections do not commute, and the rotation their product produces turns from the second line towards the first. Composing four gives a rotation by $2(\alpha - \beta + \gamma - \delta)$ — the signs alternating because the products pair up.

An odd number of reflections composes to a reflection rather than a rotation, in a line at the alternating sum itself rather than at twice it. That is why a Möbius band’s condition reads the alternating sum is a multiple of a straight angle rather than twice it is, and why a cylinder’s, which needs an even count, reads as a condition on twice the sum.

The alternation is therefore a fact about reflections and not about paper, and it shows up in every condition of this subject that comes from composing them.

What the admissible set looks like #

One equation among three angles leaves a two-dimensional family, and inside the square of possible first-and-second angles it is a curve.

That has a consequence worth stating plainly: a triple of angles chosen at random folds nothing. Not usually nothing — nothing, with probability one, because a curve has no area. Whatever length the strip is given and wherever the creases are put, the linear parts do not match, and the mismatch is not small: it is a whole reflection out.

Angles first, then positions #

Once the alternating sum is right, the rest of the condition is two linear equations in the crease positions, which is a solve rather than a search.

That two-stage structure — a discrete-looking condition on the angles, then a linear condition on the positions — is what makes bands tractable in a subject where almost nothing is. It is also what makes the count of admissible drawings a strange quantity: there is a curve of admissible angle triples, and above each point of that curve, a line of admissible position triples, and a half-line of admissible lengths.

The ranking the condition does not do #

The alternating sum says which triples are admissible. It says nothing at all about which is best, and the difference between them is large.

That ranking comes from the fitting rather than the closing, and it is a reminder that the conditions of this subject stack rather than combine. The alternating sum is an equation. The positions are an equation. Fitting is an inequality, and it is the inequality that decides which of the infinitely many closing drawings a person would actually make.

A worked instance in degrees #

Take the condition at face value and try three angles by hand.

Sixty, a hundred and twenty, sixty: $60 - 120 + 60 = 0$. Admissible.

Forty-five, ninety, forty-five: $45 - 90 + 45 = 0$. Admissible.

Seventy, a hundred and ten, seventy: $70 - 110 + 70 = 30$. Not a multiple of a straight angle, so not admissible — and no length of strip and no placement of those three creases will ever produce a Möbius band. The mismatch is a reflection thirty degrees out of place, and it is thirty degrees out of place at every length.

Ninety, ninety, ninety: $90 - 90 + 90 = 90$. Not admissible, which is the square-creased case and the reason no band creased straight across the strip folds at any odd count.

Thirty, sixty, thirty: $30 - 60 + 30 = 0$. Admissible, and it needs a long strip, because a crease at thirty degrees spans nearly two widths along the band and three of them have to fit in order.

The arithmetic is that easy, and it is the whole of the first stage. Anybody with a protractor and a strip can decide in ten seconds whether a proposed set of angles has any chance, which is not a sentence that can be written about many conditions in this subject.

The second angle is free and the third is not #

There is a small asymmetry in the condition that is easy to misread.

Given the first angle, the second may be anything; given both, the third is determined up to a straight angle. So the admissible set is parameterised by two free choices and a determined third, which is why the picture above is a plot of the first two with a mark wherever the third exists.

The mark is present at almost every pair, because some third angle always exists — the equation $\phi_3 = \phi_2 - \phi_1$ always has a solution. What restricts the picture is that the third angle has to be an angle a crease can actually have: a crease at nought or a hundred and eighty degrees runs along the strip rather than across it, and does not divide the band into panels at all.

So the admissible region is the whole square minus two thin strips near its edges, and the curve the earlier section spoke of is the curve in the space of triples rather than in the space of pairs. Both descriptions are right and they are descriptions of different spaces, which is worth being careful about: the set of admissible triples is two-dimensional inside a three-dimensional space, which is measure zero there, and its shadow on the space of first-and-second angles is almost everything.

That distinction is exactly the one the genericity argument turns on, and getting it the wrong way round makes a band look far easier to fold than it is.

What happens at four creases and beyond #

The essay has been about three creases because three is the smallest odd number that can work. The condition generalises without changing shape and the consequences shift.

At five creases the alternating sum is $\phi_1 - \phi_2 + \phi_3 - \phi_4 + \phi_5$, and there are four free angles with the fifth determined. The admissible set is bigger and the solve for positions is still two equations, now in five unknowns, so the solution set is three-dimensional and there is real freedom in where the creases go. Bands with five creases fold on shorter strips than the arithmetic might suggest, because the extra freedom can be spent on fitting.

At even counts on a Möbius band, nothing works, and it is refused before the angles are reached: the parity of the composition is wrong whatever the angles are.

