The Golden Ratio: The Story of Phi, the World's Most Astonishing Number - Mario Livio (2003)

7. PAINTERS AND POETS HAVE EQUAL LICENSE

Painting isn't an aesthetic operation; it's a form of magic designed
as a mediator between this strange hostile world and us.

—PABLO PICASSO (1881–1973)

The Renaissance produced a significant change in direction in the history of the Golden Ratio. No longer was this concept confined to mathematics. Now the Golden Ratio found its way into explanations of natural phenomena and into the arts.

We have already encountered claims that the architectural design of various structures from antiquity, such as the Great Pyramid and the Parthenon, had been based on the Golden Ratio. A closer examination of these claims revealed, however, that in most cases they could not be substantiated. The introduction of the notion of the existence of a “Divine Proportion” and the general recognition of the importance of mathematics for perspective made it more conceivable that some artists would start using scientifically based methods in general and the Golden Ratio in particular in their works. Contemporary painter and draftsman David Hockney argues in his book Secret Knowledge (2001), for example, that starting with around 1430, artists began secretly using cameralike devices, including lenses, concave mirrors, and the camera obscura, to help them create realistic-looking paintings. But did artists really use the Golden Ratio? And if they did, was the Golden Ratio's application restricted to the visual arts or did it penetrate into other areas of artistic endeavor?

THE ARTIST'S SECRET GEOMETRY?

Many of the assertions concerning the employment of the Golden Ratio in painting are directly associated with the presumed aesthetic properties of the Golden Rectangle. I shall discuss the reality (or falsehood) of such a canon for aesthetics later in the chapter. For the moment, however, I shall concentrate on the much simpler question: Did any pre-and Renaissance painters actually base their artistic composition on the Golden Rectangle? Our attempt to answer this question takes us back to the thirteenth century.

The “Ognissanti Madonna” (also known as “Madonna in Glory,” Figure 70; currently in the Uffizi Gallery in Florence) is one of the greatest panel paintings by the famous Italian painter and architect Giotto di Bondone (1267–1337). Executed between 1306 and 1310, the painting shows a half-smiling, enthroned Virgin caressing the knee of the Child. The Madonna and Child are surrounded by angels and saints arranged in some sort of perspectival “hierarchy.” Many books and articles on the Golden Ratio repeat the statement that both the painting as a whole and the central figures can be inscribed precisely in Golden Rectangles (Figure 71).

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Figure 70

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Figure 71

A similar claim is made about two other paintings with the same general subject: the “Madonna Rucellai” (painted in 1285) by the great Sienese painter Duccio di Buoninsegna, known as Duccio (ca. 1255–1319), and the “Santa Trinita Madonna” by the Florentine painter Cenni di Pepo, known as Cimabue (ca. 1240–1302). As fate would have it, currently the three paintings happen to be hanging in the same room in the Uffizi Gallery in Florence. The dimensions of the Ognissanti,” “Rucellai,” and “Santa Trinita” Madonnas give height to width ratios of 1.59, 1.55, and 1.73, respectively. While all three numbers are not too far from the Golden Ratio, two of them are actually closer to the simple ratio of 1.6 rather than to the irrational number ö. This fact could indicate (if anything) that the artists followed the Vitruvian suggestion for a simple proportion, one that is the ratio of two whole numbers, rather than the Golden Ratio. The inner rectangle in the “Ognissanti Madonna” (Figure 71) leaves us with an equally ambiguous impression. Not only are the boundaries of the rectangle drawn usually (e.g., in Trudi Hammel Garland's charming book Fascinating Fibonaccis) with rather thick lines, making any measurement rather uncertain, but, in fact, the upper horizontal side is placed somewhat arbitrarily.

Remembering the dangers of having to rely on measured dimensions alone, we may wonder if there exist any other reasons to suspect that these three artists might have desired to include the Golden Ratio in their paintings. The answer to this question appears to be negative, unless they were driven toward this ratio by some unconscious aesthetic preference (a possibility that will be discussed later in the chapter). Recall that the three Madonnas were painted more than two centuries before the publication of The Divine Proportion brought the ratio to wider attention.

The French painter and author Charles Bouleau expresses a different view in his 1963 book The Painter's Secret Geometry. Without referring to Giotto, Duccio, or Cimabue specifically, Bouleau argues that Pacioli's book represented an end to an era rather than its beginning. He asserts that The Divine Proportion merely “reveals the thinking of long centuries of oral tradition” during which the Golden Ratio “was considered as the expression of perfect beauty.” If this were truly the case, then Cimabue, Duccio, and Giotto indeed might have decided to use this accepted standard for perfection. Unfortunately, I find no evidence to support Bouleau s statement. Quite to the contrary; the documented history of the Golden Ratio is inconsistent with the idea that this proportion was particularly revered by artists in the centuries preceding the publication date of Pacioli's book. Furthermore, all the serious studies of the works of the three artists by art experts (e.g., Giotto by Francesca Flores D'Arcais; Cimabue by Luciano Bellosi) give absolutely no indication whatsoever that these painters might have used the Golden Ratio—the latter claim appears only in the writings of Golden Number enthusiasts and is based solely on the dubious evidence of measured dimensions.

Another name that invariably turns up in almost every claim of the appearance of the Golden Ratio in art is that of Leonardo da Vinci. Some authors even attribute the invention of the name “the Divine Proportion” to Leonardo. The discussion usually concentrates on five works by the Italian master: the unfinished canvas of “St. Jerome,” the two versions of “Madonna of the Rocks,” the drawing of “a head of an old man,” and the famous “Mona Lisa.” I am going to ignore the “Mona Lisa” here for two reasons: It has been the subject of so many volumes of contradicting scholarly and popular speculations that it would be virtually impossible to reach any unambiguous conclusions; and the Golden Ratio is supposed to be found in the dimensions of a rectangle around Mona Lisa's face. In the absence of any clear (and documented) indication of where precisely such a rectangle should be drawn, this idea represents just another opportunity for number juggling. I shall return, however, to the more general topic of proportions in faces in Leonardo's paintings, when I shall discuss the drawing “a head of an old man.”

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Figure 72

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Figure 73

The case of the two versions of “Madonna of the Rocks” (one in the Louvre in Paris, Figure 72, and the other in the National Gallery in London, Figure 73) is not particularly convincing. The ratio of the height to width of the painting thought to have been executed earlier (Figure 72) is about 1.64 and of the later one 1.58, both reasonably close to ö but also close to the simple ratio of 1.6.

