Paul Larson of Temple University claimed in 1978 that he discovered the Golden Ratio in the earliest notated western music-the "Kyrie" chants from the collection of Gregorian chants known as Liber Usualis. The thirty Kyrie chants in the collection span a period of more than six hundred years, starting from the tenth century. Larson stated that he found a significant "event" (e.g., the beginning or ending of a musical phrase) at the Golden Ratio separation of 105 of the 146 sections of the Kyries he had a.n.a.lyzed. However, in the absence of any supporting historical justification or convincing rationale for the use of the Golden Ratio in these chants, I am afraid that this is no more than another exercise in number juggling.

In general, counting notes and pulses often reveals various numerical correlations between different sections of a musical work, and the a.n.a.lyst faces an understandable temptation to conclude that the composer introduced the numerical relationships. Yet, without a firmly doc.u.mented basis (which is lacking in many cases), such a.s.sertions remain dubious.

In 1995, mathematician John F. Putz of Alma College in Michigan examined the question of whether Mozart (17561791) had used the Golden Ratio in the twenty-nine movements from his piano sonatas that consist of two distinct sections. Generally, these sonatas consist of two parts: the Exposition, in which the musical theme is first introduced, and the Development and Recapitulation, in which the main theme is further developed and revisited. Since musical pieces are divided into equal units of time called measures measures (or (or bars) bars), Putz examined the ratios of the numbers of measures in the two sections of the sonatas. Mozart, who "talked of nothing, thought of nothing but figures" during his school days (according to his sister's testimony), is probably one of the better candidates for the use of mathematics in his compositions. In fact, several previous articles had claimed that Mozart's piano sonatas do reflect the Golden Ratio. Putz's first results appeared to be very promising. In the Sonata No. 1 in C Major Sonata No. 1 in C Major, for example, the first movement consists of sixty-two measures in the Development and Recapitulation and thirty-eight in the Exposition. The ratio 62/38 = 1.63 is quite close to the Golden Ratio. However, a thorough examination of all the data basically convinced Putz that Mozart did not not use the Golden Ratio in his sonatas, nor is it clear why the simple matter of measures would give a pleasing effect. It therefore appears that while many believe that Mozart's music is truly "divine," the "Divine Proportion" is not a part of it. use the Golden Ratio in his sonatas, nor is it clear why the simple matter of measures would give a pleasing effect. It therefore appears that while many believe that Mozart's music is truly "divine," the "Divine Proportion" is not a part of it.

A famous composer who might have used the Golden Ratio quite extensively was the Hungarian Bela Bartok (18811945). A virtuoso pianist and folklorist, Bartok blended elements from other composers that he admired (including Strauss, Liszt, and Debussy) with folk music, to create his highly personal music. He once said that "the melodic world of my string quartets does not differ essentially from that of folk songs." The rhythmical vitality of his music, combined with a well-calculated formal symmetry, united to make him one of the most original twentieth-century composers.

The Hungarian musicologist Erno Lendvai investigated Bartok's music painstakingly and published many books and articles on the subject. Lendvai testifies that "from stylistic a.n.a.lyses of Bartok's music I have been able to conclude that the chief feature of his chromatic technique is obedience to the laws of Golden Section in every movement."

According to Lendvai, Bartok's management of the rhythm of the composition provides an excellent example of his use of the Golden Ratio. By a.n.a.lyzing the fugue movement of Bartok's Music for Strings, Percussion and Celesta Music for Strings, Percussion and Celesta, for example, Lendvai shows that the eighty-nine measures of the movement are divided into two parts, one with fifty-five measures and the other with thirty-four measures, by the pyramid peak (in terms of loudness) of the movement. Further divisions are marked by the placement and removal of the sordini (the mutes for the instruments) and by other textural changes (Figure 87). All the numbers of measures are Fibonacci numbers, with the ratios between major parts (e.g., 55/34) being close to the Golden Ratio. Similarly, in Sonata for Two Pianos and Percussion Sonata for Two Pianos and Percussion, the various themes develop in Fibonacci/Golden Ratio order in terms of the numbers of semitones (Figure 88).

Figure 87

Figure 88 Some musicologists do not accept Lendvai's a.n.a.lyses. Lendvai himself admits that Bartok said nothing or very little about his own compositions, stating: "Let my music speak for itself; I lay no claim to any explanation of my works." The fact that Bartok did not leave any sketches to indicate that he derived rhythms or scales numerically makes any a.n.a.lysis suggestive at best. Also, Lendvai actually dodges the question of whether Bartok used the Golden Ratio consciously. Hungarian musicologist Laszlo Somfai totally discounts the notion that Bartok used the Golden Ratio, in his 1996 book Bela Bartok: Composition, Concepts and Autograph Sources. Bela Bartok: Composition, Concepts and Autograph Sources. On the basis of a thorough a.n.a.lysis (which took three decades) of some 3,600 pages, Somfai concludes that Bartok composed without any preconceived musical theories. Other musicologists, including Ruth Tatlow and Paul Griffiths, also refer to Lendvai's study as "dubious." On the basis of a thorough a.n.a.lysis (which took three decades) of some 3,600 pages, Somfai concludes that Bartok composed without any preconceived musical theories. Other musicologists, including Ruth Tatlow and Paul Griffiths, also refer to Lendvai's study as "dubious."

