The rectangle that keeps its shape

A sheet of A4 folded in half is a sheet of A5. Not approximately, not by convention of the paper mill — the halved sheet has the same proportions as the whole one, and so does the half of that, and the half of that.

No other rectangle behaves like this. Halve an ordinary sheet and the result is a different shape: squatter or leaner, depending on which way round it started. The A series is the family that survives the operation, and the operation is a fold.

What halving does to a shape #

Take a rectangle whose long side is r times its short side, with r somewhere between 1 and 2 so that it is a portrait sheet rather than a landscape one. Fold it in half across the long side. The short side is untouched and the long side is now half what it was, so the new long side is whichever of those two is larger.

At proportion r the sides are 1 and r. After the fold they are 1 and r/2 — and since r is under 2, the half-length is under 1, so the short side has become the long one. The new proportion is 1 divided by r/2, which is 2/r.

That is the whole of the map, and it has one property that decides everything else: applying it twice gives 2/(2/r), which is r again. Halving a rectangle is an operation that undoes itself on the second application. Not approximately, and not eventually.

The two curves cross where r and 2/r are the same number, which means r² = 2 and r = 1.414214 to six places. That crossing is the only rectangle the fold leaves alone. Every other sheet lands on the falling curve somewhere other than the crossing, and the second fold sends it straight back.

The fold that does it has a name #

The fold in question is not an arbitrary crease. It brings one short edge of the sheet exactly onto the other, which is one of the seven ways a fold can be specified by bringing existing things into coincidence — the third of them, a line onto a line.

This matters more than it looks. The proportion is not being measured and it is not being aimed at; it is the consequence of an operation a folder can perform without a ruler and without knowing what the answer is going to be. The edges are the references and a fold needs something to align — here the sheet supplies its own.

What a compass and straightedge reach is a different question with a different answer, and it is a whole subject belonging to another site in this fleet. Nothing below derives anything about it. The claim here is narrower and is about a fold: an operation on shapes, one of whose inputs comes out unchanged.

A fixed point is not a limit #

The natural way to read the crossing is as something the folding converges on — start anywhere, fold enough times, and the shape settles down near 1.414. That reading is wrong, and the way in which it is wrong is the sharpest thing in the subject.

A sheet at proportion 1.3 halves to 1.538461538461538. Halve it again and it returns to 1.300000000000000 — the same number to every digit the arithmetic carries. It will do that a thousand times. It is not approaching anything; it is orbiting between two shapes, and the orbit has period two.

So √2 is not what halving converges on. It is what halving cannot move. A folder who starts with the wrong rectangle does not gradually acquire the right one by folding; the wrong rectangle stays exactly as wrong as it was, and announces the fact every second fold.

There is a quieter consequence hiding in the pair of values. The two members of every orbit multiply to exactly 2, because one of them is 2 divided by the other — so their geometric mean is √2, always, whatever the starting shape. Every rectangle in the world is already straddling the answer. What distinguishes the A series is not being near the mean of its own orbit but having an orbit with only one member in it.

Folding into more than two #

Nothing about the argument required the number two. Fold a rectangle into k equal parts across its long side and each part has the short side unchanged and the long side divided by k, so the proportion goes from r to k/r. The same reasoning applies: the shape survives when r = k/r, which is r = √k.

The four-part case is worth a moment because it is the one that can be verified without arithmetic. A rectangle twice as long as it is wide, folded into quarters across the long side, gives four rectangles each twice as long as they are wide. That is √4 = 2, and it is the family’s only whole number.

The three-part answer, √3 = 1.7321, is the proportion of a sheet that folds into three of itself. A square metre of it measures 760 × 1316 mm. No standards body has ever printed it, which is the point: the A series is not the only proportion a fold fixes, only the one somebody standardised.

Two of the family need no construction #

The family runs over every whole number, and its members are not equally available to a folder — which is worth separating out, because the availability is what decided which one became a standard.