The dating and authenticity of the two “Madonna of the Rocks” also put an interesting twist on the claims about the presence of the Golden Ratio. Experts who studied the two paintings concluded that, without a doubt, the Louvre version was done entirely by Leonardo's hand, while the execution of the National Gallery version might have been a collaborative effort and is still the source of some debate. The Louvre version is thought to be one of the first works that Leonardo produced in Milan, probably between 1483 and 1486. The National Gallery painting, on the other hand, usually is assumed to have been completed around 1506. The reason that these dates may be of some significance is that Leonardo met Pacioli for the first time in 1496, in the Court of Milan. The seventy-first chapter of the Divina (the end of the first portion of the book) was, in Pacioli's words: “Finished this day of December 14, at Milan in our still cloister the year 1497.” The first version (and the one with no doubts about authenticity) of the “Madonna of the Rocks” was therefore completed about ten years before Leonardo had the opportunity to hear directly from the horse's mouth about the “divine proportion.” The claim that Leonardo used the Golden Ratio in “Madonna of the Rocks” therefore amounts to believing that the artist adopted this proportion even before he started his collaboration with Pacioli. While this is not impossible, there is no evidence to support such an interpretation.

Either version of “Madonna of the Rocks” represents one of Leonardo's most accomplished masterpieces. Perhaps in no other painting did he apply better his poetic formula: “every opaque body is surrounded and clothed on its surface by shadows and light.” The figures in the paintings literally open themselves to the emotional participation of the spectator. To claim that these paintings derive any part of their strength from the mere ratio of their dimensions trivializes Leonardo's genius unnecessarily. Let us not fool ourselves; the feeling of awe we experience when facing “Madonna of the Rocks” has very little to do with whether the dimensions of the paintings are in a Golden Ratio.

A similar uncertainty exists with respect to the unfinished “St. Jerome” (Figure 74; currently in the Vatican museum). Not only is the painting dated to 1483, long before Pacioli's move to Milan, but the claim made in some books (e.g., in David Bergamini and the editors of Life Magazine's Mathematics) that “a Golden Rectangle fits so neatly around St. Jerome” requires quite a bit of wishful thinking. In fact, the sides of the rectangle miss the body (especially on the left side) and head entirely, while the arm extends well beyond the rectangle's side.

The last example for a possible use of the Golden Ratio by Leonardo is the drawing of “a head of an old man” (Figure 75; the drawing is currently in the Galleria dell'Accademia in Venice). The profile and diagram of proportions were drawn in pen some time around 1490. Two studies of horsemen in red chalk, which are associated with Leonardo's “Battle of Anghiari,” were added to the same page around 1503–1504.

While the overlying grid leaves very little doubt that Leonardo was indeed interested in various proportions in the face, it is very difficult to draw any definitive conclusions from this study. The rectangle in the middle left, for example, is approximately a Golden Rectangle, but the lines are drawn so roughly that we cannot be sure. Nevertheless, this drawing probably comes the closest to a demonstration that Leonardo used rectangles to determine dimensions in his paintings and that he might have even considered the application of the Golden Ratio to his art.

Leonardo's interest in proportions in the face may have another in-

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Figure 74

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Figure 75

teresting manifestation. In an article that appeared in 1995 in the Scientific American, art historian and computer graphics artist Lillian Schwarz presented an interesting speculation. Schwarz claimed that in the absence of his model for the “Mona Lisa,” Leonardo used his own facial features to complete the painting. Schwarz's suggestion was based on a computer-aided comparison between various dimensions in Mona Lisa's face and the respective dimensions in a red chalk drawing that is considered by many (but not all) to be Leonardo's only self-portrait.

However, as other art analysts have pointed out, the similarity in the proportions may simply reflect the fact that Leonardo used the same formulae of proportion (which may or may not have included the Golden Ratio) in the two portraits. In fact, Schwarz herself notes that even in his grotesques—a collection of bizarre faces with highly exaggerated chins, noses, mouths, and foreheads—Leonardo used the same proportions in the face as in the “head of an old man.”

If there exist serious doubts regarding whether Leonardo himself, who was not only a personal friend of Pacioli but also the illustrator for the Oivina, used the Golden Ratio in his paintings, does this mean that no other artist ever used it? Definitely not. With the surge of Golden Ratio academic literature toward the end of the nineteenth century, the artists also started to take notice. Before we discuss artists who did use the Golden Ratio, however, another myth still needs to be dispelled.

In spite of many existing claims to the contrary, the French pointillist Georges Seurat (1859–1891) probably did not use the Golden Ratio in his paintings. Seurat was interested in color vision and color combination, and he used the pointillist (multidotted) technique to approximate as best as he could the scintillating, vibratory quality of light. He was also concerned late in life with the problem of expressing specific emotions through pictorial means. In a letter he wrote in 1890, Seurat describes succinctly some of his views:

Art is harmony. Harmony is the analogy of contradictions and of similars, in tone, shade, line, judged by the dominant of those and under the influence of a play of light in arrangements that are gay, light, sad. Contradictions are…, with respect to line, those that form a right angle.… Gay lines are lines above the horizontal;… calm is the horizontal; sadness lines in the downward direction.

Seurat used these ideas explicitly in “The Parade of a Circus” (sometimes called “The Side Show;” Figure 76; currently in The Metropolitan Museum of Art, New York). Note in particular the right angle formed by the balustrade and the vertical line to the right of the middle of the painting. The entire composition is based on principles that Seurat adopted from art theorist David Sutter's book ha philosophie des Beaux-Arts appliquée à la peinture (The philosophy of the fine arts applied to painting; 1870). Sutter wrote: “when the dominant is horizontal, a succession of vertical objects can be placed on it because this series will concur with the horizontal line.”

