The Golden Ratio.

Mario Livio.

In memory of my father Robin Livio

PREFACE

The Golden Ratio is a book about one number-a very special number. You will encounter this number, 1.61803..., in lectures on art history, and it appears in lists of "favorite numbers" compiled by mathematicians. Equally striking is the fact that this number has been the subject of numerous experiments in psychology. is a book about one number-a very special number. You will encounter this number, 1.61803..., in lectures on art history, and it appears in lists of "favorite numbers" compiled by mathematicians. Equally striking is the fact that this number has been the subject of numerous experiments in psychology.I became interested in the number known as the Golden Ratio fifteen years ago, as I was preparing a lecture on aesthetics in physics (yes, this is not an oxymoron), and I haven't been able to get it out of my head since then.Many more colleagues, friends, and students than I would be able to mention, from a mult.i.tude of disciplines, have contributed directly and indirectly to this book. Here I would like to extend special thanks to Ives-Alain Bois, Mitch Feigenbaum, Hillel Gauchman, Ted Hill, Ron Lifschitz, Roger Penrose, Johanna Postma, Paul Steinhardt, Pat Thiel, Anne van der Helm, Divakar Viswanath, and Stephen Wolfram for invaluable information and extremely helpful discussions.I am grateful to my colleagues Daniela Calzetti, Stefano Casertano, and Ma.s.simo Stiavelli for their help with translations from Latin and Italian; to Claus Leitherer and Hermine Landt for help with translations from German; and to Patrick G.o.don for his help with translations from French. Sarah Stevens-Rayburn, Elizabeth Fraser, and Nancy Hanks provided me with valuable bibliographical and linguistic support. I am particularly grateful to Sharon Toolan for her a.s.sistance with the preparation of the ma.n.u.script.My sincere grat.i.tude goes to my agent, Susan Rabiner, for her relentless encouragement before and during the writing of this book.I am deeply indebted to my editor at Doubleday Broadway, Gerald Howard, for his careful reading of the ma.n.u.script and his insightful comments. I am also grateful to Rebecca Holland, Publishing Manager at Doubleday Broadway, for her unflagging a.s.sistance during the production of this book.Finally, it is due only to the continuous inspiration and patient support provided by Sofie Livio that this book got written at all.

Numberless are the world's wonders.-SOPHOCLES (495405 (495405 B.C. 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 Poe's The Mystery of Marie Roget 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 circ.u.mstances. 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 circ.u.mference 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 (17071788), 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 s.p.a.ced by a fixed distance. A needle of length equal precisely to the s.p.a.cing be tween 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 Figure 1)? Surprisingly, the answer turns out to be the number 2/. There fore, 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 intellec tual thriller with that t.i.tle. 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 intellec tual thriller with that t.i.tle.

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 Dali's famous painting "Sacrament of the Last Supper," the magnificent spiral sh.e.l.ls 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 quant.i.ty 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. 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 ent.i.tled "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: 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 ent.i.tled "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.

Figure 2 In other words, if we look at Figure 2 Figure 2, line AB AB is certainly longer than the segment is certainly longer than the segment AC; AC; at the same time, the segment at the same time, the segment AC AC is longer than is longer than CB. CB. If the ratio of the length of If the ratio of the length of AC AC to that of to that of CB CB is the same as the ratio of is the same as the ratio of AB AB to to AC 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 (18791955) 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 AC to to CB CB in in Figure 2 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. 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 ,,,...; ,,,...; known collectively as known collectively as rational numbers) 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- 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 repet.i.tion 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 doc.u.mented historical material. The precise date for the discovery of numbers that are neither whole nor fractions, known as irrational numbers irrational numbers, is not known with any certainty. Nevertheless, some researchers do place the discovery in the fifth century B.C. 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 AC and and CB CB in in Figure 2 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 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 AC and 19 times in and 19 times in CB. CB. Two such lengths that have no common measure are called Two such lengths that have no common measure are called incommensurable. incommensurable. The discovery that the Golden Ratio is an irrational number was therefore, at the same time, a discovery of incommensurability. In 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 On the Pythagorean Life (ca. (ca. A.D. A.D. 300), the philosopher and historian Iamblichus, a descendant of a n.o.ble Syrian family, describes the violent reaction to this discovery: 300), the philosopher and historian Iamblichus, a descendant of a n.o.ble 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 a.s.sociation 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 o, 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. 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 a.s.sistants. 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. 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 a.s.sistants. 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) 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 Francois La.s.serre's The Birth of Mathematics in the Age of Plato The Birth of Mathematics in the Age of Plato, and Carl B. Boyer's A History of Mathematics 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 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 (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 Encyclopaedia Britannica in 1875. Sully refers to the "interesting experimental enquiry... inst.i.tuted 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 ent.i.tled "The Golden Section" (by E. Ackermann) that appeared in 1895 in the in 1875. Sully refers to the "interesting experimental enquiry... inst.i.tuted 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 ent.i.tled "The Golden Section" (by E. Ackermann) that appeared in 1895 in the American Mathematical Monthly American Mathematical Monthly and, around the same time, in the 1898 book and, around the same time, in the 1898 book Introduction to Algebra Introduction to Algebra by the well-known teacher and author G. Chrystal (18511911). 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 by the well-known teacher and author G. Chrystal (18511911). 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 Ill.u.s.tre Nouveau Larousse Ill.u.s.tre 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. 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 a.s.sociated (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 denned as a geometrical proportion, perhaps we should not be too astonished to discover that this proportion is found in some geometrical shapes.