Figure 89 In the interesting book Debussy in Proportion Debussy in Proportion, Roy Howat of Cambridge University argues that the French composer Claude Debussy (18621918), whose harmonic innovations had a profound influence on generations of composers, used the Golden Ratio in many of his compositions. For example, in the solo piano piece Reflets dans l'eau Reflets dans l'eau (Reflections in the water), a part of the series (Reflections in the water), a part of the series Images Images, the first rondo reprise occurs after bar 34, which is at the Golden Ratio point between the beginning of the piece and the onset of the climactic section after bar 55. Both 34 and 55 are, of course, Fibonacci numbers, and the ratio 34/21 is a good approximation for the Golden Ratio. The same structure is mirrored in the second part, which is divided in a 24/15 ratio (equal to the ratio of the two Fibonacci numbers 8/5, again close to the Golden Ratio; Figure 89 Figure 89). Howat finds similar divisions in the three symphonic sketches La Mer La Mer (The sea), in the piano (The sea), in the piano piece Jardins sous la Pluie piece Jardins sous la Pluie (Gardens under the rain), and other works. (Gardens under the rain), and other works.

I must admit that given the history of La Mer La Mer, I find it somewhat difficult to believe that Debussy used any mathematical design in the composition of this particular piece. He started La Mer La Mer in 1903, and in a letter he wrote to his friend Andre Messager he says: "You may not know that I was destined for a sailor's life and that it was only quite by chance that fate led me in another direction. But I have always retained a pa.s.sionate love for her [the sea]." By the time the composition in 1903, and in a letter he wrote to his friend Andre Messager he says: "You may not know that I was destined for a sailor's life and that it was only quite by chance that fate led me in another direction. But I have always retained a pa.s.sionate love for her [the sea]." By the time the composition of La Mer of La Mer was finished, in 1905, Debussy's whole life had been literally turned upside down. He had left his first wife, "Lily" (real name Rosalie Texier), for the alluring Emma Bardac; Lily attempted suicide; and both she and Bardac brought court actions against the composer. If you listen to was finished, in 1905, Debussy's whole life had been literally turned upside down. He had left his first wife, "Lily" (real name Rosalie Texier), for the alluring Emma Bardac; Lily attempted suicide; and both she and Bardac brought court actions against the composer. If you listen to La Mer- La Mer-perhaps Debussy's most personal and pa.s.sionate work-you can literally hear not only a musical portrait of the sea, probably inspired by the work of the English painter Joseph Mallord William Turner, but also an expression of the tumultuous period in the composer's life.

Since Debussy didn't say much about his compositional technique, we must maintain a clear distinction between what may be a forced interpretation imposed on the composition and the composer's real and conscious intention (which remains unknown). To support his a.n.a.lysis, Howat relies primarily on two pieces of circ.u.mstantial evidence: Debussy's close a.s.sociation with some of the symbolist painters who are known to have been interested in the Golden Ratio, and a letter Debussy wrote in August 1903 to his publisher, Jacque Durand. In that letter, which accompanied the corrected proofs of Jardins sous la Pluie of Jardins sous la Pluie, Debussy talks about a bar missing in the composition and explains: "However, it's necessary, as regards number; the divine number." The implication here is that not only was Debussy constructing his harmonic structure with numbers in general but that the "divine number" (a.s.sumed to refer to the Golden Ratio) played an important role.

Howat also suggests that Debussy was influenced by the writings of the mathematician and art critic Charles Henry, who had great interest in the numerical relationships inherent in melody, harmony, and rhythm. Henry's publications on aesthetics, such as the Introduction a une esthetique scientifique Introduction a une esthetique scientifique (Introduction to a scientific aesthetic; 1885), gave a prominent role to the Golden Ratio. (Introduction to a scientific aesthetic; 1885), gave a prominent role to the Golden Ratio.

We shall probably never know with certainty whether this great pillar of French modernism truly intended to use the Golden Ratio to control formal proportions. One of his very few piano students, Mademoiselle Worms de Romilly, wrote once that he "always regretted not having worked at painting instead of music." Debussy's highly original musical aesthetic may have been aided, to a small degree, by the application of the Golden Ratio, but this was certainly not the main source of his creativity.

Just as a curiosity, the names of Debussy and Bartok are related through an amusing anecdote. During a visit of the young Hungarian composer to Paris, the great piano teacher Isidore Philipp offered to introduce Bartok to the composer Camille Saint-Saens, at the time a great celebrity. Bartok declined. Philipp then offered him to meet with the great organist and composer Charles-Marie Widor. Again Bartok declined. "Well," said Philipp, "if you won't meet Saint-Saens and Widor, who is there that you would like to know?" "Debussy," replied Bartok. "But he is a horrid man," said Philipp. "He hates everybody and will certainly be rude to you. Do you want to be insulted by Debussy?" "Yes," Bartok replied with no hesitation.

The introduction of recording technologies and computer music in the twentieth century accelerated precise numerical measurements and thereby encouraged number-based music. The Austrian composer Al-ban Berg (18851935), for example, constructed his Kammerkonzert entirely around the number 3: There are units of thirty bars, on three themes, with three basic "colors" (piano, violin, wind). The French composer Olivier Messiaen (19081992), who was largely driven by a deep Catholic faith and a love for nature, also used numbers consciously (e.g., to determine the number of movements) in rhythmic constructions. Nevertheless, when asked specifically in 1978 about the Golden Ratio, he disclaimed use of it.