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Figure 76

Golden Ratio aficionados often present analyses of “The Parade” (as well as other paintings, such as “The Circus”) to “prove” the use of ö. Even in the beautiful book Mathematics, by Bergamini and the editors of Life Magazine, we find: “La Parade, painted in the characteristic multi-dotted style of the French impressionist Georges Seurat, contains numerous examples of Golden proportions.” The book goes even further with a quote (attributed to “one art expert”) that Seurat “attacked every canvas by the Golden Section.” Unfortunately, these statements are unfounded. This myth was propagated by the Romanian born prelate and author Matila Ghyka (1881–1965), who was also the “art expert” quoted by Bergamini. Ghyka published two influential books, Esthétique des proportions dans la nature et dans les arts (Aesthetics of proportions in nature and in the arts; 1927) and Le Nombre d'Or: Rites et rythmes pytagoriciens dans le développement de la civilisation occidentale (The golden number, Pythagorean rites and rhythms in the development of Western civilization; 1931). Both books are composed of semimystical interpretations of mathematics. Alongside correct descriptions of the mathematical properties of the Golden Ratio, the books contain a collection of inaccurate anecdotal materials on the occurrence of the Golden Ratio in the arts (e.g., the Parthenon, Egyptian temples, etc.). The books have been almost inexplicably influential.

Concerning “The Parade” specifically, while it is true that the horizontal is cut in proportions close to the Golden Ratio (in fact, the simple ratio eight-fifths), the vertical is not. An analysis of the entire composition of this and other paintings by Seurat, as well as paintings by the Symbolist painter Pierre Puvis de Chavannes (1824–1898), led even a Golden Ratio advocate like painter and author Charles Bouleau to conclude that “I do not think we can, without straining the evidence to regard his [Puvis de Chavannes s] compositions as based on the Golden Ratio. The same applies to Seurat.” A detailed analysis in 1980 by Roger Herz-Fischler of all of Seurat s writings, sketches, and paintings reached the same conclusion. Furthermore, the mathematician, philosopher, and art critic Charles Henry (1859–1926) stated firmly in 1890 that the Golden Ratio was “perfectly ignored by contemporary artists.”

Who, then, did use the Golden Ratio either in actual paintings or in the theory of painting? The first prominent artist and art theorist to employ the ratio was probably Paul Sérusier (1864–1927). Sérusier was born in Paris, and after studying philosophy he entered the famous art school Académie Julian. A meeting with the painters Paul Gaugin and Émile Bernard converted him to their expressive use of color and symbolist views. Together with the post-Impressionist painters Pierre Bon-nard, Édouard Vuillard, Maurice Denis, and others he founded the group called the Nabis, from the Hebrew word meaning “prophets.” The name was inspired by the group's half-serious, half-burlesque pose regarding their new style as a species of religious illumination. The composer Claude Debussy was also associated with the group. Sérusier probably heard about the Golden Ratio for the first time during one of his visits (between 1896 and 1903) to his friend the Dutch painter Jan Verkade (1868–1946). Verkade was a novice in the Benedictine monastery of Beuron, in South Germany. There groups of monk-painters were executing rather dull religious compositions based on “sacred measures,” following a theory of Father Didier Lenz. According to Father Lenz's theory, the great art works of antiquity (e.g., Noah's Ark, Egyptian works, etc.) were all based on simple geometrical entities such as the circle, equilateral triangle, and hexagon. Sérusier found the charm of this theory captivating, and he wrote to Verkade: “as you can imagine, [I] have talked a great deal about your measures.” The painter Maurice Denis (1870–1943) wrote biographical notes on Sérusier, from which we learn that those “measures” employed by Father Lenz included the Golden Ratio. Even though Sérusier admits that his initial studies of the mathematics of Beuron were “not all plain sailing,” the Golden Ratio and the story of its potential association with the Great Pyramid and Greek artworks made it also into Sérusier's important art theory book L'ABC de la Peinture(The ABC of painting).

While Sérusier's interest in the Golden Ratio appears to have been more philosophical than practical, he did make use of this proportion in some of his works, mainly to “verify, and occasionally to check, his inventions of shapes and his composition.”

Following Sérusier, the concept of the Golden Ratio propagated into other artistic circles, especially that of the Cubists. The name “Cubism” was coined by art critic Louis Vauxcelles (who, by the way, had also been responsible for “Expressionism” and “Fauvism”) after viewing an exhibition of Georges Braque's work in 1908. The movement was inaugurated by Picasso's painting “Les Demoiselles d Avignon” and Braque's “Nude.” In revolt against the passionate use of color and form in Expressionism, Picasso and Braque developed an austere, almost monochrome style that deliberately rejected any subject matter that was likely to evoke emotional associations. Objects like musical instruments and even human figures were dissected into faceted geometrical planes, which were then combined in shifting perspectives. This analysis of solid forms for the purpose of revealing structure was quite amenable to the use of geometrical concepts like the Golden Ratio. In fact, some of the early Cubists, such as Jacques Villon and his brothers Marcel and Raymond Duchamp-Villon, together with Albert Gleizes and Francis Picabia, organized in Paris in October 1912 an entire exhibition entitled “Section d'Or” (“The Golden Section”). In spite of the highly suggestive name, none of the paintings that was exhibited actually included the Golden Section as a basis for its composition. Rather, the organizers chose the name simply to project their general interest in questions that related art to science and philosophy. Nevertheless, some Cubists, like the Spanish-born painter Juan Gris (1887–1927) and the Lithuanian-born sculptor Jacques (Chaim Jacob) Lipchitz (1891–1973) did use the Golden Ratio in some of their later works. Lipchitz wrote: “At the time, I was very interested in theories of mathematical pro portions, like the other cubists, and I tried to apply them to my sculptures. We all had a great curiosity for that idea of a golden rule or Golden Section, a sys tem which was reputed to lay under the art and architecture of ancient Greece.” Lipchitz helped Juan Gris in the construction of the sculpture “Arlequin” (currently in the Philadelphia Museum of Art; Figure 77), in which the two artists used Kepler's triangle (which is based on the Golden Ratio; see Figure 61) for the production of the desired proportions.

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Figure 77

Another artist who used the Golden Ratio in the early 1920s was the Italian painter Gino Severini (1883–1966). Severini attempted in his work to reconcile the somewhat conflicting aims of Futurism and Cubism. Futurism represented an effort by a group of Italian intellectuals from literary arts, the visual arts, theater, music, and cinema to bring about a cultural rejuvenation in Italy. In Severini's words: “We choose to concentrate our attention on things in motion, because our modern sensibility is particularly qualified to grasp the idea of speed.” The first painters' Futurist manifesto was signed in 1910, and it strongly urged the young Italian artists to “profoundly despise all forms of imitation.” While still a Futurist himself, Severini found in Cubism a “notion of measure” that fit his ambition of “making, by means of painting, an object with the same perfection of craftsmanship as a cabinet maker making furniture.” This striving for geometrical perfection led Severini to use the Golden Section in his preparatory drawings for several paintings (e.g., “Maternity,” currently in a private collection in Rome; Figure 78).

