The proportion a band asks for

Two numbers, both square roots, both proportions of paper, and they are different kinds of quantity.

√2 is the A series: a rectangle whose sides are in that ratio halves across the long side into two rectangles of the same ratio, which is why an A4 sheet is half an A3 and the whole series nests.

√3 is what a Möbius band needs to be folded flat into a triangle: a strip at least $\sqrt3$ times as long as it is wide, creased at sixty degrees in three places.

A shape #

The A series’ number is a shape, and what makes it one is that the property fails on either side.

A rectangle at $\sqrt2$ halves into its own proportion. A rectangle at 1.4 halves into 1.43, and at 1.42 into 1.41; each is close and neither is the same shape, so the nesting drifts and the series does not close.

So the property picks out one proportion exactly, and a sheet either is that shape or is not.

A bound #

The band’s number is a minimum, and what makes it one is that the property holds on one side and fails on the other.

A strip at $\sqrt3$ folds into a triangle. A strip at 1.8 folds too, with room to spare, and the creases are further apart. A strip at 1.7 does not fold at all — the creases the closure requires do not fit between the ends in order.

So the property is monotone in the length, the number is where it switches on, and every longer strip works.

What each buys #

The distinction is not academic; it decides what to do with the number.

A shape is something to cut paper to. The A series exists because a paper mill cuts to it and the nesting is the benefit; a sheet slightly off is slightly wrong.

A bound is something to check paper against. Any strip above $\sqrt3$ makes a Möbius triangle, so a folder cuts whatever is convenient and confirms it is long enough.

And there is a practical asymmetry: a bound is forgiving and a shape is not. Cut a strip five per cent long and the band still folds; cut an A4 sheet five per cent long and the series stops nesting.

Why the band’s number is a minimum at all #

The reason is worth having because it explains why the two kinds of proportion arise.

The closure condition is an equation, and it has a solution at every length. The crease positions the equation demands move as the strip grows, and they move more slowly than the strip does.

So there is a length below which the solution’s creases would have to lie outside the paper or cross each other, and above which they fit. That is a threshold, and thresholds are what monotone constraints produce.

The A series’ number comes from an equation rather than an inequality — halving must return the same proportion — and an equation with one unknown has isolated solutions rather than a half-line.

So the two kinds of proportion come from the two kinds of condition, and the subject has both.

Where the √3 comes from #

Since the number is the essay’s subject, it is worth deriving rather than asserting.

The folded band is an equilateral triangle. Its three panels are equilateral triangles of side $s$, stacked three deep, and that is what the closure condition’s solution turns out to be at the shortest length.

The strip’s width is the triangle’s height, $s\sqrt3/2$.

The strip’s area is three triangles, $3 \cdot s^2\sqrt3/4$.

Divide the second by the first and the strip’s length is $\tfrac32 s$. So

$$\frac{L}{W} = \frac{\tfrac32 s}{s\sqrt3/2} = \frac{3}{\sqrt3} = \sqrt3.$$

The solved drawing agrees to seven figures: at a width of one it puts the creases at nought and 1.1547, and $1.1547$ is $2/\sqrt3$, the side of an equilateral triangle of height one.

Note what the derivation does not do: it does not show that shorter strips fail. That comes from the fitting — the creases the closure demands would have to cross each other — and it is why the number is a threshold rather than a special value.

The number’s other life #

The same $\sqrt3$ appears in a different subject and the coincidence is worth naming carefully.

A smooth Möbius band, made from a strip bent without creases, has a minimum length-to-width ratio too, and it is also $\sqrt3$. That is a genuinely hard theorem in differential geometry, resolved only recently, and it is about a developable surface rather than about a folded one.

The two problems are related as a limit and the relation is not established here. What is computed above is the folded case, by composing three reflections and solving two linear equations; the agreement with the smooth case’s number is reported and not explained.

That the two agree is the sort of fact that deserves a sentence saying it has not been proved, and this is it.

A shape can also be a bound #

The categories are not exclusive and it is worth saying so before the essay’s distinction is over-applied.