Figure 3 This is, however, only the tip of the iceberg. According to Buddhist tradition, in one of Buddha's sermons he did not utter a single word; he merely held a flower in front of his audience. What can a flower teach us? A rose, for example, is often taken as a symbol of natural symmetry, harmony, love, and fragility. In Religion of Man Religion of Man, Indian poet and philosopher Rabindranath Tagore (18611941) writes: "Somehow we feel that through a rose the language of love reached our hearts." Suppose you want to quantify the symmetric appearance of a rose. Take a rose and dissect it, to uncover the way in which its petals overlap their predecessors. As I describe in Chapter 5, you will find that the positions of the petals are arranged according to a mathematical rule that relies on the Golden Ratio.

Turning now to the animal kingdom, we are all familiar with the strikingly beautiful spiral structures of many sh.e.l.ls of mollusks, such as the chambered nautilus (Nautilus pompilius; (Nautilus pompilius;In fact, the dancing Shiva of the Hindu myth holds such a nautilus in one of his hands, as a symbol of one of the instruments initiating creation. These sh.e.l.ls also have inspired many architectural constructions. American architect Frank Lloyd Wright (18691959), for example, based the de sign of the Guggenheim Museum in New York City on the structure of the chambered nautilus. Within the museum, the visitors ascend a spiral ramp, moving on, when their imaginative capacity is saturated by the art they see, just as the mollusk builds its spiral chambers when fully occupying its physical s.p.a.ce. We shall discover in Chapter 5 that the growth of spiral sh.e.l.ls also obeys a pattern that is governed by the Golden Ratio.

Figure 4

Figure 5 By now, we do not have to be number mysticists to begin to feel a certain awe at this property of the Golden Ratio to show up in what appear to be totally unrelated situations and phenomena. Furthermore, as I noted at the beginning of this chapter, the Golden Ratio can be found not only in natural phenomena but also in a variety of human-made objects and works of art. For example, in Salvador Dali's painting from 1955, "Sacrament of the Last Supper" (in the National Gallery, Washington D.C.; Figure 5 Figure 5), the dimensions of the painting (approximately 105 65) are in a Golden Ratio to each other. Perhaps even more important, part of a huge dodecahedron (a twelve-faced regular solid in which each side is a pentagon) is seen floating above the table and engulfing it. As we shall see in Chapter 4, regular solids (like the cube) that can be precisely enclosed by a sphere (with all their corners resting on the sphere), and the dodecahedron in particular, are intimately related to the Golden Ratio. Why did Dali choose to exhibit the Golden Ratio so prominently in this painting? His remark that "the Communion must be symmetrical" only begins to answer this question. As I show in Chapter 7, the Golden Ratio features (or is at least claimed to feature) in the works of many other artists, architects, and designers, and even in famous musical compositions. Broadly speaking, the Golden Ratio has been used in some of these works to achieve what we might term "visual (or audio) effectiveness." One of the properties contributing to such effectiveness is proportion- proportion-the size relationships of parts to one another and to the whole. The history of art shows that in the long search for an elusive canon of "perfect" proportion, one that would somehow automatically confer aesthetically pleasing qualities on all works of art, the Golden Ratio has proven to be the most enduring. But why?

A closer examination of the examples from nature and from the arts reveals that they raise questions at three different levels of increasing depth. First, there are the immediate questions: (a) Are all the appearances of phi in nature and in the arts that are cited in the literature real, or do some of those simply represent misconceptions and crankish interpretations? (b) Can we actually explain the appearance (if real) of phi in these and other circ.u.mstances? Second, given that we define "beauty," as, for example, in Webster's Unabridged Dictionary Webster's Unabridged Dictionary, "the quality which makes an object seem pleasing or satisfying in a certain way," this raises the question: Is there an aesthetic component to mathematics? And if so, what is the essence of this component? This is a serious question because, as the American architect, mathematician, and engineer Richard Buckminster Fuller (18951983) once put it: "When I am working on a problem, I never think about beauty. I think only of how to solve the problem. But when I have finished, if the solution is not beautiful, I know it is wrong." Finally, the most intriguing question is: What is it that makes mathematics so powerful and ubiquitous? What is the reason that mathematics and numerical constants like the Golden Ratio play such a central role in topics ranging from fundamental theories of the universe to the stock market? Does mathematics exist even independently of the humans who have discovered/invented it and its principles? Is the universe by its very nature mathematical? This last question can be rephrased, using a famous aphorism of the British physicist Sir James Jeans (18471946), as: Is G.o.d a mathematician?

I will attempt to address all of these questions in some detail in this book, via the fascinating story of phi. The sometimes-tangled history of this ratio spans millennia as well as continents. Equally important, I hope to tell a good human-interest story. A part of this story will be about a time when "scientists" and "mathematicians" were self-selected individuals who simply pursued questions that kindled their curiosity. These people often labored and died without knowing whether their works would change the course of scientific thought or would simply disappear without a trace.

Before we embark on this main journey, however, we have to familiarize ourselves with numbers in general and with the Golden Ratio in particular. After all, how did the initial idea of the Golden Ratio arise? What was it that led Euclid even to bother to define such a line division? My aim is to help you glean some insights into the true roots of what we might call Golden Numberism. To this goal, we will now take a brief exploratory tour through the very dawn of mathematics.

As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.-ALBERT E EINSTEIN (18791955) (18791955)I see a certain order in the universe and math is one way of making it visible.-MAY S SARTON (19121995) (19121995) No one knows for sure when humans started to count, that is, to measure mult.i.tude in a quant.i.tative way. In fact, we do not even know with certainty whether numbers like "one," "two," "three" (the cardinal numbers) preceded numbers like "first," "second," "third" (the ordinal numbers), or vice versa. Cardinal numbers simply determine the plurality of a collection of items, such as the number of children in a group. Ordinal numbers, on the other hand, specify the order and succession of specific elements in a group, such as a given date in a month or a seat number in a concert hall. Originally it was a.s.sumed that counting developed specifically to address simple day-to-day needs, which clearly argued for cardinal numbers appearing first. However, some anthropologists have suggested that numbers may have first appeared on the historical scene in relation to some rituals that required the successive appearance (in a specified order) of individuals during ceremonies. If true, this idea suggests that the ordinal number concept may have preceded the cardinal one.

Clearly, an even bigger mental leap was required to move from the simple counting of objects to an actual understanding of numbers as abstract quant.i.ties. Thus, while the first notions of numbers might have been related primarily to contrasts contrasts, a.s.sociated perhaps with survival-Is it one one wolf or a wolf or a pack pack of wolves?-the actual understanding that two hands and two nights are both manifestations of the number 2 probably took centuries to grasp. The process had to go through the recognition of similarities (as opposed to contrasts) and correspondences. Many languages contain traces of the original divorce between the simple act of counting and the abstract concept of numbers. In the Fiji Islands, for example, the term for ten coconuts is "koro," while for ten boats it is "bolo." Similarly, among the Tauade in New Guinea, different words are used for talking about pairs of males, pairs of females, and mixed pairs. Even in English, different names often are a.s.sociated with the same numbers of different aggregations. We say "a yoke of oxen" but never "a yoke of dogs." of wolves?-the actual understanding that two hands and two nights are both manifestations of the number 2 probably took centuries to grasp. The process had to go through the recognition of similarities (as opposed to contrasts) and correspondences. Many languages contain traces of the original divorce between the simple act of counting and the abstract concept of numbers. In the Fiji Islands, for example, the term for ten coconuts is "koro," while for ten boats it is "bolo." Similarly, among the Tauade in New Guinea, different words are used for talking about pairs of males, pairs of females, and mixed pairs. Even in English, different names often are a.s.sociated with the same numbers of different aggregations. We say "a yoke of oxen" but never "a yoke of dogs."

Surely the fact that humans have as many hands as they have feet, eyes, or b.r.e.a.s.t.s helped in the development of the abstract understanding of the number 2. Even there, however, it must have taken longer to a.s.sociate this number with things that are not identical, such as the fact that there are two major lights in the heavens, the Sun and the Moon. There is little doubt that the first distinctions were made between one and two and then between two and "many." This conclusion is based on the results of studies conducted in the nineteenth century among populations that were relatively unexposed to mainstream civilization and on linguistic differences in the terms used for different numbers in both ancient and modern times.

THREE IS A CROWD.

The first indication of the fact that numbers larger than two were once treated as "many" comes from some five millennia ago. In the language of Sumer in Mesopotamia, the name for the number 3, "es," served also as the mark of plurality (like the suffix s s in English). Similarly, ethnographic studies in 1890 of the natives of the islands in the Torres Strait, between Australia and Papua New Guinea, showed that they used a system known as two-counting. They used the words "urapun" for "one," "okosa" for "two," and then combinations such as "okosa-urapun" for "three" and "okosa-okosa" for "four." For numbers larger than four, the islanders used the word "ras" (many). Almost identical forms of nomenclatures were found in other indigenous populations from Brazil (the Botocudos) to South Africa (Zulus). The Aranda of Australia, for example, had "ninta" for "one," "tara" for "two," and then "tara mi ninta" for "three" and "tara ma tara" for "four," with all other numbers expressed as "many." Many of these populations were also found to have the tendency to group things in pairs, as opposed to counting them individually. in English). Similarly, ethnographic studies in 1890 of the natives of the islands in the Torres Strait, between Australia and Papua New Guinea, showed that they used a system known as two-counting. They used the words "urapun" for "one," "okosa" for "two," and then combinations such as "okosa-urapun" for "three" and "okosa-okosa" for "four." For numbers larger than four, the islanders used the word "ras" (many). Almost identical forms of nomenclatures were found in other indigenous populations from Brazil (the Botocudos) to South Africa (Zulus). The Aranda of Australia, for example, had "ninta" for "one," "tara" for "two," and then "tara mi ninta" for "three" and "tara ma tara" for "four," with all other numbers expressed as "many." Many of these populations were also found to have the tendency to group things in pairs, as opposed to counting them individually.