The colorful composer, mathematician, and teacher Joseph Schillinger (18951943) exemplified by his own personality and teachings the Platonic view of the relationship between mathematics and music. After studying at the St. Petersburg Conservatory and teaching and composing at the Kharkov and Leningrad State academies, he settled in the United States in 1928, where he became a professor of both mathematics and music at various inst.i.tutions, including Columbia University and New York University. The famous composer and pianist George Gershwin, the clarinetist and bandleader Benny Goodman, and the dance-band leader Glenn Miller were all among Schillinger's students. Schillinger was a great believer in the mathematical basis for music, and he developed a System of Musical Composition. In particular, in some pieces, successive notes in the melody followed Fibonacci intervals when counted in units of half-steps (Figure 90). To Schillinger, these Fibonacci leaps of the notes conveyed the same sense of harmony as the phyllotactic ratios of the leaves on a stem convey to the botanist. Schillinger found "music" in the most unusual places. In Joseph Schillinger: A Memoir Joseph Schillinger: A Memoir, the biographical book written by his widow Frances, the author tells the story of a party riding in a car during a rain shower. Schillinger noted: "The splashing rain has its rhythm and the windshield wipers their rhythmic pattern. That's unconscious art." One of Schillinger's attempts to demonstrate that music can be based entirely on mathematical formulation was particularly amusing. He basically copied the fluctuations of a stock market curve as they appeared in the New York Times New York Times on graph paper and, by translating the ups and downs into proportional musical intervals, showed that he could obtain a composition somewhat similar to those of the great Johann Sebastian Bach. on graph paper and, by translating the ups and downs into proportional musical intervals, showed that he could obtain a composition somewhat similar to those of the great Johann Sebastian Bach.

Figure 90 The conclusion from this brief tour of the world of music is that claims about certain composers having used the Golden Ratio in their music usually leap too swiftly from numbers generated by simple counting (of bars, notes, etc.) to interpretation. Nevertheless, there is no doubt that the twentieth century in particular produced a renewed interest in the use of numbers in music. As a part of this Pythagorean revival, the Golden Ratio also started to feature more prominently in the works of several composers.

The Viennese music critic Eduard Hanslick (18251904) expressed the relationship between music and mathematics magnificently in the book The Beautiful in Music: The Beautiful in Music: The "music" of nature and the music of man belong to two distinct categories. The translation from the former to the latter pa.s.ses through the science of mathematics. An important and pregnant proposition. Still, we should be wrong were we to construe it in the sense that man framed his musical system according to calculations purposely made, the system having arisen through the unconscious application of pre-existent conceptions of quant.i.ty and proportion, through subtle processes of measuring and counting; but the laws by which the latter are governed were demonstrated only subsequently by science.

PYTHAGORAS PLANNED IT.

With the words in the heading, the famous Irish poet William Butler Yeats (18651939) starts his poem "The Statues." Yeats, who once stated that "the very essence of genius, of whatever kind, is precision," examines in the poem the relation between numbers and pa.s.sion. The first stanza of the poem goes like this: Pythagoras planned it. Why did the people stare?

His numbers, though they moved or seemed to move In marble or in bronze, lacked character But boys and girls, pale from the imagined loveOf solitary beds, knew what they were, That pa.s.sion could bring character enough, And pressed at midnight in some public place Live lips upon a plummet-measured face.

Yeats emphasizes beautifully the fact that while the calculated proportions of Greek sculptures may seem cold to some, the young and pa.s.sionate regarded these forms as the embodiment of the objects of their love.

At first glance, nothing seems more remote from mathematics than poetry. We think that the blossoming of a poem out of the poet's sheer imagination should be as boundless as the blossoming of a red rose. Yet recall that the growth of the rose's petals actually occurs in a well-orchestrated pattern based on the Golden Ratio. Could poetry be constructed on this basis, as well?

There are at least two ways, in principle, in which the Golden Ratio and Fibonacci numbers could be linked to poetry. First, there can be poems about the Golden Ratio or the Fibonacci numbers themselves (e.g., "Constantly Mean" by Paul Bruckman; presented in Chapter 4) or about geometrical shapes or phenomena that are closely related to the Golden Ratio. Second, there can be poems in which the Golden Ratio or Fibonacci numbers are somehow utilized in constructing the form, pattern, or rhythm.

Examples of the first type are provided by a humorous poem by J. A. Lindon, by Johann Wolfgang von Goethe's dramatic poem "Faust," and by Oliver Wendell Holmes s poem "The Chambered Nautilus."

Martin Gardner used Lindon's short poem to open the chapter on Fibonacci in his book Mathematical Circus. Mathematical Circus. Referring to the recursive relation which defines the Fibonacci sequence, the poem reads: Referring to the recursive relation which defines the Fibonacci sequence, the poem reads: Each wife of Fibonacci, Eating nothing that wasn't starchy, Weighed as much as the two before her, His fifth was some signora!

Similarly, two lines from a poem by Katherine O'Brien read: Fibonacci couldn't sleep- Counted rabbits instead of sheep.

The German poet and dramatist Goethe (17431832) was certainly one of the greatest masters of world literature. His all-embracing genius is epitomized in Faust- Faust-a symbolic description of the human striving for knowledge and power. Faust, a learned German doctor, sells his soul to the devil (personified by Mephistopheles) in exchange for knowledge, youth, and magical power. When Mephistopheles finds that the pentagram's "Druidenfuss" ("Celtic wizard's foot") is drawn on Faust's threshold, he cannot get out. The magical powers attributed to the pentagram since the Pythagoreans (and which led to the definition of the Golden Ratio) gained additional symbolic meaning in Christianity, since the five vertices were a.s.sumed to stand for the letters in the name Jesus. As such, the pentagram was taken to be a source of fear for the devil. The text reads:

Mephistopheles therefore uses trickery-the fact that the pentagram had a small opening in it-to get by. Clearly, Goethe had no intention of referring to the mathematical concept of the Golden Ratio in Faust Faust, and he included the pentagram only for its symbolic qualities. Goethe expressed elsewhere his opinion on mathematics thus: "The mathematicians are a sort of Frenchmen: when you talk to them, they immediately translate it into their own language, and right away it is something entirely different."