The A series’ √2 is a shape for the halving property and it is not a bound for anything. But a rectangle can have both kinds of condition on it at once — a proportion required for one property and a minimum required for another — and a designer often faces both.

The band is exactly that case: an equation on its angles and an inequality on its length. Its angles are a shape condition and its length is a bound condition, on one object.

So the useful question is not is this number a shape or a bound but which condition produced it. An equation gives isolated values; an inequality gives a half-line; and a quantity satisfying both has some of each.

That is the distinction worth carrying, and it is a distinction about conditions rather than about numbers.

What a folder should do with each #

Practically, since the essay’s value is in the doing.

For a shape, cut carefully. The property holds at one proportion and degrades away from it, and how fast it degrades decides how careful to be. For the A series, a per cent is enough to be visible after three halvings.

For a bound, cut generously. The property holds everywhere above the threshold, so err on the side of more paper, and there is no benefit at all to hitting the number exactly.

For the Möbius band specifically: cut a strip about two widths long rather than 1.732. It folds, the creases are comfortably apart, and the folded triangle is a triangle with a little extra paper at the seam rather than a construction balanced on a boundary.

Anybody cutting to exactly $\sqrt3$ gets the degenerate case, where consecutive creases meet at the strip’s edges — which is elegant on paper and awkward in the hand.

Other bounds in the subject #

Once the category is named, several familiar numbers turn out to be bounds rather than shapes.

The paper a design needs. A packing has to fit in the square, so a design has a minimum sheet size and any larger sheet works. That is a bound.

The thinness a complex model needs. Layers accumulate, so a model has a maximum paper thickness and any thinner paper works. A bound, in the other direction.

The strip a fold-and-cut pattern needs. A minimum, likewise.

Against those, the shapes are a much shorter list: the A series’ √2, the silver and golden rectangles, and the proportions that make particular classical bases come out square. Each is an equation’s solution.

The family of square roots #

The √n rectangles are a family and it is worth saying what each member is good for, since the band’s √3 joins a list.

√1 — the square. Its own shape, and the sheet the whole subject is built on.

√2 — the A series. Halves into itself, which is the one property in the family that is about halving.

√3 — the band’s bound, and also the rectangle a √3 sheet is. Thirding it across the long side gives three rectangles of proportion $\sqrt3/3 = 1/\sqrt3$, which is the same shape turned — so a √3 rectangle thirds into itself the way a √2 one halves.

√4 = 2 — the double square. Halves into two squares.

√n in general — divides into $n$ copies of itself across the long side, which is the family’s defining property and generalises the A series.

So $\sqrt3$ is both a member of a family of shapes and the bound a band needs, and the two facts are unrelated. The band’s requirement comes from the closure condition and the family membership comes from the thirding property, and nothing connects them.

That coincidence is worth flagging, because a reader meeting $\sqrt3$ in both places may reasonably suspect a connection and there is none.

Why the A series is a shape and not a bound #

It is worth checking the other direction, since the essay’s claim is a contrast.

Suppose the A series’ property were a bound — any rectangle at least √2 long halves into its own proportion. That is false immediately: a rectangle at 2 halves into two squares, which are not rectangles of proportion 2.

The property is an equation in one unknown and its solution is isolated. A rectangle of proportion $r$ halves into rectangles of proportion $2/r$, and $2/r = r$ has exactly one positive solution.

So there is no half-line here and no threshold. There is one number, and the property fails on both sides of it, which is what makes it a shape.

The band’s condition, by contrast, is the creases fit — an inequality — and inequalities give half-lines.

That is the whole distinction, stated in terms of the conditions rather than the numbers: equations give shapes and inequalities give bounds, and both produce square roots because both involve areas.

One more thing the bound is not #

It is not an efficiency.

A strip at exactly the bound uses all its paper: the three panels are the whole strip and there is nothing spare. A longer strip has spare paper, which ends up as extra material at the seam or as trapezoidal panels rather than triangular ones.

That makes the bound sound like an optimum — the most economical band — and it is worth resisting the reading. The bound is the shortest strip on which the fold exists, and existence is a different question from economy.

