One member of a family

The rung below is about the one property of paper sizes everybody can state: fold an A4 sheet in half and the halves are A5, the same shape turned through a right angle. The ratio that does it is √2, the requirement forces it rather than anybody choosing it, and the printed sizes miss it by a rounding that accumulates down the series.

That essay treats √2 as the answer to a question. It is the answer to a specific question, and the general one has a general answer.

The general rectangle #

Take a rectangle 1 wide and r tall, and cut it across into n equal strips. Each strip is 1 wide and r/n tall, so its ratio of long side to short is n/r if r < n, and the strips are similar to the original when n/r = r, which is to say when r = √n.

That is the whole derivation. A 1 × √2 rectangle halves into 1 × √2 rectangles; a 1 × √3 divides into three; a 1 × √5 into five; and so on for every whole number.

Nothing about two is special in that argument. What is special about two is the operation: halving is the one division a folder can do without thinking, edge to edge, with no construction at all. Dividing into three needs a third to be constructed first, and a third takes work — Fujimoto’s method converging on it, or a crossing that lands on it exactly.

So the A-series is not built on the unique self-dividing rectangle. It is built on the only member of the family whose division is free.

That is worth holding onto, because the uniqueness claim is repeated so often that it has become part of what people know about paper. It is not that the claim is nearly true or true in spirit; there is a rectangle for every whole number, and the second one has a standard attached for a reason that has nothing to do with geometry.

The family is foldable #

The other thing worth checking is whether these rectangles can be made, since a proportion nobody can construct is a curiosity rather than a stationery standard.

They can, and by the plainest possible route. The diagonal of a 1 × √(n − 1) rectangle has length √n, so the roots arrive in order: fold the diagonal of the unit square for √2, lay it down and fold the diagonal of the resulting 1 × √2 for √3, and so on. Each step is one fold and one right angle.

Checked to eight, every rung of that ladder agrees with the square root it never takes: the construction forms hypotenuses and nothing else, and comparing the result with the square root taken directly gives zero difference at every value. The similarity of the divided strips is checked the same way, against the ratio the division never forms, and the worst departure over n from 2 to 8 is 4.4 × 10⁻¹⁶.

What the printed sizes do to it #

The rung below establishes that the printed A-series is not at √2 and quantifies the drift: A0 is printed at 1189 × 841, which is 1.413793, and each size is the previous long side halved and rounded down, so the deviation accumulates rather than cancelling — 1.418919 at A5, 1.423077 at A8.

The family view adds a sharper way to say what that costs. The printed A0 halves into a rectangle of a different shape, and the two numbers are 1.413793 before and 1.414634 after. Neither is √2 and they are not equal to each other, so the property the whole standard exists for fails at the first application, by about six parts in ten thousand.

That is invisible in use and it is not nothing: a design laid out to fill an A4 page exactly and then scaled to A3 does not fill it exactly, and the mismatch is a fraction of a millimetre at the edge. Printers have known this for as long as there have been printers, which is why the standard’s tolerances are specified in millimetres and why nothing is ever laid out to the paper’s edge.

The check that had to be able to fail #

Similarity is easy to verify and easy to verify vacuously, so the assertion behind these figures is built to bite.

The positive half is straightforward: over n from 2 to 8, cut the 1 × √n rectangle into n and compare each strip’s ratio with the original’s. The worst departure is 4.4 × 10⁻¹⁶, which is arithmetic noise.

The half that matters is the refusal, and it is aimed at the case a reader is most likely to believe: the printed A0 must fail. Its 1189 × 841 is 1.413793 and halving it gives 841 × 594.5, which is 1.414634 — a different rectangle, by more than a thousandth. A check that passed the printed sizes would be a check with a tolerance loose enough to pass anything, and it is fed them deliberately.

Why the other members were never standardised #

There is an obvious question the family raises: if √3 divides into three and √5 into five, why is there no B-series of thirds?

