1
PRELUDE TO A NUMBER
Numberless are the world's wonders.
—Sophocles (495-405 b.c.)
The famous British physicist Lord Kelvin (William Thomson; 1824-1907), 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 Poe's The Mystery of Marie Roget, 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 Buffon's Needle, after the French mathematician George-Louis Leclerc, Comte de Buffon (1707-1788), who posed and solved this probability problem in 1777. Leclerc asked: Suppose you have a large sheet of paper on the floor, ruled with parallel straight lines spaced by a fixed distance. A needle of length equal precisely to the spacing between the lines is thrown completely at random onto the paper. What is the probability that the needle will land in such a way that it will intersect one of the lines (e.g., as in Figure 1)? Surprisingly, the answer turns out to be the number 2/?. Therefore, in principle, you could even evaluate — by repeating this experiment many times and observing in what fraction of the total number of throws you obtain an intersection. (There exist, however, less tedious ways to find the value of pi.) Pi has by now become such a household word that film director Darren Aronofsky was even inspired to make a 1998 intellectual thriller with that title.
Less known than pi is another number, phi (f), 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 Dali's 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, Euclid of Alexandria. We shall return to Euclid and his fantastic accomplishments in Chapter 4, but at the moment let me note only that so great is the admiration that Euclid commands that, in 1923, the poet Edna St. Vincent Millay wrote a poem entitled "Euclid Alone Has Looked on Beauty Bare." Actually, even Millay's annotated notebook from her course in Euclidean geometry has been preserved. Euclid defined a proportion derived from a simple division of a line into what he called its "extreme and mean ratio." In Euclid's 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.
In other words, if we look at Figure 2, line AB is certainly longer than the segment AC; at the same time, the segment AC is longer than CB. If the ratio of the length of AC to that of CB is the same as the ratio of AB to AC, then the line has been cut in extreme and mean ratio, or in a Golden Ratio.
Who could have guessed that this innocent-looking line division, which Euclid defined for some purely geometrical purposes, would have consequences in topics ranging from leaf arrangements in botany to the structure of galaxies containing billions of stars, and from mathematics to the arts? The Golden Ratio therefore provides us with a wonderful example of that feeling of utter amazement that the famous physicist Albert Einstein (1879-1955) valued so much. In Einstein's own words: "The fairest thing we can experience is the mysterious. It is the fundamental emotion which stands at the cradle of true art and science. He who knows it not and can no longer wonder, no longer feel amazement, is as good as dead, a snuffed-out candle."
As we shall see calculated in this book, the precise value of the Golden Ratio (the ratio of AC to CB in Figure 2) is the never-ending, never-repeating number 1.6180339887 . . . , and such never-ending numbers have intrigued humans since antiquity. One story has it that when the Greek mathematician Hippasus of Metapontum discovered, in the fifth century b.c., that the Golden Ratio is a number that is neither a whole number (like the familiar 1, 2, 3, . . .) nor even a ratio of two whole numbers (like the fractions 1/2, 2/3, 3/4, . . . ; known collectively as rational numbers), this absolutely shocked the other followers of the famous mathematician Pythagoras (the Pythagoreans). The Pythagorean worldview (which will be described in detail in Chapter 2) was based on an extreme admiration for the arithmos—the intrinsic properties of whole numbers or their ratios—and their presumed role in the cosmos. The realization that there exist numbers, like the Golden Ratio, that go on forever without displaying any repetition or pattern caused a true philosophical crisis. Legend even claims that, overwhelmed with this stupendous discovery, the Pythagoreans sacrificed a hundred oxen in awe, although this appears highly unlikely, given the fact that the Pythagoreans were strict vegetarians. I should emphasize at this point that many of these stories are based on poorly documented historical material. The precise date for the discovery of numbers that are neither whole nor fractions, known as irrational numbers, is not known with any certainty. Nevertheless, some researchers do place the discovery in the fifth century b.c., which is at least consistent with the dating of the stories just described. What is clear is that the Pythagoreans basically believed that the existence of such numbers was so horrific that it must represent some sort of cosmic error, one that should be suppressed and kept secret.
The fact that the Golden Ratio cannot be expressed as a fraction (as a rational number) means simply that the ratio of the two lengths AC and CB in Figure 2 cannot be expressed as a fraction. In other words, no matter how hard we search, we cannot find some common measure that is contained, let's say, 31 times in AC and 19 times in CB. Two such lengths that have no common measure are called incommensurable. The discovery that the Golden Ratio is an irrational number was therefore, at the same time, a discovery of incommensurability. In On the Pythagorean Life (ca. a.d. 300), the philosopher and historian Iamblichus, a descendant of a noble Syrian family, describes the violent reaction to this discovery:
They say that the first [human] to disclose the nature of commensurability and incommensurability to those unworthy to share in the theory was so hated that not only was he banned from [the Pythagoreans'] common association and way of life, but even his tomb was built, as if [their] former colleague was departed from life among humankind.