The American physician and author Oliver Wendell Holmes (18091894) published a few collections of witty and charming poems. In "The Chambered Nautilus" he finds a moral in the self-similar growth of the logarithmic spiral that characterizes the mollusk's sh.e.l.l: Build thee more stately mansions, O my soul, As the swift seasons roll! Leave thy low-vaulted past!

Let each new temple, n.o.bler than the last, Shut thee from heaven with a dome more vast, Till thou at length art free, Leaving thine outgrown sh.e.l.l by life's unresting sea.

There are many examples of numerically based poetic structures. For example, the Divine Comedy Divine Comedy, the colossal literary cla.s.sic by the Italian poet Dante Alighieri (12651321), is divided into three parts, written in units of three lines, and each of the parts has thirty-three cantos (except for the first, which has thirty-four cantos, to bring the total to an even one hundred).

Poetry is probably the place in which Fibonacci numbers made their first appearance, even before Fibonacci's rabbits. One of the categories of meters in Sanskrit and Prakit poetry is known as matra-vrttas. These are meters in which the number of morae (ordinary short syllables) remains constant and the number of letters is arbitrary. In 1985, mathematician Parmanand Singh of Raj Narain College, India, pointed out that Fibonacci numbers and the relation that defines them appeared in the writings of three Indian authorities on matra-vrttas before A.D. A.D. 1202, the year in which Fibonacci's book was published. The first of these authors on metric was Acarya Virahanka, who lived sometime between the sixth and eighth centuries. Although the rule he gives is somewhat vague, he does mention mixing the variations of two earlier meters to obtain the next one, just as each Fibonacci number is the sum of the two preceding ones. The second author, Gopala, gives the rule specifically in a ma.n.u.script written between 1133 and 1135. He explains that each meter is the sum of the two earlier meters and calculates the series of meters 1, 2, 3, 5, 8, 13, 21..., which is precisely the Fibonacci sequence. Finally, the great Jain writer Acarya Hemacandra, who lived in the twelfth century and enjoyed the patronage of two kings, also stated clearly in a ma.n.u.script written around 1150 that "sum of the last and the last but one numbers [of variations] is [that] of the matra-vrtta coming next." However, these early poetic appearances of Fibonacci numbers went apparently unnoticed by mathematicians. 1202, the year in which Fibonacci's book was published. The first of these authors on metric was Acarya Virahanka, who lived sometime between the sixth and eighth centuries. Although the rule he gives is somewhat vague, he does mention mixing the variations of two earlier meters to obtain the next one, just as each Fibonacci number is the sum of the two preceding ones. The second author, Gopala, gives the rule specifically in a ma.n.u.script written between 1133 and 1135. He explains that each meter is the sum of the two earlier meters and calculates the series of meters 1, 2, 3, 5, 8, 13, 21..., which is precisely the Fibonacci sequence. Finally, the great Jain writer Acarya Hemacandra, who lived in the twelfth century and enjoyed the patronage of two kings, also stated clearly in a ma.n.u.script written around 1150 that "sum of the last and the last but one numbers [of variations] is [that] of the matra-vrtta coming next." However, these early poetic appearances of Fibonacci numbers went apparently unnoticed by mathematicians.

In her educational book Fascinating Fibonaccis Fascinating Fibonaccis, author Trudi Ham-mel Garland gives an example of a limerick in which the number of lines (5), the number of beats in each line (2 or 3), and the total number of beats (13) are all Fibonacci numbers.

A fly and a flea in a flue (3 beats) Were imprisoned, so what could they do? (3 beats) Said the fly, "Let us flee!" (2 beats) "Let us fly!" said the flea, (2 beats) So they fled through a flaw in the flue. (3 beats) We should not take the appearance of very few Fibonacci numbers as evidence that the poet necessarily had these numbers or the Golden Ratio in mind when constructing the structural pattern of the poem. Like music, poetry is, and especially was, often intended to be heard, not just read. Consequently, proportion and harmony that appeal to the ear are an important structural element. This does not mean, however, that the Golden Ratio or Fibonacci numbers are the only options in the poet's a.r.s.enal.

George Eckel Duckworth, a professor of cla.s.sics at Princeton University, made the most dramatic claim about the appearance of the Golden Ratio in poetry. In his 1962 book Structural Patterns and Proportions in Vergil's Aeneid Structural Patterns and Proportions in Vergil's Aeneid, Duckworth states that "Vergil composed the Aeneid Aeneid on the basis of mathematical proportion; each book reveals, in small units as well as in the main divisions, the famous numerical ratio known variously as the Golden Section, the Divine Proportion, or the Golden Mean ratio." on the basis of mathematical proportion; each book reveals, in small units as well as in the main divisions, the famous numerical ratio known variously as the Golden Section, the Divine Proportion, or the Golden Mean ratio."

The Roman poet Vergil (70 B.C. B.C.-19 B.C. B.C.) grew up on a farm, and many of his early pastoral poems deal with the charm of rural life. His national epic the Aeneid Aeneid, which details the adventures of the Trojan hero Aeneas, is considered one of the greatest poetic works in history. In twelve books, Vergil follows Aeneas from his escape from Troy to Carthage, through his love affair with Dido, to the establishment of the Roman state. Vergil makes Aeneas the paragon of piety, devotion to family, and loyalty to state.