Folding a sheet into two needs one fold and no references but the sheet’s own edges. Folding into four needs two such folds. Those are the only divisions a folder reaches by halving, and by the same token they are the only members of the family that can be produced with nothing on the paper beforehand.

Folding into three is a different job. A third is not reachable by halving at any depth, so the division has to be constructed — by a fold onto an existing point, or by the ladder that reaches every fraction exactly, or by an iteration that converges. Five and six are the same, and worse.

So the family splits by whether $k$ is a power of two. The self-sufficient members are $\sqrt{2}$ and 2 — and 2 is the sheet twice as long as it is wide, which quarters into four of itself and is the one a reader can verify by eye. Everything else in the family is a proportion that survives an operation the folder cannot perform without first performing another one.

That is a better account of why the A series is the standard than the fixed point alone gives. It is not merely a proportion that halving preserves; it is the only proportion preserved by a division a folder or a guillotine can make with no marks on the sheet at all, apart from the trivial one at 2.

Every orbit has the same mean #

The remark about geometric means also generalises, and it does so exactly.

Folding into $k$ sends a proportion $r$ to $k/r$, and applying that twice returns $k/(k/r) = r$ — so every rectangle orbits with period two under folding into $k$, whatever $k$ is, exactly as it does under halving. The orbit’s two members multiply to $k$, so their geometric mean is $\sqrt{k}$, which is the family’s member for that $k$.

So the structure is the same at every $k$: one fixed point, every other sheet trapped in a two-cycle straddling it, and the fixed point sitting at the geometric mean of every orbit including its own. Nothing anywhere in the family converges to anything.

That makes the closing observation sharper rather than only prettier. A rectangle folded into any number of parts is already the geometric mean of what it becomes — which sounds like an approach to the answer and is precisely the reason there is no approach: the mean is fixed from the first fold and the sheet never gets any nearer to it.

Which claim was checked, and how #

Three separate things are asserted, and each is checked against something that does not know the answer.

The first is the arithmetic. The proportion that survives being folded into k is found by bisection on r − k/r, a search that evaluates nothing but that expression and never calls a square root. It is then compared with the square root, which it was not given. For k = 2 the two agree to 2.2 × 10⁻¹⁶ — one unit in the last place of a double — and for k of 3, 4, 5 and 6 they agree exactly, to every bit. A search that had been seeded with the answer would prove nothing; this one arrives from the other direction.

The second is the paper. A0 is printed as 841 × 1189 mm, and those are whole millimetres, so the proportion they state is 1189/841 = 1.413793 and that is not √2. It is 4.2 × 10⁻⁴ away. The assertion is that the gap is no larger than rounding to the nearest millimetre can produce — a bound of 1.4 × 10⁻³, worked out from half a millimetre on each side rather than fitted to the discrepancy — and, separately, that a square metre at exactly √2 measures 840.896 × 1189.207 mm, which rounds to precisely 841 and 1189. Neither number comes from the other. A printing standard and a fixed point of a fold were computed independently and agree to the last millimetre.

The third is the refusal to converge, and it is the one that could most easily have been fudged. A probe at 1.3 is halved twice; the result must return to 1.3 to within 10⁻¹², must not have stayed put on the first fold, and the two members of the orbit must have √2 as their geometric mean to the same tolerance. What would have made it fail is easy to state: an orbit that crept, by any amount at all, in either direction. Creeping is what a limit does. This does not creep.

A different question about the same rectangle #

There is a second question a designer asks about the shape of a sheet, and it is not this one. The square is a choice holds the area fixed and asks which proportion lets a set of flaps claim the most paper — an efficiency question, answered by packing rather than by folding, with a peak that moves as the flap count changes.

The two questions have different answers and it is a coincidence when they overlap. At six flaps the A-series proportion happens to be the better sheet, by eight points of efficiency — but at a different flap count the peak sits elsewhere, and nothing about the fixed point of halving predicts where. A proportion that survives an operation and a proportion that packs well are unrelated properties of the same number, and reading either as evidence for the other is the error worth naming.