An interesting question is: Why did the languages used in these and other counting systems evolve to "four" and then stop (even though three and four were already expressed in terms of one and two)? One explanation suggests that this may simply reflect the fact that we happen to have four fingers in a similar position on our hands. Another, more subtle idea proposes that the answer lies in a physiological limit on human visual perception. Many studies show that the largest number we are able to capture at a glance, without counting without counting, is about four or five. You may remember that in the movie Rain Man Rain Man, Dustin Hoffman plays an autistic person with an unusual (in fact, highly exaggerated) perception of and memory for numbers. In one scene, all the toothpicks but four from a toothpick box scatter all over the floor, and he is able to tell at a glance that there are 246 toothpicks on the floor. Well, most people are unable to perform such feats. Anyone who ever tried to tally votes of any kind is familiar with this fact. We normally record the first four votes as straight lines, and then we cross those with a fifth line once a fifth vote is cast, simply because of the difficulty to perceive at a glance any number of lines that is larger than four. This system has been known in English pubs (where the barman counts the beers ordered) as the five barred gate. Curiously, an experiment described by the historian of mathematics Tobias Dantzig (18841956) in 1930 (in his wonderful book Number, the Language of Science) Number, the Language of Science) suggests that some birds also can recognize and discriminate among up to four objects. Dantzig's story goes as follows: suggests that some birds also can recognize and discriminate among up to four objects. Dantzig's story goes as follows: A squire was determined to shoot a crow which made its nest in the watch-tower of his estate. Repeatedly he had tried to surprise the bird, but in vain: at the approach of man the crow would leave its nest. From a distant tree it would watchfully wait until the man had left the tower and then return to its nest. One day the squire hit upon a ruse: two men entered the tower, one remained within, the other came out and went on. But the bird was not deceived: it kept away until the man within came out. The experiment was repeated on the succeeding days with two, three, then four men, yet without success. Finally, five men were sent: as before, all entered the tower, and one remained while the other four came out and went away Here the crow lost count. Unable to distinguish between four and five it promptly returned to its nest.

More pieces of evidence suggest that the initial counting systems followed the "one, two,... many" philosophy. These come from linguistic differences in the treatments of plurals and of fractions. In Hebrew, for example, there is a special form of plural for some pairs of identical items (e.g., hands, feet) or for words representing objects that contain two identical parts (e.g., pants, eyegla.s.ses, scissors) that is different from the normal plural. Thus, while normal plurals end in "im" (for items considered masculine) or "ot" (for feminine items), the plural form for eyes, b.r.e.a.s.t.s, and so on, or the words for objects with two identical parts, end in "ayim." Similar forms exist in Finnish and used to exist (until medieval times) in Czech. Even more important, the transition to fractions, which surely required a higher degree of familiarity with numbers, is characterized by a marked linguistic difference in the names of fractions other than a half. In Indo-European languages, and even in some that are not (e.g., Hungarian and Hebrew), the names for the fractions "one-third" () (), "one-fifth" () (), and so on generally derive from the names of the numbers of which these fractions are reciprocals (three, five, etc.). In Hebrew, for example, the number "three" is "shalosh" and "one-third" is "shlish." In Hungarian "three" is "Harom" and "one-third" is "Harmad." This is not true, however, for the number "half," which is not related to "two." In Romanian, for example, "two" is "doi" and "half is "jumate;" in Hebrew "two" is "shtayim" and "half is "hetsi;" in Hungarian "two" is "ketto" and "half is "fel." The implication may be that while the number was understood relatively early, the notion and comprehension of other fractions as reciprocals (namely, "one over") of integer numbers probably developed only after counting pa.s.sed the "three is a crowd" barrier.

COUNTING MY NUMBERLESS FINGERS.

Even before the counting systems truly developed, humans had to be able to record some quant.i.ties. The oldest archaeological records that are believed to be a.s.sociated with counting of some sort are in the form of bones on which regularly s.p.a.ced incisions have been made. The earliest, dating to about 35,000 B.C. B.C., is a part of a baboon's thigh bone found in a cave in the Lembedo Mountains in Africa. That bone was engraved with twenty-nine incisions. Another such "bookkeeping" record, a bone of a wolf with fifty-five incisions (twenty-five in one series and thirty in another, the first series grouped in fives), was found by archaeologist Karel Absolon in 1937 at Dolne Vestonice, Czechoslovakia, and has been dated to the Aurignacian era (about 30,000 years ago). The grouping into 5, in particular, suggests the concept of a base base, which I will discuss shortly. While we do not know the exact purpose of these incisions, they may have served as a record of a hunter's kills. The grouping would have helped the hunter to keep tally without having to recount every notch. Similarly marked bones, from the Magdalenian era (about 15,000 years ago), were also found in France and in the Pekarna cave in the Czech Republic.