Duckworth made detailed measurements of the lengths of pa.s.sages in the Aeneid Aeneid and computed the ratios of these lengths. Specifically, he measured the number of lines in pa.s.sages characterized as major (and denoted that number by and computed the ratios of these lengths. Specifically, he measured the number of lines in pa.s.sages characterized as major (and denoted that number by m m) and minor (and denoted the number by m) m), and calculated the ratios of these numbers. The identification of major and minor parts was based on content. For example, in many pa.s.sages the major or minor part is a speech and the other part (minor or major respectively) is a narrative or a description. From this a.n.a.lysis Duckworth concluded that the Aeneid Aeneid contains "hundreds of Golden Mean ratios." He also noted that an earlier a.n.a.lysis (from 1949) of another Vergil work contains "hundreds of Golden Mean ratios." He also noted that an earlier a.n.a.lysis (from 1949) of another Vergil work (Georgius I) (Georgius I) gave for the ratio of the two parts (in terms of numbers of lines), known as "Works" and "Days," a value very close to . gave for the ratio of the two parts (in terms of numbers of lines), known as "Works" and "Days," a value very close to .

Unfortunately, Roger Herz-Fischler has shown that Duckworth's a.n.a.lysis probably is based on a mathematical misunderstanding. Since this oversight is endemic to many of the "discoveries" of the Golden Ratio, I will explain it here briefly.

Suppose you have any pair of positive values m m and and M M, such that M M is larger than is larger than m. m. For example, For example, M = M = 317 could be the number of pages in the last book you read and 317 could be the number of pages in the last book you read and m = m = 160 could be your weight in pounds. We could represent these two numbers on a line (with proportional lengths), as in 160 could be your weight in pounds. We could represent these two numbers on a line (with proportional lengths), as in Figure 91 Figure 91. The ratio of the shorter to the longer part is equal to m/M = m/M = 160/317 = 0.504, while the ratio of the longer part to the whole is 160/317 = 0.504, while the ratio of the longer part to the whole is M/(M M/(M +m) = +m) = 317/477 = 0.665. You will notice that the value of 317/477 = 0.665. You will notice that the value of M/(M M/(M+m) is closer to 1/ = 0.618 than is closer to 1/ = 0.618 than m/M. m/M. We can prove mathematically that this is always the case. (Try it with the actual number of pages in your last book and your real weight.) From the definition of the Golden Ratio, we know that when a line is divided in a Golden Ratio, We can prove mathematically that this is always the case. (Try it with the actual number of pages in your last book and your real weight.) From the definition of the Golden Ratio, we know that when a line is divided in a Golden Ratio, m/M = M/(M m/M = M/(M +m) +m) precisely. Consequently, we may be tempted to think that if we examine a series of ratios of numbers, such as the lengths of pa.s.sages, for the potential presence of the Golden Ratio, it does not matter if we look at the ratio of the shorter to the longer or the longer to the whole. What I have just shown is that it definitely does matter. A too-eager Golden Ratio enthusiast wishing to demonstrate a Golden Ratio relationship between the weights of readers and the numbers of pages in the books they read may be able to do so by presenting data in the form precisely. Consequently, we may be tempted to think that if we examine a series of ratios of numbers, such as the lengths of pa.s.sages, for the potential presence of the Golden Ratio, it does not matter if we look at the ratio of the shorter to the longer or the longer to the whole. What I have just shown is that it definitely does matter. A too-eager Golden Ratio enthusiast wishing to demonstrate a Golden Ratio relationship between the weights of readers and the numbers of pages in the books they read may be able to do so by presenting data in the form M/(M M/(M+m), which is biased toward 1/ . This is precisely what happened to Duckworth. By making the unfortunate decision to use only the ratio M/(M M/(M+m) in his a.n.a.lysis, because he thought that this was "slightly more accurate," he compressed and distorted the data and made the a.n.a.lysis statistically invalid. In fact, Leonard A. Curchin of the University of Ottawa and Roger Herz-Fischler repeated in 1981 the a.n.a.lysis with Duckworth's data (but using the ratio in his a.n.a.lysis, because he thought that this was "slightly more accurate," he compressed and distorted the data and made the a.n.a.lysis statistically invalid. In fact, Leonard A. Curchin of the University of Ottawa and Roger Herz-Fischler repeated in 1981 the a.n.a.lysis with Duckworth's data (but using the ratio m/M) m/M) and showed that there is no evidence for the Golden Ratio in the and showed that there is no evidence for the Golden Ratio in the Aeneid. Aeneid. Rather, they concluded that "random scattering is indeed the case with Vergil." Furthermore, Duckworth "endowed" Vergil with the knowledge that the ratio of two consecutive Fibonacci numbers is a good approximation of the Golden Ratio. Curchin and Herz-Fischler, on the other hand, demonstrated convincingly that even Hero of Alexandria, who lived later than Vergil and was one of the distinguished mathematicians of his time, did not know about this relation between the Golden Ratio and Fibonacci numbers. Rather, they concluded that "random scattering is indeed the case with Vergil." Furthermore, Duckworth "endowed" Vergil with the knowledge that the ratio of two consecutive Fibonacci numbers is a good approximation of the Golden Ratio. Curchin and Herz-Fischler, on the other hand, demonstrated convincingly that even Hero of Alexandria, who lived later than Vergil and was one of the distinguished mathematicians of his time, did not know about this relation between the Golden Ratio and Fibonacci numbers.

Figure 91 Sadly, the statement about Vergil and continues to feature in most of the Golden Ratio literature, again demonstrating the power of Golden Numberism.

All the attempts to disclose the (real or false) Golden Ratio in various works of art, pieces of music, or poetry rely on the a.s.sumption that a canon for ideal beauty exists and can be turned to practical account. History has shown, however, that the artists who have produced works of lasting value are precisely those who have broken away from such academic precepts. In spite of the Golden Ratio's importance for many areas of mathematics, the sciences, and natural phenomena, we should, in my humble opinion, give up its application as a fixed standard for aesthetics, either in the human form or as a touchstone for the fine arts.

Understanding is, after all, what science is all about-and science is a great deal more than mere mindless computation.-ROGER P PENROSE (1931) (1931) The tangled tale of the Golden Ratio has taken us from the sixth century B.C. B.C. to contemporary times. Two intertwined trends thread these twenty-six centuries of history. On one hand, the Pythagorean motto "all is number" has materialized spectacularly, in the role that the Golden Ratio plays in natural phenomena ranging from phyllotaxis to the shape of galaxies. On the other, the Pythagorean obsession with the symbolic meaning of the pentagon has metamorphosed into what I believe is the false notion that the Golden Ratio provides a universal canon of ideal beauty. After all of this, you may wonder whether there still is room left for any further exploration of this seemingly simple division of a line. to contemporary times. Two intertwined trends thread these twenty-six centuries of history. On one hand, the Pythagorean motto "all is number" has materialized spectacularly, in the role that the Golden Ratio plays in natural phenomena ranging from phyllotaxis to the shape of galaxies. On the other, the Pythagorean obsession with the symbolic meaning of the pentagon has metamorphosed into what I believe is the false notion that the Golden Ratio provides a universal canon of ideal beauty. After all of this, you may wonder whether there still is room left for any further exploration of this seemingly simple division of a line.

THE TILED ROAD TO QUASI-CRYSTALS.

The Dutch painter Johannes Vermeer (16321675) is best known for his fantastically alluring genre paintings, which typically show one or

Figure 92

Figure 93 two figures engaged in some domestic task. In many of these paintings, a window on the viewer's left softly lights the room, and the way the light reflects off the tiled floor is purely magical. If you examine some of these paintings closely, you will find that quite a few, such as "The Concert," "A Lady Writing a Letter with Her Maid," "Love Letter" (Figure 92; located in the Rijksmuseum, Amsterdam), and "The Art of Painting" (Figure 93; located in the Kunsthistorisches Museum, Vienna), have identical floor tiling patterns, composed of black and white squares.

Figure 94 Squares, equilateral triangles, and hexagons are particularly easy to tile with, if one wants to cover the entire plane and achieve a pattern that repeats itself at regular intervals-known as periodic tiling (Figure 94). Simple, undecorated square tiles and the patterns they form have a fourfold symmetry-when rotated through a quarter of a circle (90 degrees), they remain the same. Similarly, equilateral, triangular tiles have a threefold symmetry (they remain the same when rotated by a third of a circle or 120 degrees), and hexagonal tiles have a sixfold symmetry (they remain the same when rotated by 60 degrees).

Periodic tilings also can be generated with more complex shapes. One of the most astounding monuments of Islamic architecture, the citadel-palace Alhambra in Granada, Spain, contains numerous examples of intricate tilings (Figure 95). Some of those patterns inspired the famous Dutch graphic artist M. C. Escher (18981972), who produced many imaginative examples of tilings (e.g., Figure 96 Figure 96), to which he referred as "divisions of the plane."

Figure 95 The geometrical plane figure most directly related to the Golden Ratio is, of course, the regular pentagon, which has a fivefold symmetry. Pentagons, however, cannot be used to fill the plane entirely and form a periodic tiling pattern. No matter how hard you try, unfilled gaps will remain. Consequently, it has long been thought that no tiling pattern with long-range order can also exhibit a fivefold symmetry. However, in 1974, Roger Penrose discovered two basic sets of tiles that can fit together to fill the entire plane and exhibit the "forbidden" five-fold rotational symmetry. The resulting patterns are not strictly periodic, even though they display a long-range order.

Figure 96

Figure 97

Figure 98 The Penrose tilings have the Golden Ratio written all over them. One pair of tiles that Penrose considered consists of two shapes known as a "dart" and a "kite" (Figure 97; a and b, respectively). Note that the two shapes are composed of the isosceles triangles that appear in the pentagon (Figure 25). The triangle in which the ratio of side to base is (Figure 97b) is the one known as a Golden Triangle, and the one in which the ratio of side to base is 1/ (Figure 97a) is the one known as a Golden Gnomon. The two shapes can be obtained by cutting a diamond shape or rhombus with angles of 72 degrees and 108 degrees in a way that divides the long diagonal in a Golden Ratio (Figure 98).

Penrose and Princeton mathematician John Horton Conway showed that in order to cover the whole plane with darts and kites in a nonperiodic way (as in Figure 99 Figure 99), certain matching rules must be obeyed. The latter can be ensured by adding "keys" in the form of notches and protrusions on the sides of the figures, like in the pieces of a jigsaw puzzle (Figure 100). Penrose and Conway further proved that darts and kites can fill the plane in infinitely many nonperiodic ways, with every pattern that can be discerned being surrounded by every other pattern. One of the most startling properties of any Pen-rose kite-dart tiling design is that the number of kites is about 1.618 times the number of darts. That is, if we denote by N Nkites the number of kites and the number of kites and N Ndarts the number of darts, then the number of darts, then N Nkjtes/Ndarts approaches the larger the area we take in. approaches the larger the area we take in.

Figure 99

Figure 100

Figure 101

Figure 102

Figure 103

Figure 104 Another pair of Penrose tiles that can fill the entire plane (nonperiodically) is composed of two diamonds (rhombi), one fat (obtuse) and one thin (acute; Figure 101 Figure 101). As in the kite-dart pair, each of the rhombi is composed to two Golden Triangles or Golden Gnomons (Figure 102), and special matching rules have to be obeyed (in this case described by decorating the appropriate sides or angles of the rhombi; Figure 103 Figure 103) to obtain a plane-filling pattern (as in Figure 104 Figure 104). Again, in large areas there are 1.618 times more fat rhombi than thin ones, Nr Nrfat/Ntin = . = .

The fat and thin rhombi are intimately related to the darts and kites and both, through the Golden Ratio, to the pentagon-pentagram system.

Recall that the Pythagorean interest in the Golden Ratio started with the infinite series of nested pentagons and pentagrams in Figure 105 Figure 105. All four of the Penrose tiles are hidden in this figure. Points B B and and D D mark the opposite far corners of the kite mark the opposite far corners of the kite DCBA DCBA, while points A A and and C C mark the "wings" of the dart mark the "wings" of the dart EABC EABC Similarly, you can find the fat rhombus Similarly, you can find the fat rhombus AECD AECD and the thin one (not to scale) and the thin one (not to scale) ABCF. ABCF.

Figure 105 Penrose s work on tiling has been expanded to three dimensions. In the same way that two-dimensional tiles can be used to fill the plane, three-dimensional "blocks" can be used to fill up s.p.a.ce. In 1976, mathematician Robert Ammann discovered a pair of "cubes" (Figure 106), one "squashed" and one "stretched," known as rhombohedra, that can fill up s.p.a.ce with no gaps. Ammann was further able to show that given a set of face-matching rules, the pattern that emerges is nonperiodic and has the symmetry properties of the icosahedron (Figure 20e; this is the equivalent of fivefold symmetry in three dimensions, since five symmetric edges meet at every vertex). Not surprisingly, the two rhombohedra are Golden Rhombohedra-their faces actually are identical to the rhombi of the Penrose tiles (Figure 101).

Figure 106 Penrose s tilings might have remained in the relative obscurity of recreational mathematics were it not for a dramatic discovery in 1984. Israeli materials engineer Dany Schectman and his collaborators found that the crystals of an aluminum manganese alloy exhibit both long-range order and fivefold symmetry. This was just about as shocking to crystallographers as the discovery of a herd of five-legged cows would be to zoologists. For decades, solid-state physicists and crystallographers were convinced that solids can come in only two basic forms: Either they are highly ordered and fully periodic crystals, or they are totally amorphous. In ordered crystals, like those of ordinary table salt, atoms or groups of atoms appear in precisely recurring motifs, called unit cells unit cells, which form periodic structures. For example, in salt, the unit cell is a cube, and each chlorine atom is surrounded by sodium neighbors and vice versa (Figure 107). Just as in a perfectly tiled floor, the position and orientation of each unit cell determines uniquely the entire pattern. In amorphous materials, such as gla.s.ses, on the other hand, the atoms are totally disordered. In the same way that only shapes like squares (with a fourfold symmetry), triangles (threefold symmetry), and hexagons (sixfold symmetry) can fill the entire plane with a periodic tiling, only crystals with two-, three-, four-, and six fold symmetry were thought to exist. Schectman's crystals caused complete bewilderment because they appeared both to be highly ordered (like periodic crystals) and to exhibit fivefold (or icosahedral) symmetry. Before this discovery, few people suspected that another state of matter could exist, sharing important aspects with both crystalline and amorphous substances.

Figure 107 These new kinds of crystals (since the original discovery, other alloys of aluminum have been found) are now known as quasi-crystals- quasi-crystals- they are neither amorphous like gla.s.s nor precisely periodic like salt. In other words, these unusual materials appear to have precisely the properties of Penrose tilings! But this realization by itself is of little use to physicists, who want to understand why and how the quasi-crystals form. Penrose's and Ammann's matching rules are in this case little more than a clever mathematical exercise that does not explain the behavior of real atoms or atom cl.u.s.ters. In particular, it is difficult to imagine energetics that permit precisely the existence of two types of cl.u.s.ters (like the two Ammann rhombohedra) in just the required proportion in terms of density. they are neither amorphous like gla.s.s nor precisely periodic like salt. In other words, these unusual materials appear to have precisely the properties of Penrose tilings! But this realization by itself is of little use to physicists, who want to understand why and how the quasi-crystals form. Penrose's and Ammann's matching rules are in this case little more than a clever mathematical exercise that does not explain the behavior of real atoms or atom cl.u.s.ters. In particular, it is difficult to imagine energetics that permit precisely the existence of two types of cl.u.s.ters (like the two Ammann rhombohedra) in just the required proportion in terms of density.

A clue toward a possible explanation came in 1991, when mathematician Sergei E. Burkov of the Landau Inst.i.tute of Theoretical Physics in Moscow realized that two shapes of tiles are not needed to achieve quasi-periodic tiling in the plane. Burkov showed that quasi-periodicity could be generated even using a single, decagonal (ten-sided) unit, provided that the tiles are allowed to overlap-a property forbidden in previous tiling attempts. Five years later, German mathematician Petra Gummelt of the Ernst Moritz Arndt University in Greifswald proved rigorously that Penrose tiling can be obtained by using a single "decorated" decagon combined with a specific overlapping rule. Two decagons may overlap only if shaded areas in the decoration overlap (Figure 108). The decagon is also closely related to the Golden Ratio-the radius of the circle that circ.u.mscribes a decagon with a side length of 1 unit is equal to .

Based on Gummelt's work, mathematics finally could be turned into physics. Physicists Paul Steinhardt of Princeton University and Hyeong-Chai Jeong of Sejong University in Seoul showed that the purely mathematical rules of overlapping units could be transformed into a physical picture in which "quasi-unit cells," which are really cl.u.s.ters of atoms, simply share atoms. Steinhardt and Jeong suggested that quasi-crystals are structures in which identical cl.u.s.ters of atoms (quasi-unit cells) share atoms with their neighbors, in a pat tern that is designed to maximize the cl.u.s.ter density. In other words, quasi-periodic packing produces a system that is more stable (higher density and lower energy) than otherwise. Steinhardt, Jeong, and collaborators also attempted to verify this model experimentally in 1998. They bombarded a quasi-crystal alloy of aluminum, nickel, and cobalt with X-ray and electron beams. The images of the structure obtained from the scattered beams were in remarkable agreement with the picture of overlapping decagons. This is shown in Figure 109 Figure 109, where a decagon tiling pattern is superimposed on the experimental result. More recent experiments gave results that were somewhat more ambiguous. Nevertheless, the general impression remains that quasi-crystals can be explained by the Steinhardt-Jeong model.

Figure 108

Figure 109

Figure 110 Images of the surfaces of quasi-crystals (taken in 1994 and 2001) reveal another fascinating relation to the Golden Ratio. Using a technique known as scanning tunneling microscopy (STM), scientists from the University of Basel, Switzerland, and from the Ames Laboratory at Iowa State University were able to obtain high-resolution images of the surfaces of an aluminum-copper-iron alloy and an aluminum-palladium-manganese alloy, both of which are quasi-crystals. The images show flat "terraces" (Figure 110) terminating in steps that come primarily in two heights, "high" and "low" (both measuring only a few hundred-millionths of an inch). The ratio of the two heights was found to be equal to the Golden Ratio!

Quasi-crystals are a magnificent example of a concept that started out as a purely mathematical ent.i.ty (based on the Golden Ratio) but that eventually provided an explanation of a real, natural phenomenon. What is even more amazing about this particular development is that the concept emerged out of recreational recreational mathematics. How could mathematicians have "antic.i.p.ated" later discoveries by physicists? The question becomes more intriguing yet when we recall that Durer and Kepler showed interest in tilings with fivefold symmetric shapes already in the sixteenth and seventeenth centuries. Can even the most esoteric topics in mathematics eventually find applications in either natural or human-inspired phenomena? We shall return to this question in Chapter 9. mathematics. How could mathematicians have "antic.i.p.ated" later discoveries by physicists? The question becomes more intriguing yet when we recall that Durer and Kepler showed interest in tilings with fivefold symmetric shapes already in the sixteenth and seventeenth centuries. Can even the most esoteric topics in mathematics eventually find applications in either natural or human-inspired phenomena? We shall return to this question in Chapter 9.

Another fascinating aspect of the quasi-crystals story is related to two of the main theorists involved. Both Penrose and Steinhardt spent much of their scientific careers on topics related to cosmology-the study of the universe as a whole. Penrose is the person who discovered that Einstein's theory of general relativity predicts its own defects, points in which the strength of gravity becomes infinite. These mathematical singularities correspond to the objects we call black holes black holes, which are ma.s.ses that have collapsed to such densities that their gravity is sufficiently strong to prevent any light, ma.s.s, or energy to escape from them. Observations during the past quarter century have revealed that black holes are not just imaginary theoretical concepts but actual objects that exist in the universe. Recent observations with the two large s.p.a.ce observatories, the Hubble s.p.a.ce Telescope and the Chandra X-ray Observatory, have shown that black holes are not even very rare. Rather, the centers of most galaxies harbor monstrous black holes with ma.s.ses between a few million and a few billion times the ma.s.s of our Sun. The presence of the black holes is revealed by the gravitational pull they exert on stars and gas in their neighborhood. According to the standard "big bang" model that describes the origin of our entire universe, the cosmos as a whole started its expansion from such a singularity-an extremely hot and dense state.

Paul Steinhardt was one of the key figures in the development of what is known as the inflationary model of the universe. According to this model, originally proposed by physicist Alan Guth of MIT, when the universe was only a tiny fraction of a second old (0.000 ... 1; with the "1" at the 35th decimal place), it underwent a fantastically rapid expansion, increasing in size by a factor of more than 10 decimal place), it underwent a fantastically rapid expansion, increasing in size by a factor of more than 1030 (1 followed by 30 zeros) within a fraction of a second. This model explains a few otherwise puzzling properties of our universe, such as the fact that it looks almost precisely the same in every direction-it is exquisitely isotropic. In 2001, Steinhardt and collaborators proposed a new version of the universe's very beginnings, known as the Ekpyrotic Universe (from the Greek word for "conflagration," or a sudden burst of fire). In this still very speculative model, the big bang occurred when two three-dimensional universes moving along a hidden extra dimension collided. (1 followed by 30 zeros) within a fraction of a second. This model explains a few otherwise puzzling properties of our universe, such as the fact that it looks almost precisely the same in every direction-it is exquisitely isotropic. In 2001, Steinhardt and collaborators proposed a new version of the universe's very beginnings, known as the Ekpyrotic Universe (from the Greek word for "conflagration," or a sudden burst of fire). In this still very speculative model, the big bang occurred when two three-dimensional universes moving along a hidden extra dimension collided.