Doubling a square: the diagonal is the construction

0:00Here is a square. Draw me another one with exactly twice its area. Not roughly twice. Exactly twice. Around eight hundred years before the common era, Baudhayana set this problem down in the Sulba-Sutra, a manual of geometric construction, and gave an answer that needs no measuring at all. The answer is a single segment you can already see in the square. But before that segment, the guess almost everyone makes.

0:33Double the area, so double the side. It sounds inevitable. Take a square of side one, area one, and stretch every side to two. The new square does not have area two. It has area four. You can see why by cutting it: four copies of the original square fit inside it exactly. The reason is that area is a length multiplied by a length. Double the side and you have doubled that product twice over, so the area goes up by two times two.

1:07Doubling a side never doubles an area. It quadruples it. So the side has to grow by less than double. By how much? Try one and a half. One and a half squared is two point two five, which overshoots. Try one point four, which is seven fifths. Squared that is one point nine six, which falls short. Seventeen twelfths lands inside a hundredth of two, and still misses. Ninety-nine seventieths is closer still, by a factor of three, and it misses too.

1:44You can go on hunting. No fraction whatsoever squares to exactly two, and none of sixty denominators tried here comes close enough to be equal. Which is why the answer is not a number to multiply by. It is a line to draw. Draw the square's diagonal. Now build a new square standing on that diagonal, using it as one side. That square has exactly twice the area of the one you started with.

2:16Not approximately. Exactly, and the proof does not use the diagonal's length anywhere. Nobody in this argument ever has to know how long the diagonal is. Instead, both squares are going to be cut into copies of one single triangle, and then counted. The whole construction is one line, drawn from one corner to the opposite one. Here is the device that makes the picture carry the argument. Take the original square's own top side and extend it straight across the new square. Baudhayana called that direction east-west.

2:58Take its own left side and extend that too, north-south. Look where those two lines arrive. Each one passes through two corners of the new square, so between them they reach all four. They are not drawn to hit those corners. They arrive there because the original square's side lies along a diagonal of the new square, and a diagonal runs corner to corner. The two lines cross at right angles, and they cross exactly at the new square's centre.

3:31And they have cut it into four triangles. Now go back to the original square and draw the diagonal you already have. It cuts that square into two triangles. Two triangles inside. Four triangles outside. And they are all the same triangle. So the original square is two of that piece and the new square is four of it. Four is twice two, and that is the whole proof. No lengths, no square roots, no arithmetic beyond counting to four.

4:05Area is being measured here in identical pieces, which is why the word double comes out exact. But saying they are the same triangle is a claim, and it needs a reason. Every one of the four is built the same way. Two of its sides run from the centre out to a corner. Those are halves of the new square's two diagonals, and a square's two diagonals are equal, so all eight of those half-sides are the same length.

4:37The corner between them, at the centre, is a right angle, because a square's diagonals meet at right angles. Two sides the same and the angle between them the same. That fixes the triangle completely. The third side, the one lying along the outside, is a side of the new square, and there are four of those, all equal. So the four are not merely similar looking. They are the same triangle, four times over.

5:10It is worth being strict about this, because looking alike proves nothing. Here are two right triangles of exactly the same area, two square units each, and they are not the same triangle at all. One has legs four and one. The other has legs two and two. Same area, different shape. Equal area is not congruence. And there is a second trap in the picture. The four triangles' short sides are halves of the tilted square's diagonals.

5:43Not halves of the largest square on the page. Those would be a different length entirely, and the argument would collapse. Take the sides from the right square. One more objection, and it is the one your eye keeps making. The new square is standing on a corner, and a square standing on a corner looks pinched. It looks smaller than it is. It is not. Turning a figure changes nothing whatsoever about its area.

6:14A square drawn on a slanted step of three across and four up has area twenty-five, and so does an upright square of side five. Same area, exactly, and the tilted one is still a square by the test that matters. That test is worth stating: four equal sides is not enough, because a squashed diamond has four equal sides too. A square needs its two diagonals equal as well, and the diamond fails that.

6:48Nothing stops you doing it again. Take the new square, draw its diagonal, and build on that. The first square has area one, the second two, the third four. Each is exactly double the one before. Count the triangles and you get two, then four, then eight. And look at the third square. Its area is four, which is where the doubled side was going to land at the very start.

7:16So the wrong answer was not nonsense. It was the right answer to the question asked twice. Keep going and the areas run one, two, four, eight, sixteen, for as long as you have paper. You can do all of this with scissors, and there are two ways. Take two identical paper squares. Cut the second along both its diagonals into four quarter triangles. Set those four around the first square, one against each side, and the outline you get is a square of area two.

7:53It is the square on the first square's diagonal, built out of paper. The second way is simpler. Take two identical squares and cut each along one diagonal only. Four right triangles. Put their four right angles together at a centre and their four long sides on the outside. That is a square of area two as well. Different cuts, different pieces, the same square at the end. So what is the length of that diagonal?

8:25The new square has area two, and its side is the diagonal. So the diagonal is a length whose square is two. And we already went hunting for that number and did not find it. It is not any fraction at all. Which is exactly why the counting argument matters. It never needed the number. Two pieces in the small square, four in the big one. Four is twice two, and the area doubles.

8:54The length is a question for another day, and it turns out to be a strange and important one. Baudhayana had the construction long before anybody had the number.