The square that cannot shrink

If $\sqrt2$ were a fraction $a/b$, then $a^2 = 2b^2$: a square exactly twice another square. The proof that no such pair exists is usually done with parity, and there is a proof that is done with squares.

Two squares of side $12$ have total area $288$. The large square has area $289$. So the two nearly cover it and overlap slightly — and the amount they overlap and the amount they miss differ by exactly one, which is $17^2 - 2 \times 12^2$.

That near-miss is the interesting quantity, and it is worth saying what the figure is doing with it before the argument starts. If a pair with $a^2 = 2b^2$ existed, the two small squares would cover the large one exactly: no area double-counted, none left out. So the overlap and the two corners would have equal area, and the picture would be a statement of equality between a square and two squares — the same equation again, at smaller scale. The whole proof is the observation that this cannot keep happening.

The picture as an argument #

Suppose $a^2 = 2b^2$ with $a$ and $b$ whole and $b < a < 2b$. Lay two squares of side $b$ into opposite corners of a square of side $a$. Since their combined area equals the large square exactly, the area they overlap must equal the area they miss.

The overlap is a square of side $2b - a$. The missed region is two squares of side $a - b$. So

$$(2b - a)^2 = 2(a - b)^2,$$

which is the same equation with a smaller pair. And it is genuinely smaller: $b < a < 2b$ gives $0 < 2b - a < b$ and $0 < a - b < b$.

Repeating produces an infinite strictly decreasing sequence of positive whole numbers, which cannot exist. So no such pair does, and $\sqrt2$ is not a fraction.

That is Tennenbaum’s proof, from the 1950s, and every step of it is a statement about areas of squares that can be pointed at.

The honest version of the picture #

There is a problem with drawing that argument, and this collection cannot ignore it: the squares in the proof do not exist. The proof supposes a solution, and there is none, so any drawing of “the” configuration is a drawing of something that is not there.

What the figures here do instead is draw a real pair — $17$ and $12$, whose ratio is a good approximation to $\sqrt2$ — and carry along the amount by which it fails. Call $d = a^2 - 2b^2$ the discrepancy. For $(17, 12)$ it is $+1$: the overlap square exceeds the two corner squares by exactly one unit of area.

Now the identity does the work. Expanding,

$$(2b - a)^2 - 2(a - b)^2 = -(a^2 - 2b^2) = -d,$$

so the descent flips the sign of the discrepancy and leaves its size alone.

So the contradiction is visible rather than assumed. A solution would be a pair with discrepancy zero; the descent preserves the discrepancy’s size while the squares shrink; and by the time the squares have shrunk to nothing the discrepancy is still whatever it was at the start. A discrepancy of zero would have had to be zero at every step and at the bottom, where the squares have side one and the discrepancy is $\pm 1$.

What descent is #

Fermat named the method and used it constantly: assume a solution in positive whole numbers, construct a strictly smaller one, and conclude there is none.

It is induction turned upside down, and the property it relies on is that the positive whole numbers are well-ordered — every non-empty set of them has a least element. A construction that always produces something smaller cannot be applied to a least element, so there is no least element, so the set is empty.

The method proves negatives, which is what makes it valuable and what limits it. It shows that $\sqrt2$ is not a ratio, that no Pythagorean triangle has square area, that $x^4 + y^4 = z^4$ has no solutions. Each conclusion is an absence, and none of the proofs produces anything.

It has one further use that is not negative at all, and it is the one that made Fermat’s reputation: descent can prove that something does exist, by descending on the size of a counterexample to its existence. Every prime one more than a multiple of four is a sum of two squares, and Euler’s proof takes a multiple of such a prime that is a sum of two squares, manufactures a smaller multiple with the same property, and continues until the multiplier is one. The descent terminates in the answer rather than in a contradiction.

So the method is really about the well-ordering and not about negation. A construction that always produces something smaller either runs out — proving nothing was there — or arrives at a bottom that has to be examined. Which of the two happens depends on the problem, and the arithmetic of the descent step is identical either way.

The same shape without the negation is how Euclid’s argument about primes works: build an object that must contain something new, and conclude the list was incomplete. Descent builds an object that must be smaller and concludes the set was empty. Both are constructions used to rule something out.

The proof everybody meets first #

The standard argument is different and shorter. Assume $a/b$ is in lowest terms with $a^2 = 2b^2$. Then $a^2$ is even, so $a$ is even, so $a = 2c$ and $4c^2 = 2b^2$, giving $b^2 = 2c^2$ — so $b$ is even too, contradicting lowest terms.

Both proofs are correct and they emphasise different things. The parity proof turns on unique factorisation, in the step “$a^2$ even implies $a$ even”, which is Euclid’s lemma for the prime $2$; and it generalises immediately to $\sqrt{n}$ for any non-square $n$ by counting the exponent of a prime.

The geometric proof needs no factorisation at all. What it needs is the ordering of the whole numbers and the ability to lay squares on top of each other. That makes it the argument that would have been available to a Greek geometer, and it is much closer to what the Greeks are thought to have had — incommensurability was a statement about lengths having no common measure, which is the never-terminating square-peel rather than a statement about fractions.

Where the numbers in the figures come from #

The pairs used above are not arbitrary. $(3,2)$, $(7,5)$, $(17,12)$, $(41,29)$, $(99,70)$ are the numerators and denominators of the convergents of $\sqrt2$, and their discrepancies alternate $+1, -1, +1, -1$.

That is not a coincidence: they are precisely the solutions of $a^2 - 2b^2 = \pm 1$, the Pell equation. The descent above is the map that takes one solution to the previous one, and running it backwards — $(a, b) \mapsto (a + 2b, a + b)$ — generates them all from $(1,1)$.

