3M Company. - The Golden Ratio: the Story of PHI, the Worlds Most Astonishing Number

Numberless are the worlds wonders.

S OPHOCLES (495405 B.C. )

The famous British physicist Lord Kelvin (William Thomson; 18241907), after whom the degrees in the absolute temperature scale are named, once said in a lecture: When you cannot express it in numbers, your knowledge is of a meager and unsatisfactory kind. Kelvin was referring, of course, to the knowledge required for the advancement of science. But numbers and mathematics have the curious propensity of contributing even to the understanding of things that are, or at least appear to be, extremely remote from science. In Edgar Allan Poes The Mystery of Marie Rogt, the famous detective Auguste Dupin says: We make chance a matter of absolute calculation. We subject the unlooked for and unimagined, to the mathematical formulae of the schools. At an even simpler level, consider the following problem you may have encountered when preparing for a party: You have a chocolate bar composed of twelve pieces; how many snaps will be required to separate all the pieces? The answer is actually much simpler than you might have thought, and it does not require almost any calculation. Every time you make a snap, you have one more piece than you had before. Therefore, if you need to end up with twelve pieces, you will have to snap eleven times. (Check it for yourself.) More generally, irrespective of the number of pieces the chocolate bar is composed of, the number of snaps is always one less than the number of pieces you need.

Even if you are not a chocolate lover yourself, you realize that this example demonstrates a simple mathematical rule that can be applied to many other circumstances. But in addition to mathematical properties, formulae, and rules (many of which we forget anyhow), there also exist a few special numbers that are so ubiquitous that they never cease to amaze us. The most famous of these is the number pi (), which is the ratio of the circumference of any circle to its diameter. The value of pi, 3.14159, has fascinated many generations of mathematicians. Even though it was defined originally in geometry, pi appears very frequently and unexpectedly in the calculation of probabilities. A famous example is known as Darren Aronofsky was even inspired to make a 1998 intellectual thriller with that title.

Figure 1

Less known than pi is another number, phi (), which is in many respects even more fascinating. Suppose I ask you, for example: What do the delightful petal arrangement in a red rose, Salvador Dalis famous painting Sacrament of the Last Supper, the magnificent spiral shells of mollusks, and the breeding of rabbits all have in common? Hard to believe, but these very disparate examples do have in common a certain number or geometrical proportion known since antiquity, a number that in the nineteenth century was given the honorifics Golden Number, Golden Ratio, and Golden Section. A book published in Italy at the beginning of the sixteenth century went so far as to call this ratio the Divine Proportion.

In everyday life, we use the word proportion either for the comparative relation between parts of things with respect to size or quantity or when we want to describe a harmonious relationship between different parts. In mathematics, the term proportion is used to describe an equality of the type: nine is to three as six is to two. As we shall see, the Golden Ratio provides us with an intriguing mingling of the two definitions in that, while defined mathematically, it is claimed to have pleasingly harmonious qualities.

The first clear definition of what has later become known as the Golden Ratio was given around 300 B.C . by the founder of geometry as a formalized deductive system, Euclids words:

A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser.