2.1 The Pythagorean Theorem - proof (The road to reality by Sir Roger Penrose)

Let us consider the issue of geometry. What, indeed, are the different ‘kinds of geometry’ that were alluded to in the last chapter? To lead up to this issue, we shall return to our encounter with Pythagoras and consider that famous theorem that bears his name for any right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides (Fig. 2.1). What reasons do we have for believing that this assertion is true? How, indeed, do we ‘prove’ the Pythagorean Theorem? Many arguments are known. I wish to consider two such, chosen for their particular transparency, each of which has a different emphasis.

For the first, consider the pattern illustrated in Fig. 2.2. It is composed entirely of squares of two different sizes. It may be regarded as ‘obvious’ that this pattern can be continued indefinitely and that the entire plane is thereby covered in this regular repeating way, without gaps or overlaps, by squares of these two sizes. The repeating nature of this pattern is made manifest by the fact that if we mark the centers of the larger squares, they form the vertices of another system of squares, of a somewhat greater size than either, but tilted at an angle to the original ones (Fig. 2.3) and which alone will cover the entire plane. Each of these tilted squares is marked in exactly the same way, so that the markings on these squares fit together to form the original two-square pattern. The same would apply if, instead of taking the centers of the larger of the two squares of the original pattern, we chose any other point, together with its set of corresponding points throughout the pattern. The new pattern of tilted squares is just the same as before but moved along without rotation—i.e. by means of a motion referred to as a translation. For simplicity, we can now choose our starting point to be one of the corners in the original pattern (see Fig. 2.4).

It should be clear that the area of the tilted square must be equal to the sum of the areas of the two smaller squares—indeed the pieces into which the markings would subdivide this larger square can, for any starting point for the tilted squares, be moved around, without rotation, until they fit together to make the two smaller squares (e.g. Fig. 2.5). Moreover, it is evident from Fig. 2.4 that the edge-length of the large tilted square is the hypotenuse of a right-angled triangle whose two other sides have lengths equal to those of the two smaller squares. We have thus established the Pythagorean theorem: the square on the hypotenuse is equal to the sum of the squares on the other two sides.

The above argument does indeed provide the essentials of a simple proof of this theorem, and, moreover, it gives us some ‘reason’ for believing that the theorem has to be true, which might not be so obviously the case with some more formal argument given by a succession of logical steps without clear motivation. It should be pointed out, however, that there are several implicit assumptions that have gone into this argument. Not the least of these is the assumption that the seemingly obvious pattern of repeating squares shown in Fig. 2.2 or even in Fig. 2.6 is actually geometrically possible—or even, more critically, that a square is something geometrically possible! What do we mean by a ‘square’ after all? We normally think of a square as a plane figure, all of whose sides are equal and all of whose angles are right angles. What is a right angle? Well, we can imagine two straight lines crossing each other at some point, making four angles that are all equal. Each of these equal angles is then a right angle.

Let us now try to construct a square. Take three equal line segments AB, BC, and CD, where ABC and BCD are right angles, D and A being on the same side of the line BC, as in Fig. 2.7. The question arises: is AD the same length as the other three segments? Moreover, are the angles DAB and CDA also right angles? These angles should be equal to one another by a left–right symmetry in the figure, but are they actually right angles? This only seems obvious because of our familiarity with squares, or perhaps because we can recall from our schooldays some statement of Euclid that can be used to tell us that the sides BA and CD would have to be ‘parallel’ to each other, and some statement that any ‘transversal’ to a pair of parallels has to have corresponding angles equal, where it meets the two parallels. From this, it follows that the angle DAB would have to be equal to the angle complementary to ADC (i.e. to the angle EDC, in Fig. 2.7, ADE being straight) as well as being, as noted above, equal to the angle ADC. An angle (ADC) can only be equal to its complementary angle (EDC) if it is a right angle. We must also prove that the side AD has the same length as BC, but this now also follows, for example, from properties of transversals to the parallels BA and CD. So, it is indeed true that we can prove from this kind of Euclidean argument that squares, made up of right angles, actually do exist. But there is a deep issue hiding here.