From Scientific American
Most schoolchildren learn the importance of the number ϖ, or pi (3.14159...), early on in their mathematical training. The area of a circle? ϖ (r2). Circumference? 2ϖr. But for astrophysicist Livio, the number φ, or phi (1.61803...), holds more wonder. Known as the Golden Ratio, Golden Number and other names related to gold or divinity, phi can be used to describe the characteristics of seashells, sunflowers, paintings and pentagrams. Livio traces the history of phi's discovery in tandem with the development of arithmetic, algebra and higher mathematics, examining along the way whether its value was factored into the construction of the Egyptian pyramids or the Parthenon of Greece. Practical examples placed throughout the story, requiring nothing more than a handheld calculator or some scratch paper, help to illustrate phi's relevance to mathematics and the world at large. Editors of Scientific American
Numbers aficionados will delight in astrophysicist Livio's history of an irrational number whose fame is second only to that of pi. It's called the golden ratio and was discovered by Euclid more than 2,000 years ago. It seems that for any line divided into two unequal segments, the resultant lengths of the two segments and the original line can be formed into a ratio that equals phi, or 1.618 . . . This curiosity of plane and solid geometry might have remained just an oddity had the ratio not cropped up in unusual places, from the structure of crystals to botany to the shape of spiral galaxies. This unending surprise drives Livio's narrative, which he spices with profiles of people obsessed by this ubiquitous number. Some have tried to prove that the ratio was the design principle for the Parthenon; Kepler was crazy about phi; and there's a whole mathematical community devoted to Fibonacci numbers, whose permutations produce phi again and again. Livio's encyclopedic selection of subjects, supported by dozens of illustrations, will snare anyone with a recreational interest in mathematics. Gilbert Taylor
Copyright © American Library Association. All rights reserved
From the Inside Flap
Throughout history, thinkers from mathematicians to theologians have pondered the mysterious relationship between numbers and the nature of reality. In this fascinating book, Mario Livio tells the tale of a number at the heart of that mystery: phi, or 1.6180339887...This curious mathematical relationship, widely known as "The Golden Ratio," was discovered by Euclid more than two thousand years ago because of its crucial role in the construction of the pentagram, to which magical properties had been attributed. Since then it has shown a propensity to appear in the most astonishing variety of places, from mollusk shells, sunflower florets, and rose petals to the shape of the galaxy. Psychological studies have investigated whether the Golden Ratio is the most aesthetically pleasing proportion extant, and it has been asserted that the creators of the Pyramids and the Parthenon employed it. It is believed to feature in works of art from Leonardo da Vinci's Mona Lisa to Salvador Dali's The Sacrament of the Last Supper, and poets and composers have used it in their works. It has even been found to be connected to the behavior of the stock market!
The Golden Ratio is a captivating journey through art and architecture, botany and biology, physics and mathematics. It tells the human story of numerous phi-fixated individuals, including the followers of Pythagoras who believed that this proportion revealed the hand of God; astronomer Johannes Kepler, who saw phi as the greatest treasure of geometry; such Renaissance thinkers as mathematician Leonardo Fibonacci of Pisa; and such masters of the modern world as Goethe, Cezanne, Bartok, and physicist Roger Penrose. Wherever his quest for the meaning of phi takes him, Mario Livio reveals the world as a place where order, beauty, and eternal mystery will always coexist.
About the Author
Mario Livio is head of the Science Division at the Hubble Space Telescope Institute, where he studies a broad range of subjects in astrophysics, particularly the rate of expansion of the universe. He is the author of one previous book, The Accelerating Universe (2000). He is a frequent public lecturer at such venues as the Smithsonian Institution and the Hayden Planetarium. He lives in Baltimore, Maryland.
Excerpt. © Reprinted by permission. All rights reserved.
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 pro...
