To double a square you build a new one on its diagonal. Plato's slave boy does it in a single stroke — and that diagonal is the first length ever shown to be no fraction at all. Here is the proof, as a picture that shrinks.
In Plato's Meno, Socrates draws a square and asks a boy who has learned no geometry to double it. The boy guesses wrong — double the side, and you quadruple the area — and then, led by questions, finds the answer that was there all along: build the new square on the diagonal of the old one. The diagonal of a unit square encloses exactly twice the area. Its length is what we now write √2.
So the diagonal doubles the square. The next question is the one that broke the Pythagoreans: is that diagonal a fraction of the side? Is there some unit — however fine — that measures both the side and the diagonal a whole number of times? In symbols: are there whole numbers a and b with a² = 2b² — a square that is exactly twice another square? The famous proof argues from odd and even. This page gives the other proof — the one that needs no algebra, only a picture you can take apart.
Suppose, for contradiction, that a² = 2b² for some positive whole numbers — so a square of side a has exactly the area of two squares of side b. Lay those two b-squares into opposite corners of the a-square. Because two b-squares hold the whole area but the big square is narrower than two b's laid side by side, the two squares must overlap in the middle — and leave two corners bare.
The instrument below does exactly this. Step through the candidates: each is a pair (a, b) whose ratio is a better and better approximation to √2. Watch the overlap square and the two bare corner squares. The whole proof lives in one comparison: is the overlap exactly twice the corner?
Count the area two ways. The big square is a². The two b-squares cover 2b², but their overlap — a square of side 2b−a — got counted twice, and the two bare corners — each a square of side a−b — never got counted at all. Bookkeeping that is true for any a and b:
a² = 2b² − (2b−a)² + 2(a−b)².
Now feed in the assumption a² = 2b². The 2b² cancels the a², and what is left is exact:
(2b−a)² = 2(a−b)².
Read that line. It says the overlap square is twice the corner square — which is to say (2b−a,\; a−b) is another pair of whole numbers solving the very same equation x² = 2y². And because b < a < 2b for any real solution, the new pair is made of strictly smaller positive whole numbers. The overlap is a smaller copy of the same impossible thing.
So from one solution the picture hands you a smaller one — and from that, a smaller one still, forever. But you cannot descend through the positive whole numbers forever; below 1 there is nowhere to land. A chain that must always step down and can never reach bottom cannot exist. Therefore the first solution cannot exist either. No square is exactly twice another, and √2 is irrational. ∎
This is infinite descent, and here it is not a turn of phrase: the descend button literally produces the smaller square, and you can keep pressing until the figure runs out of room. A true solution would never run out. That it always does is the contradiction, drawn.
There is no exact solution — but there are glorious near-misses, and the instrument steps through the best of them. 17² = 289 and 2·12² = 288: the square on 17 overshoots two squares on 12 by a single unit. 99² = 9801 and 2·70² = 9800: off by one again. These pairs are the convergents of √2, and for every one of them
a² − 2b² = ±1.
The ratio a/b closes on √2 as tightly as you like — but the area gap never reaches 0. It sits at one, flipping sign, all the way out. That is the irrationality you can hold in your hand: the approximation gets perfect; the equation never does.
And here is the turn. The descent map that proves impossibility — (a,b) → (2b−a,\; a−b) — has an exact inverse, (a,b) → (a+2b,\; a+b). Run it up from the seed (1,1) and it manufactures the convergents one after another: 1/1, 3/2, 7/5, 17/12, 41/29, 99/70, … — the best rational approximations √2 has, each the closest fraction to it for its size. The same machine that cannot bottom out among the integers is the machine that builds the best fractions. Descent is the proof; ascent is the gift. Press ascend and watch the table fill.
Before there was algebra there was anthyphairesis — “mutual subtraction,” the same engine as Euclid's algorithm for a common measure, run on lengths. To test whether the diagonal and the side of a square share a common unit, you lay the side along the diagonal as many times as it fits (once), then lay the remainder against the side, and repeat — and if it ever comes out even, that remainder is your common measure.
For the square's diagonal and side it never comes out even. The first subtraction leaves a gap; the figure left behind is a smaller square in the very same proportion as the one you started with — so the process repeats, identically, forever. The quotients it reads off are 1, 2, 2, 2, …, which is to say
√2 = 1 + 1/(2 + 1/(2 + 1/(2 + ⋯)))
— the continued fraction that never ends. A length with a terminating mutual subtraction is a fraction; √2's subtraction is endlessly self-similar, so it is no fraction. This is the same truth as the overlapping squares, told as a process instead of a picture — and it is the seam this layer shares with its neighbours: the odd-and-even proof on one side, and on the other the number whose continued fraction is all ones, the hardest of all to approximate. √2, all twos, is its quieter cousin: irrational by the same mechanism, just not quite so stubborn.
One equation, three roads to it. Parity. A shrinking picture. A subtraction that won't stop. They share no common measure with each other either — and they all arrive.