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Figure 78

Russian Cubist painter Maria Vorobëva, known as Marevna, provides an interesting instance of the role of the Golden Ratio in Cubist art. Marevna's 1974 book, Life with the Painters of La Ruche, is a fascinating account of the lives and works of her personal friends—a group that included the painters Picasso, Modigliani, Soutine, Rivera (with whom she had a daughter), and others in Paris of the 1920s. Although Marevna does not give any specific examples and some of her historical comments are inaccurate, the text implies that Picasso, Rivera, and Gris had used the Golden Ratio as “another way of dividing planes, which is more complex and attracts experienced and inquisitive minds.”

Another art theorist who had great interest in the Golden Ratio at the beginning of the twentieth century was the American Jay Hambidge (1867–1924). In a series of articles and books, Hambidge defined two types of symmetry in classical and modern art. One, which he called “static symmetry,” was based on regular figures like the square and equilateral triangle, and was supposed to produce lifeless art. The other, which he dubbed “dynamic symmetry,” had the Golden Ratio and the logarithmic spiral in leading roles. Hambidge's basic thesis was that the use of “dynamic symmetry” in design leads to vibrant and moving art. Few today take his ideas seriously.

One of the strongest advocates for the application of the Golden Ratio to art and architecture was the famous Swiss-French architect and painter Le Corbusier (Charles-Édouard Jeanneret, 1887–1965).

Jeanneret was born in La Chaux-de-Fonds, Switzerland, where he studied art and engraving. His father worked in the watch business as an enameler, while his mother was a pianist and music teacher who encouraged her son toward a musician's dexterity as well as more abstract pursuits. He began his studies of architecture in 1905 and eventually became one of the most influential figures in modern architecture. In the winter of 1916–1917, Jeanneret moved to Paris, where he met Amédée Ozenfant, who was well connected in the Parisian haut monde of artists and intellectuals. Through Ozenfant, Jeanneret met with the Cubists and was forced to grapple with their inheritance. In particular, he absorbed an interest in proportional systems and their role in aesthetics from Juan Gris. In the autumn of 1918, Jeanneret and Ozenfant exhibited together at the Galérie Thomas. More precisely, two canvases by Jeanneret were hung alongside many more paintings by Ozenfant. They called themselves “Purists,” and entitled their catalog Après le Cubisme (After cubism). Purism invoked Piero della Francesca and the Platonic aesthetic theory in its assertion that “the work of art must not be accidental, exceptional, impressionistic, inorganic, protestatory, picturesque, but on the contrary, generalized, static, expressive of the invariant.”

Jeanneret did not take the name “Le Corbusier” (co-opted from ancestors on his mother's side called Lecorbesier) until he was thirty-three, well installed in Paris, and confident of his future path. It was as if he wanted basically to repress his faltering first efforts and stimulate the myth that his architectural genius bloomed suddenly into full maturity.

Originally, Le Corbusier expressed rather skeptical, and even negative, views of the application of the Golden Ratio to art, warning against the “replacement of the mysticism of the sensibility by the Golden Section.” In fact, a thorough analysis of Le Corbusier's architectural designs and “Purist” paintings by Roger Herz-Fischler shows that prior to 1927, Le Corbusier never used the Golden Ratio. This situation changed dramatically following the publication of Matila Ghyka's influential book Aesthetics of Proportions in Nature and in the Arts, and his Golden Number, Pythagorean Rites and Rhythms (1931) only enhanced the mystical aspects of ö even further. Le Corbusier's fascination with Aesthetics and with the Golden Ratio had two origins. On one hand, it was a consequence of his interest in basic forms and structures underlying natural phenomena. On the other, coming from a family that encouraged musical education, Le Corbusier could appreciate the Pythagorean craving for a harmony achieved by number ratios. He wrote: “More than these thirty years past, the sap of mathematics has flown through the veins of my work, both as an architect and painter; for music is always present within me.” Le Corbusier's search for a standardized proportion culminated in the introduction of a new proportional system called the “Modulor.”

The Modulor was supposed to provide “a harmonic measure to the human scale, universally applicable to architecture and mechanics.” The latter quote is in fact no more than a rephrasing of Protagoras' famous saying from the fifth-century B.C. “Man is the measure of all things.” Accordingly, in the spirit of the Vitruvian man (Figure 53) and the general philosophical commitment to discover a proportion system equivalent to that of natural creation, the Modulor was based on human proportions (Figure 79).

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Figure 79

A six-foot (about 183-centimeter) man, somewhat resembling the familiar logo of the “Michelin man,” with his arm upraised (to a height of 226 cm; 7′5″), was inserted into a square (Figure 80). The ratio of the height of the man (183 cm; 6′) to the height of his navel (at the midpoint of 113 cm; 3' 8.5″) was taken to be precisely in a Golden Ratio. The total height (from the feet to the raised arm) was also divided in a Golden Ratio (into 140 cm and 86 cm) at the level of the wrist of a downward-hanging arm. The two ratios (113/70) and (140/86) were further subdivided into smaller dimensions according to the Fibonacci series (each number being equal to the sum of the preceding two; Figure 81). In the final version of the Modulor (Figures 79 and 81), two scales of interspiraling Fibonacci dimensions were therefore introduced (the “red and the blue series”).

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Figure 80

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Figure 81

Le Corbusier suggested that the Modulor would give harmonious proportions to everything, from the sizes of cabinets and door handles, to buildings and urban spaces. In a world with an increasing need for mass production, the Modulor was supposed to provide the model for standardization. Le Corbusier's two books, Le Modulor (published in 1948) and Modulor II (1955), received very serious scholarly attention from architectural circles, and they continue to feature in any discussion of proportion. Le Corbusier was very proud of the fact that he had the opportunity to present the Modulor even to Albert Einstein, in a meeting at Princeton in 1946. In describing that event he says: “I expressed myself badly, I explained ‘Modulor’ badly, I got bogged down in the morass of ‘cause and effect.’” Nevertheless, he received a letter from Einstein, in which the great man said this of the Modulor: “It is a scale of proportions which makes the bad difficult and the good easy.”