The distinction shows because there is nothing to economise. A band is not a design competing for paper against other flaps; it is one object, and using a little more strip costs nothing except the strip.

So the number is a feasibility threshold and calling it an optimum would import a criterion nothing here is optimising. It happens to be the point of maximum efficiency as well, and that is a coincidence of there being only one construction rather than a fact worth building on.

The vocabulary this collection uses #

A note on how the terms are used here, since proportion has been carrying two meanings.

A paper proportion in the A-series sense is a shape: the ratio a sheet is cut to, chosen for a property that holds only at that ratio.

A required proportion is a bound: the minimum ratio a construction needs, above which everything works.

The collection’s essays on the rectangle that keeps its shape and one member of a family are about the first, and both are careful to say that the property is what picks the number.

Nothing here has been about the second until now, because nothing here had a construction with a length threshold in it. The band is the first, and it will not be the last: any construction on a strip has a minimum strip, and the collection has simply not been asking.

So the anchor gains a second kind of entry, and the useful discipline is to say which kind a number is when quoting it.

Making the band, with the bound in mind #

The practical version, since the essay is about how to use a number.

Cut a strip about two widths long — twenty-four centimetres by twelve is comfortable. That is well above the bound, so the fold will be forgiving.

Mark the long edges at a third and two thirds along. Fold at sixty degrees from a corner to the first mark; unfold. Fold from that mark to the second, slanting the other way; unfold. Fold from the second to the far corner, slanting back.

Join the ends with a half twist and let the creases collapse. It goes flat into a triangle with a little spare paper at the seam.

Now try it at 1.7 widths. The same three creases can be marked and the join does not meet: the solution the closure wants has its creases running off the ends.

Two strips, one either side of the bound, and the difference is visible before anything is folded. That is what a bound looks like in the hand, and it is nothing like what a shape looks like — an A4 sheet cut slightly wrong folds perfectly well and merely fails to nest.

A third kind, briefly #

For completeness there is a third category and the subject has a few examples.

A ratio that has to be avoided — a proportion at which something degenerates. The Yoshimura drawn at exactly √3 of a half-column sits on a knife edge nine decimal places wide, where its vertex tables hold thirty labellings rather than eight and the search costs three times as much.

That is neither a shape to cut to nor a bound to clear. It is a value to stay away from, and staying away from it by any amount is enough.

Three categories: cut to it, clear it, avoid it. Every proportion in this subject is one of the three, and knowing which decides how much care the number deserves.

What the collection has computed #

The bound is computed rather than quoted, which is the reason to trust it.

The closure condition is solved for the crease positions at each length, by a linear solve, and the resulting drawing is tested for whether its creases lie inside the strip and in order. Feasibility is monotone in the length — checked, by confirming that the band is too short at 1.6 widths and comfortable at 2.4 — and a bisection then locates the threshold.

It comes out at 1.7320508, against $\sqrt3 = 1.7320508$.

And the ranking over other angle triples is computed the same way: the next-cheapest admissible triple needs 1.9225 widths, and most of the admissible curve needs two and a half or more. So the equilateral band’s bound is a genuine minimum with a margin, rather than one convention among several.

The band has one of each #

The neatest part of the band’s case is that it exhibits both kinds of condition on one object.

Its angles satisfy an equation: the alternating sum has to be a multiple of a straight angle, so the admissible triples form a curve and almost every triple folds nothing.

Its length satisfies an inequality: at least $\sqrt3$ widths, and every longer strip works.

So a folder wanting a Möbius triangle has to be exact about the angles and only approximately right about the length, which is a very unusual division of care in this subject and is worth knowing before cutting anything.

The two numbers, side by side #

√2 is a shape. Only that proportion halves into itself; the property is an equation; a sheet is that shape or is not; and the number is something to cut to.

√3 is a bound. Every strip at least that long folds; the property is an inequality; the number is a threshold; and it is something to check against.

Both are paper proportions and both are square roots, and treating the second as though it were the first — cutting a strip to exactly $\sqrt3$ and expecting the fold to be somehow better — is a mistake the vocabulary invites.