A bone that has been subjected to much speculation is the Ishango bone found by archaeologist Jean de Heinzelin at Ishango near the border between Uganda and Zaire (Figure 6). That bone handle of a tool, dating to about 9000 B.C. B.C., displays three rows of notches arranged, respectively, in the following groups: (i) 9, 19, 21, 11; (ii) 19, 17, 13, 11; (iii) 7, 5, 5, 10, 8, 4, 6, 3. The sum of the numbers in the first two rows is 60 in each, which led some to speculate that they may represent a record of the phases of the Moon in two lunar months (with the possibility that some incisions may have been erased from the third row, which adds up only to 48). More intricate (and far more speculative) interpretations also have been proposed. For example, on the basis of the fact that the second row (19, 17, 13, 11) contains sequential primes (numbers that have no divisors except for 1 and the number itself), and the first row (9, 19, 21, 11) contains numbers that are different by 1 from either 10 or 20, de Heinzelin concluded that the Ishango people had some rudimentary knowledge of arithmetic and even of prime numbers. Needless to say, many researchers find this interpretation somewhat far-fetched.

Figure 6 The Middle East has produced another interesting recording system, dating to the period between the ninth and second millennia B.C. B.C. In places ranging from Anatolia in the north to Sudan in the south, archaeologists have discovered h.o.a.rds of little toylike objects of different shapes made of clay. They are in the form of disks, cones, cylinders, pyramids, animal shapes, and others. University of Texas at Austin archaeologist Denise Schmandt-Besserat, who studied these objects in the late 1970s, developed a fascinating theory. She believes that these clay objects served as pictogram tokens in the market, symbolizing the types of objects being counted. Thus, a small clay sphere might have stood for some quant.i.ty of grain, a cylinder for a head of cattle, and so on. The mideastern prehistoric merchants could therefore, according to Schmandt-Besserat's hypothesis, conduct the accounting of their business by simply lining up the tokens according to the types of goods being transacted. In places ranging from Anatolia in the north to Sudan in the south, archaeologists have discovered h.o.a.rds of little toylike objects of different shapes made of clay. They are in the form of disks, cones, cylinders, pyramids, animal shapes, and others. University of Texas at Austin archaeologist Denise Schmandt-Besserat, who studied these objects in the late 1970s, developed a fascinating theory. She believes that these clay objects served as pictogram tokens in the market, symbolizing the types of objects being counted. Thus, a small clay sphere might have stood for some quant.i.ty of grain, a cylinder for a head of cattle, and so on. The mideastern prehistoric merchants could therefore, according to Schmandt-Besserat's hypothesis, conduct the accounting of their business by simply lining up the tokens according to the types of goods being transacted.

Whatever type of symbols was used for different numbers-incisions on bones, clay tokens, knots on strings (devices called quipu, used by the Inca), or simply the fingers-at some point in history humans faced the challenge of being able to represent and manipulate large numbers. For practical reasons, no symbolic system that has a uniquely different name or different representing object for every number can survive for long. In the same way that the letters in the alphabet represent in some sense the minimal number of characters with which we can express our entire vocabulary and all written knowledge, a minimal set of symbols with which all the numbers can be characterized had to be adopted. This necessity led to the concept of a "base" set-the notion that numbers can be arranged hierarchically, according to certain units. We are so familiar in everyday life with base 10 that it is almost difficult to imagine that other bases could have been chosen.

The idea behind base 10 is really quite simple, which does not mean it did not take a long time to develop. We group our numbers in such a way that ten units at a given level correspond to one unit at a higher level in the hierarchy. Thus 10 "ones" correspond to 1 "ten," 10 "tens" correspond to 1 "hundred," 10 "hundreds" correspond to 1 "thousand," and so on. The names for the numbers and the positioning of the digits also reflect this hierarchical grouping. When we write the number 555, for example, although we repeat the same cipher three times, it means something different each time. The first digit from the right represents 5 units, the second represents 5 tens, or 5 times ten, and the third 5 hundreds, or 5 times ten squared. This important rule of position, the place-value system place-value system, was first invented by the Babylonians (who used 60 as their base, as described below) around the second millennium B.C. B.C., and then, over a period of some 2,500 years, was reinvented, in succession, in China, by the Maya in Central America, and in India.

Of all Indo-European languages, Sanskrit, originating in northern India, provides some of the earliest written texts. In particular, four of the ancient scriptures of Hinduism, all having the Sanskrit word "veda" (knowledge) in their t.i.tle, date to the fifth century B.C. B.C. The numbers 1 to 10 in Sanskrit all have different names: eka, dvau, trayas, catvaras, panca, sat, sapta, astau, nava, dasa. The numbers 11 to 19 are all simply a combination of the number of units and 10. Thus, 15 is "panca-dasa," 19 is "nava-dasa," and so on. English, for example, has the equivalent "teen" numbers. In case you wonder, by the way, where "eleven" and "twelve" in English came from, "eleven" derives from "an" (one) and "lif" (left, or remainder) and "twelve" from "two" and "lif" (two left). Namely, these numbers represent "one left" and "two left" after ten. Again as in English, the Sanskrit names for the tens ("twenty," "thirty," etc.) express the unit and plural tens (e.g., 60 is sasti), and all Indo-European languages have a very similar structure in their vocabulary for numbers. So the users of these languages quite clearly adopted the base 10 system. The numbers 1 to 10 in Sanskrit all have different names: eka, dvau, trayas, catvaras, panca, sat, sapta, astau, nava, dasa. The numbers 11 to 19 are all simply a combination of the number of units and 10. Thus, 15 is "panca-dasa," 19 is "nava-dasa," and so on. English, for example, has the equivalent "teen" numbers. In case you wonder, by the way, where "eleven" and "twelve" in English came from, "eleven" derives from "an" (one) and "lif" (left, or remainder) and "twelve" from "two" and "lif" (two left). Namely, these numbers represent "one left" and "two left" after ten. Again as in English, the Sanskrit names for the tens ("twenty," "thirty," etc.) express the unit and plural tens (e.g., 60 is sasti), and all Indo-European languages have a very similar structure in their vocabulary for numbers. So the users of these languages quite clearly adopted the base 10 system.