So the three constructions in this corner of the collection are one construction. Peeling squares off a rectangle gives the continued fraction $[1; 2, 2, 2, \ldots]$; its convergents are the Pell solutions; and folding those squares into smaller ones is the descent. The reason $\sqrt2$’s approximations are so regular and the reason the descent is so clean are the same reason, which is that the leftover strip at each stage is similar to the strip before it.

Run the descent backwards and it becomes a generator. From $(a, b)$ the map $(a + 2b,\ a + b)$ produces the next solution, so $(1,1) \to (3,2) \to (7,5) \to (17,12) \to (41,29)$, and every solution of the Pell equation arises this way. The Greeks knew these as the side and diagonal numbers and used them exactly as approximations to $\sqrt2$; Theon of Smyrna describes the recurrence in the second century, with no equation and no proof, as a rule for generating better and better pairs.

That is worth noticing for what it says about the proof. The same recurrence, read downwards, shows that no exact solution exists; read upwards, it produces the best possible approximations to the thing that does not exist. A construction and an impossibility proof are the same map traversed in opposite directions.

What the picture cannot show #

Every figure here is a single step of an argument that needs infinitely many, and the infinity is where the contradiction lives.

More precisely: the drawings show that the discrepancy is preserved and the squares shrink. What they cannot show is the well-ordering principle, which is the fact doing the actual refuting. No picture rules out an infinite descending sequence of positive whole numbers; that is an axiom about the integers, not an observation about squares, and it is the only part of the proof that is not visible.

There is a second thing missing, and it is more interesting. The figures show $\sqrt2$, and the argument does not obviously say anything about $\sqrt3$ or $\sqrt5$. Tennenbaum’s construction does adapt — the $\sqrt3$ version uses triangles and the $\sqrt5$ version uses a pentagon-like arrangement — but each needs its own drawing, and there is no single picture covering every non-square $n$. The parity proof covers all of them in one line. That is the standing trade: a picture explains one case completely and a formula covers every case opaquely.

Other roots, other pictures #

The construction adapts, and how it adapts says something about why the picture is specific where the algebra is general.

For $\sqrt3$ the equation is $a^2 = 3b^2$ and the natural figure is an equilateral triangle of side $a$ with three triangles of side $b$ laid into its corners. They overlap in three smaller triangles and miss a central one, the areas balance in the same way, and a smaller solution falls out. For $\sqrt5$ the arrangement involves a pentagon and is genuinely harder to draw.

Each case is a separate construction with a separate drawing, and there is no picture that covers $\sqrt{n}$ in general. The parity argument covers every non-square $n$ in a line: if $n$ is not a perfect square, some prime divides $n$ to an odd power, and comparing that prime’s exponent on both sides of $a^2 = nb^2$ gives an odd number equal to an even one.

So the two proofs trade in opposite directions. The picture is complete about one case and silent about the rest; the algebra is complete about every case and explains none of them. This collection’s bias is towards the first, and the reason is worth restating: a reader who has followed the shrinking squares knows why no solution exists, in a way that the exponent count does not supply, and the general theorem is easier to accept once one case has been understood rather than merely verified.

There is one further generalisation that keeps the geometry. The same overlapping-squares figure with the two small squares of different sizes proves that $\sqrt{n}$ is irrational whenever $n$ lies strictly between two consecutive squares, which is every non-square $n$ — the construction is due to Conway and Guy, and it recovers the general theorem geometrically at the cost of a figure nobody would call clean.

What it did to Greek mathematics #

The discovery that a square’s diagonal has no common measure with its side is usually dated to the Pythagoreans, and the anecdote about Hippasus being drowned for revealing it is a late invention. The mathematical consequence, however, is real and enormous.

Greek geometry treated ratios of magnitudes as fundamental, and a theory of ratio that assumed every two lengths have a common measure had just been shown to be wrong. Eudoxus’ theory of proportion, preserved in Book V of the Elements, is the repair: it defines equality of ratios by comparing multiples, without ever asserting that a common measure exists, and it works for incommensurable magnitudes.

That definition is essentially Dedekind’s construction of the real numbers, arrived at two thousand two hundred years earlier and then not built on. What the Greeks did not do was turn ratios into numbers; magnitudes stayed geometric, and arithmetic stayed a separate subject. The gap between the two was not closed until the nineteenth century, and closing it was the same work as making sense of limits.

The consequence for how Greek mathematics reads is direct. Book II of the Elements proves algebraic identities — what would now be written $(a+b)^2 = a^2 + 2ab + b^2$ — as statements about rectangles, and it does so because a rectangle is a legitimate object where a product of two incommensurable lengths is not. The dissection proof of Pythagoras has the same character. What looks like a stylistic preference for pictures is a foundational necessity: geometry was the only place where the quantities could be handled at all.

So the collection’s habit of drawing arguments has an ancestor with a much stronger motive. The Greeks drew their proofs because the alternative did not yet exist; this site draws them because the alternative exists and hides the reason.

Where the ladder goes next #

The immediate continuation is the general statement: $\sqrt n$ is irrational for every non-square $n$, and more strongly, a root of a monic polynomial with whole-number coefficients is either a whole number or irrational. That covers every square root, cube root and combination of them in one theorem, and its proof is the parity argument generalised.

The harder direction is transcendence — showing that a number satisfies no polynomial at all, which is what $\pi$ and $e$ do. Irrationality is a statement about degree one; transcendence is a statement about every degree; and the gap between them is measured by how well a number can be approximated by fractions, which is where Liouville found the first transcendental number by building one that was approximated too well to be algebraic.