Part of the answer is the folding one — thirds are work — and part of it is arithmetic that runs the other way. A halving series doubles the number of sheets at every step, which suits a world where a sheet is cut down repeatedly and where a run of anything is a power of two. A thirds series would produce nine sheets in two steps and twenty-seven in three, which is a coarser ladder with fewer useful stops.

There is also a third reason and it is the one that decides it. A halving series can be produced by folding and a thirds series cannot. A printer cutting stock in half needs a guillotine and a mark at the middle, which is the easiest mark there is; cutting into three needs the mark constructed, and a mark constructed once per sheet is a mark that can drift. The standard that survived is the one whose production step is self-checking.

Where the model stops #

Similar is not the same as useful. A 1 × √5 rectangle is long and thin — the strips it divides into are long and thin — and a shape’s usefulness for print, for a book or for a screen has almost nothing to do with whether it divides into copies of itself. The property is a piece of geometry that a standard can be built on and it is not a reason to prefer a shape.

Every ratio here is irrational except when n is a square. √4 is 2 and √9 is 3, so the fourth and ninth members are rectangles of whole-number ratio: a 1 × 2 divides across into four 1 × ½ strips, each of ratio two, and a 1 × 3 into nine. The property holds exactly as it does everywhere else in the family; what is missing is the irrationality that makes the second member feel like a discovery. That is a degeneracy rather than an exception, and it is worth noticing because a family whose members were all irrational would be a suspicious family.

The construction is exact and the manufacture is not. Everything above is about ideal rectangles. Real paper is cut to a tolerance, expands with humidity along the grain and not across it, and arrives from the mill with a bias. The rung below measures what rounding to whole millimetres costs; the mill’s own variation is larger.

Nothing here is about the golden rectangle, which has a different self-similarity: remove a square and what remains is similar to the original. That is a subtraction property rather than a division property, it produces a different number, and the two are routinely confused because both are described as rectangles that reproduce themselves.

The family does not extend to non-integers. A rectangle of ratio √2.5 divides into two and a half copies of itself, which is not a division. The property is about whole numbers of parts, and the roots that matter are the roots of whole numbers — which is why the family is a list rather than a continuum.

The division is into strips and not into rectangles generally. A √n rectangle divides into n copies by parallel cuts. Whether it divides into n copies some other way, or into copies of a different rectangle, are separate questions with separate answers and none of them is asked here.

The B and C series, which are the family’s other answer #

There are two more ISO series and they make the point from a direction the mathematics does not.

The B series sits between the A sizes: Bn is the geometric mean of An and A(n−1), so B4 is between A4 and A3. The C series does the same between A and B, and it is what envelopes are made in — a C4 envelope holds an A4 sheet. Every one of them has ratio √2, because the geometric mean of two rectangles of the same shape is a rectangle of that shape.

So the standard did not need another member of the family; it needed more sizes of the same member, and it got them by interpolating rather than by changing the ratio. That is exactly what the family view predicts is the sensible move: the ratio is fixed by the halving operation, and everything else a standard wants — intermediate sizes, envelope sizes — is obtained without touching it.

What the family is actually for #

The reason to have the general statement is not that anybody will use a √5 rectangle. It is that it separates two things the single case runs together.

The A-series is described as being built on a unique property, and that description gets repeated because it is more striking than the truth. Uniqueness is doing rhetorical work: it suggests the standard was discovered rather than chosen, that √2 was forced by the requirement, and that any other choice would have failed.

What was actually forced was a pairing. The requirement was similarity under the operation printers already performed, and halving is that operation. Given halving, √2 is forced — the rung below shows exactly that — and given a different operation, a different ratio is forced just as firmly. The standard’s foundation is a printing practice, not a number.

That is a small correction to a familiar story, and it is the kind this site keeps making: a result presented as an inevitability turns out to be an inevitability given a choice nobody mentions. The square origami paper is sold on is the same shape of fact, and so is the assumption that a base has one axis.