There is very little doubt that the almost universal popularity of base 10 stems simply from the fact that we happen to have ten fingers. This possibility was already raised by the Greek philosopher Aristotle (384322 B.C. B.C.) when he wondered (in Problemata): Problemata): "Why do all men, barbarians and Greek alike, count up to ten and not up to any other number?" Base 10 really offers no other superiority over, say, base 13. We could even argue theoretically that the fact that 13 is a prime number, divisible only by 1 and itself, gives it an advantage over 10, because most fractions would be irreducible in such a system. While, for example, under base 10 the number 36/100 also can be expressed as 18/50 or 9/25, such multiple representations would not exist under a prime base like 13. Nevertheless, base 10 won, because ten fingers stood out in front of every human's eyes, and they were easy to use. In some Malay-Polynesian languages, the word for "hand," "lima," is actually the same as the word for "five." Does this mean that all the known civilizations chose 10 as their base? Actually, no. "Why do all men, barbarians and Greek alike, count up to ten and not up to any other number?" Base 10 really offers no other superiority over, say, base 13. We could even argue theoretically that the fact that 13 is a prime number, divisible only by 1 and itself, gives it an advantage over 10, because most fractions would be irreducible in such a system. While, for example, under base 10 the number 36/100 also can be expressed as 18/50 or 9/25, such multiple representations would not exist under a prime base like 13. Nevertheless, base 10 won, because ten fingers stood out in front of every human's eyes, and they were easy to use. In some Malay-Polynesian languages, the word for "hand," "lima," is actually the same as the word for "five." Does this mean that all the known civilizations chose 10 as their base? Actually, no.

Of the other bases that have been used by some populations around the world, the most common was base 20, known as the vigesimal base. In this counting system, which was once popular in large portions of Western Europe, the grouping is based on 20 rather than 10. The choice of this system almost certainly comes from combining the fingers with the toes to form a larger base. For the Inuit (Eskimo) people, for example, the number "twenty" is expressed by a phrase with the meaning "a man is complete." A number of modern languages still have traces of a vigesimal base. In French, for example, the number 80 is "quatre-vingts" (meaning "four twenties"), and an archaic form of "six-vingts" ("six twenties") existed as well. An even more extreme example is provided by a thirteenth-century hospital in Paris, which is still called L'opital de Quinze-Vingts (The Hospital of Fifteen Twenties), because it was originally designed to contain 300 beds for blind veterans. Similarly, in Irish, 40 is called "daichead," which is derived from "da fiche" (meaning "two times twenty"); in Danish, the numbers 60 and 80 ("tresindstyve" and "firsindstyve" respectively, shortened to "tres" and "firs") are literally "three twenties" and "four twenties."

Probably the most perplexing base found in antiquity, or at any other time for that matter, is base 60-the s.e.xagesimal system. This was the system used by the Sumerians in Mesopotamia, and even though its origins date back to the fourth millennium B.C. B.C., this division survived to the present day in the way we represent time in hours, minutes, and seconds as well as in the of the circle (and the subdivision of into minutes and seconds). Sixty as a base for a number system taxes the memory considerably, since such a system requires, in principle, a unique name or symbol for all the numbers from 1 to 60. Aware of this difficulty, the ancient Sumerians used a certain trick to make the numbers easier to remember-they inserted 10 as an intermediate step. The introduction of 10 allowed them to have unique names for the numbers 1 to 10; the numbers 10 to 60 (in units of 10) represented combinations of names. For example, the Sumerian word for 40, "nimin," is a combination of the word for 20, "ni," and the word for 2, "min." If we write the number 555 in a purely s.e.xagesimal system, what we mean is 5 (60)2+5 (60)+5, or 18,305 in our base 10 notation.

Many speculations have been advanced as to the logic or circ.u.mstances that led the Sumerians to choose the unusual base of 60. Some are based on the special mathematical properties of the number 60: It is the first number that is divisible by 1, 2, 3, 4, 5, and 6. Other hypotheses attempt to relate 60 to concepts such as the number of months in a year or days in a year (rounded to 360), combined somehow with the numbers 5 or 6. Most recently, French math teacher and author Georges Ifrah argued in his superb 2000 book, The Universal History of Numbers The Universal History of Numbers, that the number 60 may have been the consequence of the mingling of two immigrant populations, one of which used base 5 and the other base 12. Base 5 clearly originated from the number of fingers on one hand, and traces for such a system can still be found in a few languages, such as in the Khmer in Cambodia and more prominently in the Saraveca in South America. Base 12, for which we find many vestiges even today-for example, in the British system of weights and measures-may have had its origins in the number of joints in the four fingers (excluding the thumb; the latter being used for the counting).