Le Corbusier translated his theory of the Modulor into practice in many of his projects. For example, in his notes for the impressive urban layout of Chandigarh, India, which included four major government buildings—a Parliament, a High Court, and two museums—we find: “But, of course, the Modulor came in at the moment of partitioning the window area. … In the general section of the building which involves providing shelter from the sun for the offices and courts, the Modulor will bring textural unity in all places. In the design of the frontages, the Modulor (texturique) will apply its red and blue series within the spaces already furnished by the frames.”

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Figure 82

Le Corbusier was certainly not the last artist to be interested in the Golden Ratio, but most of those after him were fascinated more by the mathematical-philosophical-historical attributes of the ratio than by its presumed aesthetic properties. For example, the British abstract artist Anthony Hill used a Fibonacci series of dimensions in his 1960 “Constructional Relief” (Figure 82). Similarly, the contemporary Israeli painter and sculptor Igael Tumarkin has deliberately included the formula for the value of image in one of his paintings.

An artist who transformed the Fibonacci sequence into an important ingredient of his art is the Italian Mario Merz. Merz was born in Milan in 1925, and in 1967 he joined the art movement labeled Arte Povera (Poor Art), which also included the artists Michelangelo Pisto-letto, Luciano Fabro, and Jannis Kounellis. The name of the movement (coined by the critic Germano Celant) was derived from the desire of its members to use simple, everyday life materials, in a protest against what they regarded as a dehumanized, consumer-driven society. Merz started to use the Fibonacci sequence in 1970, in a series of “conceptual” works that include the numbers in the sequence or various spirals. Merz's desire to utilize Fibonacci numbers was based on the fact that the sequence underlies so many growth patterns of natural life. In a work from 1987 entitled “Onda d'urto” (Shock wave), he has a long row of stacks of newspapers, with the Fibonacci numbers glowing in blue neon lights above the stacks. The work “Fibonacci Naples” (from 1970) consists of ten photographs of factory workers, building in Fibonacci numbers from a solitary person to a group of fifty-five (the tenth Fibonacci number).

False claims about artists allegedly using the Golden Ratio continue to spring up almost like mushrooms after the rain. One of these claims deserves some special attention, since it is repeated endlessly.

The Dutch painter Piet Mondrian (1872–1944) is best known for his geometric, nonobjective style, which he called “neoplasticism.” In particular, much of his art is characterized by compositions involving only vertical and horizontal lines, rectangles, and squares, and employing only primary colors (and sometimes black or grays) against a white background, as in “Broadway Boogie-Woogie” (Figure 83; in

The Museum of Modern Art, New York). Curved lines, three-dimensionality, and realistic representation were entirely eliminated from his work.

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Figure 83

Not surprisingly, per haps, Mondrian's geometrical compositions attracted quite a bit of Golden Numberist speculation. In Mathematics, David Bergamini admits that Mondrian himself “was vague about the design of his paintings,” but nevertheless claims that the linear abstraction “Place de la Concorde” incorporates overlapping Golden Rectangles. Charles Bouleau was much bolder in The Painter's Secret Geometry, asserting that “the French painters never dared to go as far into pure geometry and the strict use of the golden section as did the cold and pitiless Dutchman Piet Mondrian.” Bouleau further states that in “Broadway Boogie-Woogie,” “the horizontals and verticals which make up this picture are nearly all in the golden ratio.” With so many lines to choose from in this painting, it should come as no surprise that quite a few can be found at approximately the right separations. Having spent quite some time reading the more serious analyses of Mondrian's work and not having found any mention of the Golden Ratio there, I became quite intrigued by the question: Did Mondrian really use the Golden Ratio in his compositions or not? As a last resort I decided to turn to the real expert—Yves-Alain Bois of Harvard University, who coauthored the book Mondrian that accompanied the large retrospective exhibit of the artist's work in 1999. Bois's answer was quite categorical: “As far as I know, Mondrian never used a system of proportion (if one excepts the modular grids he painted in 1918–1919, but there the system is deduced from the format of the paintings themselves: they are divided in 8 × 8 units).” Bois added: “I also vaguely remember a remark by Mondrian mocking arithmetic computations with regard to his work.” He concluded: “I think that the Golden Section is a complete red herring with regard to Mondrian.”

All of this intricate history does leave us with a puzzling question. Short of intellectual curiosity, for what reason would so many artists even consider employing the Golden Ratio in their works? Does this ratio, as manifested for example in the Golden Rectangle, truly contain some intrinsic, aesthetically superior qualities? The attempts to answer this question alone resulted in a multitude of psychological experiments and a vast literature.

THE SENSES DELIGHT IN THINGS DULY PROPORTIONED

With the words in the title of this section, Italian scholastic philosopher St. Thomas Aquinas (ca. 1225–1274) attempted to capture a fundamental relationship between beauty and mathematics. Humans seem to react with a sense of pleasure to “forms” that possess certain symmetries or obey certain geometrical rules.

In our examination of the potential aesthetic value of the Golden Ratio, we will concentrate on the aesthetics of very simple, nonrepresentational forms and lines, not on complex visual materials and works of art. Furthermore, in most of the psychological experiments I shall describe, the term “beautiful” was actually shunned. Rather, words like “pleasing” or “attractive” have been used. This avoids the need for a definition of “beautiful” and builds on the fact that most people have a pretty good idea of what they like, even if they cannot quite explain why.

Numerous authors have claimed that the Golden Rectangle is the most aesthetically pleasing of all rectangles. The more modern interest in this question was largely initiated by a series of rather crankish publications by the German researcher Adolph Zeising, which started in 1854 with Neue Lehre von den Proportionen des menschlichen Körpers (The latest theory of proportions in the human body) and culminated in the publication (after Zeising's death) of a massive book, Der Goldne Schnitt (The golden section), in 1884. In these works, Zeising combined his own interpretation of Pythagorean and Vitruvian ideas to argue that “the partition of the human body, the structure of many animals which are characterized by well-developed building, the fundamental types of many forms of plants,… the harmonics of the most satisfying musical accords, and the proportionality of the most beautiful works in architecture and sculpture” are all based on the Golden Ratio. To him, therefore, the Golden Ratio offered the key to the understanding of all proportions in “the most refined forms of nature and art.”

