Alex's Adventures in Numberland - Alex Bellos [34]
In some commentaries about Pythagoras it is said that before he founded the Brotherhood he travelled on a fact-finding mission to Egypt. If he had spent any time on an Egyptian building site he would have seen that the labourers used a trick to create a right angle that was an application of the theorem that would later gain his name. A rope was marked with knots spread out at a distance of 3, 4 and 5 units. Since 32 + 42 = 52, when the rope was stretched around three posts, with a knot at each post, it formed a triangle with one right angle.
Pythagoras’s Theorem.
Rope-stretching was the most convenient way to achieve right angles, which were needed so that bricks, or giant stone blocks such as those used to construct the Pyramids, could be stacked next to and on top of each other. (The word hypotenuse comes from the Greek for ‘stretched against’.) The Egyptians could have used many other numbers in addition to 3, 4 and 5 to get real right angles. In fact, there is an infinite number of numbers a, b and c such that a2 + b2 = c2. They could have marked out their rope into sections of 5, 12 and 13, for example since 25 + 144 = 169, or 8, 15 and 17, since 64 + 225 = 289, or even 2772, 9605 and 9997 (7,683,984 + 92,256,025 = 9940,009) though that would hardly have been practical. The numbers 3, 4, 5 are best suited to the task. As well as being the triple with the lowest value, it is also the only one whose digits are consecutive integers. Due to its rope-stretching heritage, the right-angled triangle with sides that are in the ratio 3:4:5 is known as an Egyptian triangle. It is a pocket-sized right-angle-generating machine that is a jewel of our mathematical patrimony, an intellectual artefact of great power, elegance and concision.
The Egyptian equivalent of a set square was a rope divided in the ratio 3:4:5, which provides a right angle when tied around three posts.
The squares mentioned in Pythagoras’s Theorem can be understood as numbers and also as pictures – literally the squares drawn on the sides of the triangle. Imagine that in the following image the squares are made of gold. You are not engaged to a member of the Pythagorean Brotherhood, so acquiring gold is desirable. Either you can take the two smaller squares, or you can have the one largest square. Which would you prefer?
The mathematician Raymond Smullyan said that when he put this question to his students, half the class wanted the big single square and the other half wanted the double. Both sides were stunned when he told them that it would make no difference.
This is true because, as the theorem states, the combined area of the two smaller squares is equal to the area of the large square. All right-angled triangles can be extended in this way to produce three squares such that the area of the large one can be divided exactly into the areas of the two smaller ones. It is not the case that the square on the hypotenuse is sometimes the sum of the squares of the other two sides, and sometimes not. The fit is perfect at all times.
It is not known if Pythagoras really discovered his theorem, even though his name has been attached to it since classical times. Whether or not he did, it vindicates his world-view, demonstrating a remarkable harmony in the mathematical universe. In fact, the theorem reveals a relationship between more than just the squares on the sides of a right-angled triangle. The area of a semicircle on the hypotenuse, for example, is equal to the sum of the areas of the semicircles on the other two sides. A pentagon on the hypotenuse is equal to the sum of pentagons on the other two sides, and this holds for hexagons, octagons and, indeed, any regular or irregular shape. If, say, three Mona Lisas were drawn on a right-angled triangle, then the area of big Mona is equal to the combined area of the two smaller ones.
For me, the real delight in Pythagoras’s Theorem is in the realization of why it must be true. The simplest proof is as follows. It dates back to the