On a cylinder the roles swap. Even counts are the ones that can work; the alternating sum condition becomes a condition on a rotation rather than on a reflection; and two parallel creases square across the strip satisfy it trivially, which is why a paper tube flattens with two creases along its length and a tube of any circumference will do.

The general statement covering all of these is one sentence: the composed linear part, determined by the alternating sum, has to equal the linear part of the sheet’s own identification. Everything else is which sheet and how many creases.

Why nobody states it about vertices this way #

Kawasaki’s condition is normally stated as the alternating sum of the sectors is nought, or equivalently as the odd sectors and the even sectors each add to a straight angle, and it is normally proved by an argument about the paper closing up rather than by composing reflections.

The reflection proof is not better and it is more general, which is the only reason to prefer it here. It says: walk round the vertex, compose the reflections, require the identity; the linear part of the composition is a rotation by twice the alternating sum of the sectors, so the sum must be a multiple of a straight angle; and the sectors are positive and add to a full turn, which pins it to nought.

Written that way, the condition on a band is the same proof with the vertex’s loop replaced by the band’s, and the identity replaced by the gluing map. Written the usual way, the two look unrelated, and the second one has to be discovered rather than derived.

That is most of the argument for keeping the general form around even where the special case is enough: the general form is what makes a second instance recognisable when it turns up.

The condition, checked #

The alternating sum is asserted rather than trusted, and the assertion has a case it must reject.

Every band drawn here has its composition computed and compared against the map its own gluing demands, and the comparison returns a distance in widths of the strip rather than a verdict. A triple whose alternating sum is right and whose positions were solved reads nought to rounding. A triple whose alternating sum is thirty degrees out is asked for and refused, and the refusal is checked: if the machinery ever reported seventy, a hundred and ten and seventy as folding, the figures would stop being drawn.

That is the shape of assertion this collection prefers. A check that has never rejected anything is a comment about the code rather than a statement about the mathematics, and a condition this cheap to state deserves at least one case that it correctly declines.

Where the condition is silent #

Three things it does not say, each of which has caught somebody.

It says nothing about which sheet. A cylinder’s gluing map has a linear part that is the identity, so its condition on an even number of creases is that the alternating sum is a multiple of a straight angle and twice it is a multiple of a full turn — which for parallel creases is automatic. The condition takes a different form on the two sheets and it is the same equation.

It says nothing about the letters. Mountain and valley never appear above. The composition treats both alike, because folding along a line reflects the paper across it whichever way the fold goes, and the whole of this argument is about where the paper ends up rather than which way it went.

It says nothing about a sheet with vertices. A band’s creases meet nothing, so the sheet has exactly one loop and one equation. Add a vertex and there are as many loops as the panel graph has independent cycles, and requiring all of them is the full flat-foldability problem, which is hard.

An angle is a direction, not an orientation #

One more piece of care, because it is where a sign gets lost.

A crease is a line, and a line has a direction only up to reversal: the crease running at sixty degrees is the same crease as the one running at two hundred and forty. So the angles in the alternating sum are defined modulo a straight angle, not modulo a full turn.

That is why the condition reads a multiple of a straight angle rather than nought, and it is why the admissible triples come in families: sixty, a hundred and twenty, sixty is the same drawing as sixty, a hundred and twenty, two hundred and forty, and the sum changes by a straight angle between the two ways of writing it.

The consequence for computation is small and the consequence for reading the condition is not. A reader who fixes each angle in the range from nought to a hundred and eighty — which is the natural thing to do, since a crease crossing the strip has an angle in that range — will find the alternating sum of three of them lying between minus a hundred and eighty and plus three hundred and sixty, and the condition admits nought, a hundred and eighty, and minus a hundred and eighty as the same answer.

Getting that wrong makes a third of the admissible triples invisible.

Two conditions on one drawing #

A last placement, since the essay has two alternating sums in it and the collection has more.

Kawasaki’s condition at a vertex and the band’s condition on its crease directions are the two alternating sums here. The big-little-big lemma reads angles too and is not an alternating sum — it is a comparison, picking out the strictly smallest sector, and it says nothing at all when no sector is strictly smallest.

So of the subject’s three angle conditions, two are alternating sums arising from compositions of reflections and one is not, and the odd one out is the one that can be vacuous.

What is worth carrying #

Three sentences.

A composition of reflections has a linear part that depends only on the directions of the lines, through an alternating sum, and a translation that depends on where they are.

A closed walk on a sheet has to compose to the motion that walk’s own identification demands, which on a disc is the identity and on a glued sheet is not.

Put those together and every alternating-sum condition in this subject is the same condition asked about a different loop — which is why Kawasaki’s, discovered at a vertex, and a band’s, discovered on a sheet with no vertices at all, are written with the same arithmetic and could hardly look less alike.