One of the founders of modern psychology, Gustav Theodor Fechner (1801–1887), took it upon himself to verify Zeising's pet theory. Fechner is considered a pioneer of experimental aesthetics. In one of his early experiments, he conducted a public opinion poll in which he asked all the visitors to the Dresden Gallery to compare the beauty of two nearly identical Madonna paintings (the “Darmstadt Madonna” and the “Dresden Madonna”) that were exhibited together. Both paintings were attributed to the German painter Hans Holbein the Younger (1497–1543), but there was a suspicion that the “Dresden Madonna” was actually a later copy. That particular experiment resulted in a total failure—out of 11,842 visitors, only 113 answered the questionnaire, and even those were mostly art critics or people who had formed previous judgments.

Fechner's first experiments with rectangles were performed in the 1860s, and the results were published in the 1870s and eventually summarized in his 1876 book, Vorschule der Aesthetik (Introduction to aesthetics). He rebelled against a top-down approach to aesthetics, which starts with the formulation of abstract principles of beauty, and rather advocated the development of experimental aesthetics from the bottom up. The experiment was quite simple: Ten rectangles were placed in front of a subject who was asked to select the most pleasing one and the least pleasing one. The rectangles varied in their length-to-width ratios from a square (a ratio of 1.00) to an elongated rectangle (a ratio of 2.5). Three of the rectangles were more elongated than the Golden Rectangle, and six were closer to a square. According to Fechner's own description of the experimental setting, subjects often waited and wavered, rejecting one rectangle after another. Meanwhile the experimenter would explain that they should carefully select the most pleasing, harmonic, and elegant rectangle. In Fechner's experiment, 76 percent of all choices centered on the three rectangles having the ratios 1.75, 1.62, and 1.50, with the peak at the Golden Rectangle (1.62). All other rectangles received less than 10 percent of the choices each.

Fechner's motivation for studying the subject was not free of prejudice. He himself admitted that the inspiration for the research came to him when he “saw the vision of a unified world of thought, spirit and matter, linked together by the mystery of numbers.” While nobody accuses Fechner of altering the results, some speculate that he may have subconsciously produced circumstances that would favor his desired outcome. In fact, Fechner's unpublished papers reveal that he conducted similar experiments with ellipses, and having failed to discover any preference for the Golden Ratio, he did not publish the results.

Fechner further measured the dimensions of thousands of printed books, picture frames in galleries, windows, and other rectangularly shaped objects. His results were quite interesting, and often amusing. For example, he found that German playing cards tended to be somewhat more elongated than the Golden Rectangle, while French playing cards were less so. On the other hand, he found the average height-to-width ratio of forty novels from the public library to be near ö. Paintings (the area inside the frame) were actually found to be “significantly shorter” than a Golden Rectangle. Fechner made the following (politically incorrect by today's standards) observation about window shapes: “Only the window shapes of the houses of peasants seem often to be square, which is consistent with the fact that people with a lower level of education prefer this form more than people with a higher education.” Fechner further claimed that the point at which the transverse piece crosses the upright post in graveyard crosses divides the post, on the average, in a Golden Ratio.

Many researchers repeated similar experiments over the twentieth century, with varying results. Overly eager Golden Ratio enthusiasts usually report only those experiments that seem to support the idea of an aesthetic preference for the Golden Rectangle. However, more careful researchers point out the very crude nature and methodological defects of many of these experiments. Some found that the results depended, for example, on whether the rectangles were positioned with their long side horizontally or vertically, on the size and color of the rectangles, on the age of the subjects, on cultural differences, and especially on the experimental method used. In an article published in 1965, American psychologists L. A. Stone and L. G. Collins suggested that the preference for the Golden Rectangle indicated by some of the experiments was related to the area of the human visual field. These researchers found that an “average rectangle” of rectangles drawn within and around the binocular visual field of a variety of subjects has a length-to-width ratio of about 1.5, not too far from the Golden Ratio. Subsequent experiments, however, did not confirm Stone and Collins s speculation. In an experiment conducted in 1966 by H. R. Schiffman of Rutgers University, subjects were asked to “draw the most aesthetically pleasing rectangle” that they could on a sheet of paper. After completion, they were instructed to orient the figure either horizontally or vertically (with respect to the long side) in the most pleasing position. While Schiffman found an overwhelming preference for a horizontal orientation, consistent with the shape of the visual field, the average ratio of length to width was about 1.9—far from both the Golden Ratio and the visual field's “average rectangle.”

The psychologist Michael Godkewitsch of the University of Toronto cast even greater doubts about the notion of the Golden Rectangle being the most pleasing rectangle. Godkewitsch first pointed out the important fact that average group preferences may not reflect at all the most preferred rectangle for each individual. Often something that is most preferred on the average is not chosen first by anyone. For example, the brand of chocolate that everybody rates second best may on the average be ranked as the best, but nobody will ever buy it! Consequently, first choices provide a more meaningful measure of preference than mean preference rankings. Godkewitsch further noted that if preference for the Golden Ratio is indeed universal and genuine, then it should receive the largest number of first choices, irrespective of which other rectangles the subjects are presented with.

Godkewitsch published in 1974 the results of a study that involved twenty-seven rectangles with length-to-width ratios in three ranges. In one range the Golden Rectangle was next to the most elongated rectangle, in one it was in the middle, and in the third it was next to the shortest rectangle. The results of the experiment showed, according to Godkewitsch, that the preference for the Golden Rectangle was an artifact of its position in the range of rectangles being presented and of the fact that mean preference rankings (rather than first choices) were used in the earlier experiments. Godkewitsch concluded that “the basic question whether there is or is not, in the Western world, a reliable verbally expressed aesthetic preference for a particular ratio between length and width of rectangular shapes can probably be answered negatively. Aesthetic theory has hardly any rationale left to regard the Golden Section as a decisive factor in formal visual beauty.”

Not all agree with Godkewitsch's conclusions. British psychologist Chris McManus published in 1980 the results of a careful study that used the method of paired comparisons, whereby a judgment is made for each pair of rectangles. This method is considered to be superior to other experimental techniques, since there is good evidence that ranking tends to be a process of successive paired comparisons. McManus concluded that “there is moderately good evidence for the phenomenon which Fechner championed, even though Fechner's own method for its demonstration is, at best, highly suspect owing to methodological artifacts.” McManus admitted, however, that “whether the Golden Section per se is important, as opposed to similar ratios (e.g. 1.5, 1.6 or even 1.75), is very unclear.”

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Figure 84

You can test yourself (or your friends) on the question of which rectangle you prefer best. Figure 84 shows a collection of forty-eight rectangles, all having the same height, but with their widths ranging from 0.4 to 2.5 times their height. University of Maine mathematician George Markowsky used this collection in his own informal experiments. Did you pick the Golden Rectangle as your first choice? (It is the fifth from the left in the fourth row.)