The same story, three times on this site #

The shape of this correction has now appeared often enough here to be worth naming.

A property is stated with a uniqueness claim attached — the √2 rectangle is the self-dividing one, the square is the sheet origami is folded from, a base has one axis — and in each case the uniqueness turns out to be conditional on an unstated operation or convention. The rectangle is unique given halving. The square is conventional given a paper market. The single axis is a hypothesis of the design method rather than a fact about folding.

None of the three is a mistake exactly. Each is a true statement with its condition dropped, and the condition is dropped because it is so universally satisfied that nobody notices it is there. What the general version buys is knowing where the boundary of the claim is — which matters the moment somebody wants to work outside it, and matters not at all until then.

The family is larger than a list #

The derivation above cuts the rectangle into strips, which is the division a printer makes. Allow a grid of cuts instead — p columns and q rows — and the answer stops being indexed by whole numbers, in two separate ways.

A rectangle 1 wide and r tall, cut into p columns and q rows, gives parts 1/p wide and r/q tall. There are two ways for such a part to be similar to the whole.

The first is that it stands the same way up, which needs $rp/q = r$, so $p = q$. Every rectangle in the world divides into four copies of itself, and nine, and sixteen, by cutting it into a square grid. That is trivially true and it is worth saying out loud, because it is the fact the uniqueness story most needs a reader not to think of. Self-division into similar copies is not a rare property. It is universal, and the A-series’ version of it is interesting only because the copies come out turned.

The second way is the turned one, and it needs $q/(rp) = r$, which gives $r = \sqrt{q/p}$ with the rectangle dividing into $pq$ parts. Setting $p = 1$ recovers the strips: $r = \sqrt{n}$, into n. But $p$ need not be one. A rectangle of ratio $\sqrt{3/2}$ — about 1.2247 — divides into six parts, two columns by three rows, each of them the same rectangle on its side. So does $\sqrt{5/3}$ into fifteen, and $\sqrt{5/2}$ into ten.

So the self-dividing rectangles are not a list indexed by the whole numbers. They are indexed by the positive rationals, one shape for each, and they are dense in the line: between any two ratios a stationer might consider there are infinitely many that divide into turned copies of themselves.

Which makes the choice narrower and clearer #

That sounds as though it dilutes the A-series’ claim further, and in one sense it does. It also sharpens what the claim actually was, which is more useful.

Among a dense set of self-dividing ratios, √2 is picked out by a condition that has nothing to do with similarity: it is the only one whose division is into two parts, and two is the number a sheet can be divided into by folding it edge to edge with no mark, no measurement and no construction. Every other member of the dense family needs a grid laid out first, and a grid laid out is a grid that can be laid out wrong.

Read that way the standard’s foundation is not a geometric fact at all. The geometry supplies a dense set of candidates, and the production step picks one of them. Change the production step — a mill that trisects, a machine that cuts a two-by-three grid in one pass — and $\sqrt{3}$ or $\sqrt{3/2}$ becomes the sensible standard by the same reasoning that gave √2, with exactly as much inevitability and exactly as little.

Where the ladder goes next #

The obvious continuation is the rectangle that divides into unequal similar copies — a shape cut into pieces all similar to the original but not to each other. Those exist, they are studied under other names, and whether any of them is foldable from a square in a small number of folds is a question this machinery could answer.

There is also the question the B and C series answer practically and nobody answers geometrically: which pairs of shapes nest, in the sense that one holds the other with a constant margin. That is a question about two rectangles rather than one, the standard solves it by geometric means, and whether folding has anything to say about it is unexplored.

Nearer to hand is the constraint the family sits under. Every rectangle here divides by parallel cuts, and a folder does not cut — a folder folds, which produces layers rather than pieces. What a √n rectangle’s fold into n does to the layers, and whether the resulting stack is one of the tidy ones, is a question about a folded state rather than about a proportion, and the machinery for it is already on the site.