A crease with no vertex to belong to

Crease density is the total length of crease divided by the area of paper: how much line is on the sheet. It is a material quantity, it predicts how a sheet behaves under load, and it is one of the few numbers in this subject that a manufacturer can measure directly.

Everything else about a crease is measured somewhere else entirely — at the vertex it runs into, where the angles are and where the conditions live.

A band of paper has the first and none of the second.

The object #

A rectangular strip with creases running across it from one long edge to the other, and its two ends glued.

Each crease has two ends and both are on the boundary. No crease meets any other. So the sheet has crease length, it has density, and it has not one interior vertex at any crease count on either gluing.

Every condition holds #

The four conditions this subject checks are developability, Kawasaki, Maekawa and the big-little-big lemma. Each reads the creases meeting at one interior point of the paper.

On a band there is no such point, so all four hold, vacuously, on every band at every crease count.

That is not a defect in the conditions. It is the exact statement of what a local check is: a claim that nothing goes wrong at any point, which on a sheet with no points where anything could go wrong is true and empty.

And the sheet still refuses #

Half of the bands measured have no flat folded state.

A cylinder needs an even number of creases round it and a Möbius band needs an odd number, because the seam contributes a factor to the parity. The ones with the wrong count refuse, and nothing at any vertex could have said so, since there are no vertices.

So a sheet can have density, satisfy every local condition, and not fold — which puts the vertex where it belongs in the account: as the place where most of the difficulty happens to live rather than as the place where difficulty is.

What density does capture #

The measure is not diminished by any of this and it is worth saying what it is for.

Crease density is about the material. A sheet with a lot of line on it is weakened, bends more easily in the directions its creases run, and behaves as something between a solid and a mechanism. That is a physical statement and every crease contributes to it whether or not it meets another.

So a band’s density is a perfectly good number: total crease length over area, exactly as for any other sheet, and it says what it always says about how the paper will behave.

What it does not say is anything about folding, and it never claimed to.

The density of a band, computed #

Since the essay is filed under density, the number is worth having.

A band of length $L$ and width $W$ with $k$ creases square across it has crease length $kW$ and area $LW$, so its density is $k/L$ — creases per unit length, independent of the width.

At the shortest Möbius band that folds, $L = \sqrt3 W$ with three creases at sixty degrees, so each crease has length $W/\sin 60° = 2W/\sqrt3$, the total is $2\sqrt3 W$, and the density is $2\sqrt3 W / (\sqrt3 W^2) = 2/W$.

Two creases’ worth of line per unit width, which for a strip four centimetres wide is fifty lines per metre — a low density by the standards of any tessellation, and the object is nonetheless one of the more constrained sheets here.

That is the essay’s point in one number. Density measures the material and says nothing about the difficulty, and the two are as nearly independent as two quantities about one sheet can be.

Where else a crease meets nothing #

The band is the extreme case and the situation is not confined to it, which is worth showing.

A ring of paper. The annulus with radial creases has creases from the hole to the rim, meeting nothing. Same situation, same vacuous conditions, same global refusal — and it has been in this collection for a long time as the standing example of a global obstruction.

A simple pleat. A sheet folded into parallel pleats has creases running edge to edge and meeting nothing. It has no interior vertices, it folds trivially, and every condition holds vacuously — which nobody notices because it folds.

The boundary of any pattern. Creases running to the sheet’s edge have one end at a vertex and one end at nothing, and the end at nothing is where the free letters come from and where a propagation stops.

So the phenomenon is ordinary. What is unusual about a band is that it has nothing else, so there is no local structure to distract from the global condition.

The checker’s blind spot, priced #

The collection has an essay about what a checker cannot check, and the band gives it a number.

A pattern checker reads interior vertices and evaluates four conditions at each. On a band it reads nought vertices and evaluates nought conditions, and returns a pass.

That pass is not a bug. It is a correct report that no condition was violated, and it is indistinguishable — from the outside — from a pass on a pattern where sixty vertices were checked and all sixty held.

The repair the collection made is to refuse rather than pass: a pattern with no interior vertices is reported as nothing to check rather than as checked, and a generator that means it has to say so. That distinguishes an empty ledger from a full one, which is the whole of what was wrong.

It is a small change and the failure it prevents is exactly this essay’s object: a three-crease ring, certified by every check the subject has, that cannot be folded by anybody.

What a manufacturer measures #

Since density is the material quantity, it is worth saying what it is used for and why the folding conditions do not enter.

A creased sheet is weaker in bending across its creases and stiffer along them, and how much depends on how much line there is and how it is distributed. That is what density predicts, and it predicts it for any crease whatever — meeting another or not, folded or not, in a pattern that folds flat or one that does not.

So a manufacturer specifying a creased material cares about density and does not care whether the pattern is flat-foldable. Those are different products: one is a material with anisotropic stiffness and the other is a mechanism.

The band is a good illustration of the divergence, because it has a perfectly ordinary density and its folding behaviour is decided by a parity that no material property could see.

What is left to measure #

If a band has no vertices, the question is what quantities it does have, and the list is short and interesting.

Density, as above.

A crossing count. How many creases a loop that cannot be shrunk crosses, which is the parity that decides everything.

Crease angles, which have no vertex to be sectors of, and which turn out to satisfy a condition of their own — an alternating sum.

A composed motion. The reflections in the creases, multiplied, which has to equal the map that glues the sheet.

That is a complete list for a band, and only the first is a density. The rest are global, and the collection had none of them before there was a sheet with no vertices to force the issue.