Incidentally, strange bases pop up in the most curious places. In Lewis Carroll's Alice's Adventures in Wonderland Alice's Adventures in Wonderland, to a.s.sure herself that she understands the strange occurrences around her, Alice says: "I'll try if I know all the things I used to know. Let me see: four times five is twelve, and four times six is thirteen, and four times seven is-oh dear! I shall never get to twenty at that rate!" In his notes to Carroll's book, The Annotated Alice The Annotated Alice, the famous mathematical recreation writer Martin Gardner provides a nice explanation for Alice's bizarre multiplication table. He proposes that Alice is simply using bases other than 10. For example, if we use base 18, then 45 = 20 will indeed be written as 12, because 20 is 1 unit of 18 and 2 units of 1. What lends plausibility to this explanation is of course the fact that Charles Dodgson ("Lewis Carroll" was his pen name) lectured on mathematics at Oxford.

OUR NUMBERS, OUR G.o.dS.

Irrespective of the base that any of the ancient civilizations chose, the first group of numbers to be appreciated and understood at some level was the group of whole numbers (or natural natural numbers). These are the familiar 1, 2, 3, 4,... Once humans absorbed the comprehension of these numbers as abstract quant.i.ties into their consciousness, it did not take them long to start to attribute special properties to numbers. From Greece to India, numbers were accredited with secret qualities and powers. Some ancient Indian texts claim that numbers are almost divine, or "Brahma-natured." These ma.n.u.scripts contain phrases that are nothing short of worship to numbers (like "hail to one"). Similarly, a famous dictum of the Greek mathematician Pythagoras (whose life and work will be described later in this chapter) suggests that "everything is arranged according to number." These sentiments led on one hand to important developments in number theory but, on the other, to the development of numbers). These are the familiar 1, 2, 3, 4,... Once humans absorbed the comprehension of these numbers as abstract quant.i.ties into their consciousness, it did not take them long to start to attribute special properties to numbers. From Greece to India, numbers were accredited with secret qualities and powers. Some ancient Indian texts claim that numbers are almost divine, or "Brahma-natured." These ma.n.u.scripts contain phrases that are nothing short of worship to numbers (like "hail to one"). Similarly, a famous dictum of the Greek mathematician Pythagoras (whose life and work will be described later in this chapter) suggests that "everything is arranged according to number." These sentiments led on one hand to important developments in number theory but, on the other, to the development of numerology- numerology-the set of doctrines according to which all aspects of the universe are a.s.sociated with numbers and their idiosyncrasies. To the numerologist, numbers were fundamental realities, drawing symbolic meanings from the relation between the heavens and human activities. Furthermore, essentially no number that was mentioned in the holy writings was ever treated as irrelevant. Some forms of numerology affected entire nations. For example, in the year 1240 Christians and Jews in Western Europe expected the arrival of some messianic king from the East, because it so happened that the year 1240 in the Christian calendar corresponded to the year 5000 in the Jewish calendar. Before we dismiss these sentiments as romantic naivete that could have happened only many centuries ago, we should recall the extravagant hoopla surrounding the ending of the last millennium.

One special version of numerology is the Jewish Gematria (possibly based on "geometrical number" in Greek), or its Muslim and Greek a.n.a.logues, known as Khisab al Jumal ("calculating the total"), and Isopsephy (from the Greek "isos," equal, and "psephizein," to count), respectively. In these systems, numbers are a.s.signed to each letter of the alphabet of a language (usually Hebrew, Greek, Arabic, or Latin). By adding together the values of the const.i.tuent letters, numbers are then a.s.sociated with words or even entire phrases. Gematria was especially popular in the system of Jewish mysticism practiced mainly from the thirteenth to the eighteenth century known as cabala. Hebrew scholars sometimes used to amaze listeners by calling out a series of apparently random numbers for some ten minutes and then repeating the series without an error. This feat was accomplished simply by translating some pa.s.sage of the Hebrew scriptures into the language of Gematria.

One of the most famous examples of numerology is a.s.sociated with 666, the "number of the Beast." The "Beast" has been identified as the Antichrist. The text in the Book of Revelations (13:18) reads: "This calls for wisdom: let anyone with understanding calculate the number of the beast, for it is the number of a man. Its number is six hundred and sixty-six." The phrase "it is the number of a man" prompted many of the Christian mystics to attempt to identify historical figures whose names in Gematria or Isopsephy give the value 666. 666. These searches led to, among others, names like those of Nero Caesar and the emperor Diocletian, both of whom persecuted Christians. In Hebrew, Nero Caesar was written as (from right to left): xqw oexp, and the numerical values a.s.signed in Gematria to the Hebrew letters (from right to left)-50, 200, 6, 50; 100, 60, 200-add up to These searches led to, among others, names like those of Nero Caesar and the emperor Diocletian, both of whom persecuted Christians. In Hebrew, Nero Caesar was written as (from right to left): xqw oexp, and the numerical values a.s.signed in Gematria to the Hebrew letters (from right to left)-50, 200, 6, 50; 100, 60, 200-add up to 666. 666. Similarly, when only the letters that are also Roman numerals (D, I, C, L, V) are counted in the Latin name of Emperor Diocletian, DIOCLES AVGVSTVS, they also add up to Similarly, when only the letters that are also Roman numerals (D, I, C, L, V) are counted in the Latin name of Emperor Diocletian, DIOCLES AVGVSTVS, they also add up to 666 666 (500+ 1+ 100+ 50+ 5+ 5+ 5). Clearly, all of these a.s.sociations are not only fanciful but also rather contrived (e.g., the spelling in Hebrew of the word Caesar actually omits a letter, of value 10, from the more common spelling). (500+ 1+ 100+ 50+ 5+ 5+ 5). Clearly, all of these a.s.sociations are not only fanciful but also rather contrived (e.g., the spelling in Hebrew of the word Caesar actually omits a letter, of value 10, from the more common spelling).