GOLDEN MUSIC

Every string quartet and symphony orchestra today still uses Pythagoras' discovery of whole-number relationships among the different musical tones. Furthermore, in the ancient Greek curriculum and up to medieval times, music was considered a part of mathematics, and musicians concentrated their efforts on the understanding of the mathematical basis of tones. The concept of the “music of the spheres” represented a glorious synthesis of music and mathematics, and in the imaginations of philosophers and musicians, it wove the entire cosmos into one grand design that could be perceived only by the gifted few. In the words of the great Roman orator and philosopher Cicero (ca. 106–43 B.C.): “The ears of mortals are filled with this sound, but they are unable to hear it.… You might as well try to stare directly at the Sun, whose rays are much too strong for your eyes.” Only in the twelfth century did music break away from adherence to mathematical prescriptions and formulae. However, even as late as the eighteenth century, the German rationalist philosopher Gottfried Wilhelm Leibnitz (1646–1716) wrote: “Music is a secret arithmetical exercise and the person who indulges in it does not realize that he is manipulating numbers.” Around the same time, the great German composer Johann Sebastian Bach (1685–1750) had a fascination for the kinds of games that can be played with musical notes and numbers. For example, he encrypted his signature in some of his compositions via musical codes. In the old German musical notation, B stood for B-flat and H stood for B-natural, so Bach could spell out his name in musical notes: B-flat, A, C, B-natural. Another encryption Bach used was based on Gematria.

Taking A = 1, B = 2, C = 3, and so on, B-A-C-H = 14 and J-S-B-A-C-H = 41 (because I and J were the same letter in the German alphabet of Bach's time). In his entertaining book Bachanalia (1994), mathematician and Bach enthusiast Eric Altschuler gives numerous examples for the appearances of 14s (encoded BACH) and 41s (encoded JSBACH) in Bach's music that he believes were put there deliberately by Bach. For example, in the first fugue, the C Major Fugue, Book One of Bach's Well Tempered Clavier, the subject has fourteen notes. Also, of the twenty-four entries, twenty-two run all the way to completion and a twenty-third runs almost all the way to completion. Only one entry—the fourteenth—doesn't run anywhere near completion. Altschuler speculates that Bach's obsession with encrypting his signature into his compositions is similar to artists incorporating their own portraits into their paintings or Alfred Hitchcock making a cameo appearance in each of his movies.

Given this historical relationship between music and numbers, it is only natural to wonder whether the Golden Ratio (and Fibonacci numbers) played any role either in the development of musical instruments or in the composition of music.

The violin is an instrument in which the Golden Ratio does feature frequently. Typically, the violin soundbox contains twelve or more arcs of curvature (which make the violin's curves) on each side. The flat arc at the base often is centered at the Golden Section point up the center line.

Some of the best-known violins were made by Antonio Stradivari (1644–1737) of Cremona, Italy. Original drawings (Figure 85) show that Stradivari took special care to place the “eyes” of the f-holes geometrically, at positions determined by the Golden Ratio. Few (if any) elieve that it is the application of the Golden Ratio that gives a Stradivarius violin its superior quality. More often such elements as varnish, sealer, wood, and general craftsmanship are cited as the potential “secret” ingredient. Many experts agree that the popularity of eighteenth-century violins in general stems from their adaptability for use in large concert halls. Most of these experts will also tell you that there is no “secret” in Stradivarius violins—these are simply inimitable works of art, the sum of all the parts that make up their superb craftsmanship.

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Figure 85

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Figure 86

Another musical instrument often mentioned in relation to Fibonacci numbers is the piano. The octave on a piano keyboard consists of thirteen keys, eight white keys and five black keys (Figure 86). The five black keys themselves form one group of two keys and another of three keys. The numbers 2, 3, 5, 8, and 13 happen all to be consecutive Fibonacci numbers. The primacy of the C major scale, for example, is partly due to the fact that it is being played on the piano's white keys. However, the relationship between the piano keyboard and Fibonacci numbers is very probably a red herring. First, note that the chromatic scale (from C to B in the figure), which is fundamental to western music, is really composed of twelve, not thirteen, semitones. The same note, C, is played twice in the octave, to indicate the completion of the cycle. Second, and more important, the arrangement of the keys in two rows, with the sharp and flats being grouped in twos and threes in the upper row, dates back to the early fifteenth century, long before the publication of Pacioli's book and even longer before any serious understanding of Fibonacci numbers.

In the same way that Golden Numberists claim that the Golden Ratio has special aesthetic qualities in the visual arts, they also attribute to it particularly pleasing effects in music. For example, books on the Golden Ratio are quick to point out that many consider the major sixth and the minor sixth to be the most pleasing of musical intervals and that these intervals are related to the Golden Ratio. A pure musical tone is characterized by a fixed frequency (measured in the number of vibrations per second) and a fixed amplitude (which determines the instantaneous loudness). The standard tone used for tuning is A, which vibrates at 440 vibrations per second. A major sixth can be obtained from a combination of A with C, the latter note being produced by a frequency of about 264 vibrations per second. The ratio of the two frequencies 440/264 reduces to 5/3, the ratio of two Fibonacci numbers. A minor sixth can be obtained from a high C (528 vibrations per second) and an E (330 vibrations per second). The ratio in this case, 528/330, reduces to 8/5, which is also a ratio of two Fibonacci numbers and already very close to the Golden Ratio. (The ratios of successive Fibonacci numbers approach the Golden Ratio.) However, as in painting, note that in this case, too, the concept of a “most pleasing musical interval” is rather ambiguous.

Fixed-note instruments like the piano are tuned according to the “tempered scale” popularized by Bach, in which each semitone has an equal frequency ratio to the next semitone, making it easy to play in any key. The ratio of two adjacent frequencies in a well-tempered instrument is 2 (the twelfth root of two). How was this number derived? Its origins actually can be traced to ancient Greece. Recall that an octave is obtained by dividing a string into two equal parts (a frequency ratio of 2:1), and a fifth is produced by a frequency ratio of 3:2 (basically using two-thirds of a string). One of the questions that intrigued the Pythagoreans was whether by repeating the procedure for creating the fifth (applying the 3/2 frequency ratio consecutively) one could generate an integer number of octaves. In mathematical terms, this means asking: Are there any two integers n and m such that (3/2)n is equal to 2m? As it turns out, while no two integers satisfy this equality precisely, n = 12 and m = 7 come pretty close, because of the coincidence that 21/12 nearly equal to 31/19 (the nineteenth root of 3). The twelve frequencies of the octave are therefore all approximate powers of the basic frequency ratio 21/12. Incidentally, you may be amused to note that the ratio of 19/12 is equal to 1.58, not too far from ö.