Two quantities that look alike #

Crease density and vertex density are both quantities per unit area and they behave completely differently, which is worth separating since the essay is about their divergence.

Crease density is length over area. It is continuous in the pattern: nudge a crease and it changes a little. It is defined for any drawing, including one with no vertices at all.

Vertex density — vertices per unit area — is a count over an area. It is discontinuous: two creases that nearly meet contribute nothing, and the same two moved to meet contribute one.

That discontinuity is why the second is not a material quantity. A sheet with two creases that nearly touch behaves, mechanically, exactly like one where they touch, and the folding behaviour is completely different.

So the subject has one quantity that is continuous and about the material, and one that is discontinuous and about the folding, and a band is the case where the second is nought and the first is not.

The shortest statement #

Three sentences, since the essay’s content is a separation rather than a result.

A crease has a length, and lengths add up to a density, which is about the material.

A crease has ends, and ends at a vertex carry conditions, which are about folding.

A crease with both ends on the boundary has the first and not the second — and the sheet it is on can still refuse, for a reason that is about the sheet rather than about any crease.

Where the density measure was defined #

Worth a note on provenance, since the essay is partly about what a measure was for.

Crease density was introduced in this collection to answer a manufacturing question: how much line a pattern puts on a sheet, and therefore how much the material is weakened and how much work a machine has to do. It is measured as a rate and reported per pattern, and it is one of the very few numbers here that transfers directly to a factory.

Nothing about it was ever a claim about folding, and the essay is not a correction to it. What the band does is provide a case where the two are maximally separated — a sheet with an entirely ordinary density whose folding behaviour is decided by a parity — and having such a case is useful precisely because it stops the two from being confused when they usually track each other.

On a tessellation they do track each other, roughly: denser patterns have more vertices, more conditions and harder searches. That correlation is real, it is not causal, and a band is the counter-example that shows so.

What a band is good for #

Having spent the essay on what a band lacks, it is worth naming what makes it useful.

It is the only object in this collection where the global conditions can be studied with nothing else in the way. No vertices, no local conditions, no propagation, no search: one loop, one parity, one composed motion.

That is why the whole of the closure condition’s general form was worked out on bands. On a pattern with vertices the condition holds round every loop of the panel graph and there are many; on a band there is one, and the equation can be written down and solved.

A band is therefore the subject’s simplest non-trivial object, in the specific sense of having exactly one thing that can go wrong. Almost everything else here has several, interacting.

Why the vertex became the unit #

It is worth asking why the subject organises itself round vertices at all, given that the interesting conditions here are not at any.

Because on a disc of paper, almost everything is. A pattern of any complexity has creases meeting, the conditions at those meetings are strong, and a pattern all of whose vertices pass usually folds — usually, not always, which is the subject’s most-repeated caution.

And because a vertex condition is checkable. It reads a handful of numbers at a point and answers, which makes it the natural unit for a checker.

So the vertex is the unit for good reasons and it is not the whole of the subject, and a band is the cleanest available demonstration of the gap.

Making one, and feeling the difference #

Two strips settle the essay’s claim in about three minutes, and doing it is worth the time because satisfies every condition and does not fold is a sentence that invites disbelief.

Cut two strips of paper, each about thirty centimetres by three. Join the ends of each into a loop with tape, without any twist.

Crease the first at four places, square across. Crease the second at three.

Press each flat. The four-crease loop goes flat immediately, into a four-layer strip. The three-crease loop resists, and pressing harder produces a fourth crease somewhere.

Now look for a vertex on either of them. There is none: every crease runs from one long edge to the other and meets nothing. Both loops have the same kind of crease, the same density of line per area to within a third, and opposite verdicts.

The difference is a count of how many creases a finger travelling once round the loop crosses, which is not a property of any point of the paper and is not a property of the material.

What the density of each one is #

Completing the comparison with the numbers.

The four-crease loop: four creases, each three centimetres long, twelve centimetres of line on ninety square centimetres of paper. About point one three centimetres of line per square centimetre.

The three-crease loop: nine centimetres of line on the same ninety. About point one.

Two densities within a third of each other, one object that folds and one that cannot. That is the whole of the essay in two strips of paper, and it is the reason the collection keeps the two vocabularies apart.

The measure and the mechanism #

There is a general observation here about measures, and it is worth stating because this subject has several of them.

A measure is chosen for what it predicts. Crease density predicts mechanical behaviour and it is the right measure for that. It is not a measure of anything about folding, and the reason is not that it is a bad measure but that folding is not a quantity per unit area.

Whether a sheet folds is a yes or no about the whole sheet. It is not additive, it is not local, and it does not have a density. Two halves of a sheet can each fold and the whole fail, which is the content of the local-versus-global distinction and is exactly the property a density cannot have.

So the subject has measures for the material and verdicts for the folding, and the two families do not convert into each other. A band is a clean case because its density is ordinary and its verdict is decided by a parity, and there is no arithmetic relating them at all.

That is worth carrying whenever a quantity in this subject is being asked to do more than it was defined for.

A crease that belongs to the sheet #

The right description of a band’s crease is that it belongs to the sheet rather than to a point.

It divides the band into panels. It contributes to a loop’s crossing count. Its angle enters an alternating sum. Its reflection enters a composed motion. Every one of those is a statement about the crease and the sheet together, and none of them is local.

That is a category the subject does not usually have, and it is not exotic: a crease running from one edge of a square to the other, with nothing crossing it, is in exactly the same position and appears in plenty of ordinary patterns.

What is different about a band is that all its creases are like that, so the local vocabulary has nothing to say and the global vocabulary has to say everything.