Getting the proportion onto real paper #

A fold that halves a sheet is easy because the sheet supplies both references. Dividing into three is not, and the difference is where the practical part of this subject lives.

Haga’s theorem is the machinery: one fold onto an existing point produces exact rational divisions elsewhere on the sheet, and from those any whole number of equal parts can be reached without a ruler. The result is exact in the same sense the halving is — a consequence of the fold rather than of care.

The alternative to exactness is an algorithm. Folding a strip into thirds starts from a guess and halves the error into place, which converges rather than constructs — a genuine limit, and worth setting beside the two-cycle above precisely because it is the opposite kind of behaviour.

That contrast is not decoration. Two operations that both consist of nothing but folding a sheet in half behave in opposite ways: one contracts an error towards a fixed point, and the other refuses to move a proportion at all. What separates them is which quantity is being watched, and an error that is folded too is the general form of the question.

Where the paper stops being the mathematics #

The figures above draw proportions. That is all they draw, and it is worth saying what they therefore cannot show.

They cannot show area: every sheet in the family view is a square metre, and a picture of proportions alone would not distinguish A0 from A8. They cannot show that a fold has to land on the line — a crease a millimetre off centre produces two rectangles of different proportion, neither of them the parent’s. And they cannot show thickness, because the sheet in every one of them is a rectangle with none.

The idealisation is worth naming exactly: a sheet of zero thickness, halved on a line with no width, losing nothing to the bend. Real paper honours none of the three.

The halving bound is the sharpest of these. A0 halved eight times is A8, and eight halvings of a single sheet in one direction need a strip about ten metres long — so the A series is a series of cuts, not of folds, past about the fourth step. The proportion is preserved by an operation that a real sheet cannot perform more than a handful of times.

The rounding tells the same story from the other end, and it is the detail that makes the standard honest rather than embarrassing. Each A size is the previous one’s long side halved and rounded down to whole millimetres: 1189 → 594, 841 → 420, 594 → 297, and so on to 74 × 105 at A7. The printed proportions therefore wander — 1.413793 at A0, 1.414286 at A2 and A4, 1.418919 at A5 and A7, 1.423077 at A8. They oscillate around √2 and the oscillation grows going down the series, because a whole millimetre is a larger share of a small sheet. The mathematics is exact and the paper is rounded, and the discrepancy is entirely the rounding’s. Paper is not ideal is the standing form of that observation on this site.

Who noticed it, and when #

The proportion is generally credited to Georg Christoph Lichtenberg, who set it out in a letter to Johann Beckmann in 1786 — the observation that a rectangle whose sides are as 1 to √2 keeps its shape when halved, arrived at as a curiosity about paper sizes rather than as a theorem.

It took a long time to become a standard. Walter Porstmann’s DIN 476, published in Germany in 1922, is the version the modern series descends from: the √2 proportion, the square-metre base sheet, and the naming scheme that makes A4 four halvings below A0. ISO 216 adopted it internationally much later. The gap between the observation and the standard is around 136 years, which is the sort of interval this site keeps finding — the name is rarely the date, and the naming usually happens long after somebody first wrote the thing down.

What is worth resisting is the tidy version in which the standard was derived from the mathematics. The two agree, and the agreement was checked above rather than assumed, but a committee choosing a sheet size has several constraints and only one of them is a fixed point. The square-metre base sheet is not forced by anything; it is a decision, and it is the decision that turns a proportion into 841 × 1189.

Where the ladder goes next #

The immediate continuation is the references. A fold needs things already on the paper to align, and what a folder can construct is bounded by what can be referred to — the halving of a rectangle is the easiest case there is, because the sheet’s own edges are the references and no earlier fold is needed to produce them.

The other direction is the sheet as a design variable. Which shape of paper a set of flaps wants is a question about packing, answered at constant area, and the answer is spiky rather than smooth. Neither question knows about the other, which is the useful thing to carry away: a rectangle can be fixed by an operation, efficient for a purpose, or convenient to cut, and there is no reason for one number to be all three.