In the professional mathematical literature, the common symbol for the Golden Ratio is the Greek letter tau (from the Greek solag, to-mi, which means "the cut" or "the section"). However, at the beginning of the twentieth century, the American mathematician Mark Barr gave the ratio the name of phi, the first Greek letter in the name of Phidias, the great Greek sculptor who lived around 490 to 430 b.c. Phidias' greatest achievements were the "Athena Parthenos" in Athens and the "Zeus" in the temple of Olympia. He is traditionally also credited with having been in charge of other Parthenon sculptures, although it is quite probable that many were actually made by his students and assistants. Barr decided to honor the sculptor because a number of art historians maintained that Phidias had made frequent and meticulous use of the Golden Ratio in his sculpture. (We shall examine similar claims very scrupulously in this book.) I will use the names Golden Ratio, Golden Section, Golden Number, phi, and also the symbol interchangeably throughout, because these are the names most frequently encountered in the recreational mathematics literature.
Some of the greatest mathematical minds of all ages, from Pythagoras and Euclid in ancient Greece, through the medieval Italian mathematician Leonardo of Pisa and the Renaissance astronomer Johannes Kepler, to present-day scientific figures such as Oxford physicist Roger Penrose, have spent endless hours over this simple ratio and its properties. But the fascination with the Golden Ratio is not confined just to mathematicians. Biologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered and debated the basis of its ubiquity and appeal. In fact, it is probably fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics.
An immense amount of research, in particular by the Canadian mathematician and author Roger Herz-Fischler (described in his excellent book A Mathematical History of the Golden Number), has been devoted even just to the simple question of the origin of the name "Golden Section." Given the enthusiasm that this ratio has generated since antiquity, we might have thought that the name also has ancient origins. Indeed, some authoritative books on the history of mathematics, like Frankois Lasserre's The Birth of Mathematics in the Age of Plato, and Carl B. Boyer's A History of Mathematics, place the origin of this name in the fifteenth and sixteenth centuries, respectively. This, however, appears not to be the case. As far as I can tell from reviewing much of the historical fact-finding effort, this term was first used by the German mathematician Martin Ohm (brother of the famous physicist Georg Simon Ohm, after whom Ohm's law in electromagnetism is named), in the 1835 second edition of his book Die Reine Elementar-Mathematik (The pure elementary mathematics). Ohm writes in a footnote: "One also customarily calls this division of an arbitrary line in two such parts the golden section." Ohm's language clearly leaves us with the impression that he did not invent the term himself but rather used a commonly accepted name. Yet the fact that he did not use it in the first edition of his book (published in 1826) suggests at least that the name "Golden Section" (or, in German, "Goldene Schnitt") gained its popularity only around the 1830s. The name might have been used orally prior to that, perhaps in nonmathematical circles. There is no question, however, that following Ohm's book, the term "Golden Section" started to appear frequently and repeatedly in the German mathematical and art history literature. It may have made its debut in English in an article by James Sully on aesthetics, which appeared in the ninth edition of the Encyclopaedia Britannica in 1875. Sully refers to the "interesting experimental enquiry . . . instituted by [Gustav Theodor] Fechner [a physicist and pioneering German psychologist in the nineteenth century] into the alleged superiority of 'the golden section' as a visible proportion." (I discuss Fechner's experiments in Chapter 7.) The earliest English uses in a mathematical context appear to have been in an article entitled "The Golden Section" (by E. Ackermann) that appeared in 1895 in the American Mathematical Monthly and, around the same time, in the 1898 book Introduction to Algebra by the well-known teacher and author G. Chrystal (1851-1911). Just as a curiosity, let me note that the only definition of a "Golden Number" that appears in the 1900 edition of the French encyclopedia Nouveau Larousse Illustre is: "A number used to indicate each of the years of the lunar cycle." This refers to the position of a calendar year within the nineteen-year cycle after which the phases of the Moon recur on the same dates. Clearly the phrase took a longer time to enter the French mathematical nomenclature.
But what is all the fuss about? What is it that makes this number, or geometrical proportion, so exciting as to deserve all of this attention?
The Golden Ratio's attractiveness stems first and foremost from the fact that it has an almost uncanny way of popping up where it is least expected.
Take, for example, an ordinary apple, the fruit often associated (probably mistakenly) with the tree of knowledge that figures so prominently in the biblical account of humankind's fall from grace, and cut it through its girth. You will find that the apple's seeds are arranged in a five-pointed star pattern, or pentagram (Figure 3). Each of the five isosceles triangles that make the corners of a pentagram has the property that the ratio of the length of its longer side to the shorter one (the implied base) is equal to the Golden Ratio, 1.618. . . . But, you may think, maybe this is not so surprising. After all, since the Golden Ratio has been defined as a geometrical proportion, perhaps we should not be too astonished to discover that this proportion is found in some geometrical shapes.