Saturday, June 27, 2009

The Pythagorean theorem

The Pythagorean theorem is named after the Greek mathematician Pythagoras, who by tradition is credited with its discovery and proof, although it is often argued that knowledge of the theory predates him. (There is much evidence that Babylonian mathematicians understood the principle, if not the mathematical significance.)

An elegant proof of Pythagorean theorem can be found in Roger Penrose's popular science classic "The Road to Reality - A Complete Guide to the Laws of the Universe". His proof suits perfectly to the central theme of this blog; "Negative Matter", in other words the ability to see and transform negative spaces in order to convey further "meaning" into positive spaces.

First consider the pattern illustrated below. It is composed entirely of squares of two different sizes. It is 'obvious' that this pattern can be continued indefinitely, and the entire Euclidean plane can be covered in this repeating way with no gaps or overlaps by squares of these two sizes.

If we mark the centres of the larger squares, they form the vertices of another system of squares, at somewhat greater size than either, but tilted at an angle to the original ones.

Instead of taking the centres of the two large squares of the original pattern, we may choose 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 across either vertices without rotation (i.e. by means of motion referred as translation). For simplicity lets choose our starting point to be one of the corners in the original pattern.

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. The small square at the top-left has a small triangular portion outside the tilted square, which is identical to the one inside, if we move along the vertex of tilted square. A similar observation can be made about larger triangles, proving our assertion.

Moreover it is evident 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 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.

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