Amusingly, in 1994, a relation was "discovered" (and appeared in the Journal of Recreational Mathematics) the Journal of Recreational Mathematics) even between the "number of the Beast" and the Golden Ratio. With a scientific pocket calculator, you can use the trigonometric functions sine and cosine to calculate the value of the expression [sin even between the "number of the Beast" and the Golden Ratio. With a scientific pocket calculator, you can use the trigonometric functions sine and cosine to calculate the value of the expression [sin 666 666+ cos (6 6 (6 6 6)]. Simply enter 6)]. Simply enter 666 666 and hit the [sin] b.u.t.ton and save that number, then enter 216 (= 6 6 6) and hit the [cos] b.u.t.ton, and add the number you get to the number you saved. The number you will obtain is a good approximation of the negative of phi. Incidentally, President Ronald Reagan and Nancy Reagan changed their address in California from and hit the [sin] b.u.t.ton and save that number, then enter 216 (= 6 6 6) and hit the [cos] b.u.t.ton, and add the number you get to the number you saved. The number you will obtain is a good approximation of the negative of phi. Incidentally, President Ronald Reagan and Nancy Reagan changed their address in California from 666 666 St. Cloud Road to 668 to avoid the number St. Cloud Road to 668 to avoid the number 666 666, and 666 666 was also the combination to the mysterious briefcase in Quentin Tarantino's movie was also the combination to the mysterious briefcase in Quentin Tarantino's movie Pulp Fiction. Pulp Fiction.

One clear source of the mystical att.i.tude toward whole numbers was the manifestation of such numbers in human and animal bodies and in the cosmos, as perceived by the early cultures. Not only do humans have the number 2 exhibited all over their bodies (eyes, hands, nostrils, feet, ears, etc.), but there are also two genders, there is the Sun-Moon system, and so on. Furthermore, our subjective time is divided into three tenses (past, present, future), and, due to the fact that Earth's rotation axis remains more or less pointed in the same direction (roughly toward the North Star, Polaris, although small variations do exist, as described in Chapter 3), the year is divided into four seasons. The seasons simply reflect the fact that the orientation of Earth's axis relative to the Sun changes over the course of the year. The more directly a part of the Earth is exposed to sunlight, the longer the daylight hours and the warmer the temperature. In general, numbers acted in many circ.u.mstances as the mediators between cosmic phenomena and human everyday life. For example, the names of the seven days of the week were based on the names of the celestial objects originally considered to be planets: the Sun, the Moon, Mars, Mercury, Jupiter, Venus, and Saturn.

The whole numbers themselves are divided into odd and even, and n.o.body did more to emphasize the differences between the odd and even numbers, and to ascribe a whole menagerie of properties to these differences, than the Pythagoreans. In particular, we shall see that we can identify the Pythagorean fascination with the number 5 and their admiration for the five-pointed star as providing the initial impetus for the interest in the Golden Ratio.

PYTHAGORAS AND THE PYTHAGOREANS.

Pythagoras was born around 570 B.C. B.C. in the island of Samos in the Aegean Sea (off Asia Minor), and he emigrated sometime between 530 and 510 to Croton in the Dorian colony in southern Italy (then known as Magna Graecia). Pythagoras apparently left Samos to escape the stifling tyranny of Polycrates (died ca. 522 in the island of Samos in the Aegean Sea (off Asia Minor), and he emigrated sometime between 530 and 510 to Croton in the Dorian colony in southern Italy (then known as Magna Graecia). Pythagoras apparently left Samos to escape the stifling tyranny of Polycrates (died ca. 522 B.C. B.C.), who established Samian naval supremacy in the Aegean Sea. Perhaps following the advice of his presumed teacher, the mathematician Thales of Miletus, Pythagoras probably lived for some time (as long as twenty-two years, according to some accounts) in Egypt, where he would have learned mathematics, philosophy, and religious themes from the Egyptian priests. After Egypt was overwhelmed by Persian armies, Pythagoras may have been taken to Babylon, together with members of the Egyptian priesthood. There he would have encountered the Mesopotamian mathematical lore. Nevertheless, the Egyptian and Babylonian mathematics would prove insufficient for Pythagoras' inquisitive mind. To both of these peoples, mathematics provided practical tools in the form of "recipes" designed for specific calculations. Pythagoras, on the other hand, was one of the first to grasp numbers as abstract ent.i.ties that exist in their own right.

In Italy, Pythagoras began to lecture on philosophy and mathematics, quickly establishing an enth