Another way in which the Golden Ratio could, in principle, contribute to the satisfaction from a piece of music is through the concept of proportional balance. The situation here is somewhat trickier, however, than in the visual arts. A clumsily proportioned painting will instantly stick out in an exhibit like a sore thumb. In music, on the other hand, we have to hear the entire piece before making a judgment. Nevertheless, there is no question that experienced composers design the framework of their music in such a way that not only are the different parts in perfect balance with each other, but also each part in itself provides a fitting container for its musical argument.

We have seen many examples where Golden Ratio enthusiasts have scrutinized the proportions of numerous works in the visual arts to discover potential applications of ö. These aficionados have subjected many musical compositions to the same type of treatment. The results are very similar—alongside a few genuine utilizations of the Golden Ratio as a proportional system, there are many probable misconceptions.

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 analyzed. 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 analyst faces an understandable temptation to conclude that the composer introduced the numerical relationships. Yet, without a firmly documented basis (which is lacking in many cases), such assertions remain dubious.

In 1995, mathematician John F. Putz of Alma College in Michigan examined the question of whether Mozart (1756–1791) 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 (or 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, 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 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 Béla Bartók (1881–1945). A virtuoso pianist and folklorist, Bartók 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 Ernö Lendvai investigated Bartók's music painstakingly and published many books and articles on the subject. Lendvai testifies that “from stylistic analyses of Bartók'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, Bartók's management of the rhythm of the composition provides an excellent example of his use of the Golden Ratio. By analyzing the fugue movement of Bartók's 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, the various themes develop in Fibonacci/Golden Ratio order in terms of the numbers of semitones (Figure 88).

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Figure 87

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Figure 88

Some musicologists do not accept Lendvai's analyses. Lendvai himself admits that Bartók 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 Bartók did not leave any sketches to indicate that he derived rhythms or scales numerically makes any analysis suggestive at best. Also, Lendvai actually dodges the question of whether Bartók used the Golden Ratio consciously. Hungarian musicologist Laszlo Somfai totally discounts the notion that Bartók used the Golden Ratio, in his 1996 book Béla Bartók: Composition, Concepts and Autograph Sources. On the basis of a thorough analysis (which took three decades) of some 3,600 pages, Somfai concludes that Bartók composed without any preconceived musical theories. Other musicologists, including Ruth Tatlow and Paul Griffiths, also refer to Lendvai's study as “dubious.”

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Figure 89

In the interesting book Debussy in Proportion, Roy Howat of Cambridge University argues that the French composer Claude Debussy (1862–1918), 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 (Reflections in the water), a part of the series 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). Howat finds similar divisions in the three symphonic sketches La Mer (The sea), in the piano piece Jardins sous la Pluie (Gardens under the rain), and other works.

I must admit that given the history of 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 in 1903, and in a letter he wrote to his friend André 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 passionate love for her [the sea].” By the time the composition 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 La Mer—perhaps Debussy's most personal and passionate 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 analysis, Howat relies primarily on two pieces of circumstantial evidence: Debussy's close association 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, 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” (assumed 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 à une esthétique scientifique (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 Bartók 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 Bartók to the composer Camille Saint-Saëns, at the time a great celebrity. Bartók declined. Philipp then offered him to meet with the great organist and composer Charles-Marie Widor. Again Bartók declined. “Well,” said Philipp, “if you won't meet Saint-Saëns and Widor, who is there that you would like to know?” “Debussy,” replied Bartók. “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,” Bartók 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 (1885–1935), 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 (1908–1992), 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 (1895–1943) 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 institutions, 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, 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 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.

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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 (1825–1904) expressed the relationship between music and mathematics magnificently in the book 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 passes 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 quantity 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 (1865–1939) 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 passion. 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 love

Of solitary beds, knew what they were,
That passion 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 passionate 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. 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 (1743–1832) was certainly one of the greatest masters of world literature. His all-embracing genius is epitomized in 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 assumed 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:

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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, 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 (1809–1894) 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 shell:

Build thee more stately mansions, O my soul,
As the swift seasons roll! Leave thy low-vaulted past!
Let each new temple, nobler than the last, Shut thee from heaven with a dome more vast,
Till thou at length art free, Leaving thine outgrown shell by life's unresting sea.

There are many examples of numerically based poetic structures. For example, the Divine Comedy, the colossal literary classic by the Italian poet Dante Alighieri (1265–1321), 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. 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 manuscript 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 manuscript 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, 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 arsenal.

George Eckel Duckworth, a professor of classics 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, Duckworth states that “Vergil composed the 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.”

The Roman poet Vergil (70 B.C.—19 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, 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 passages in the Aeneid and computed the ratios of these lengths. Specifically, he measured the number of lines in passages characterized as major (and denoted that number by m) and minor (and denoted the number by m), and calculated the ratios of these numbers. The identification of major and minor parts was based on content. For example, in many passages the major or minor part is a speech and the other part (minor or major respectively) is a narrative or a description. From this analysis Duckworth concluded that the Aeneid contains “hundreds of Golden Mean ratios.” He also noted that an earlier analysis (from 1949) of another Vergil work (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 ö.

Unfortunately, Roger Herz-Fischler has shown that Duckworth's analysis 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 and M, such that M is larger than m. For example, M = 317 could be the number of pages in the last book you read and m = 160 could be your weight in pounds. We could represent these two numbers on a line (with proportional lengths), as in Figure 91. The ratio of the shorter to the longer part is equal to m/M = 160/317 = 0.504, while the ratio of the longer part to the whole is M/(M +m) = 317/477 = 0.665. You will notice that the value of M/(M+m) is closer to 1/ ö = 0.618 than 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, 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 passages, 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), which is biased toward 1/ ö. This is precisely what happened to Duckworth. By making the unfortunate decision to use only the ratio M/(M+m) in his analysis, because he thought that this was “slightly more accurate,” he compressed and distorted the data and made the analysis statistically invalid. In fact, Leonard A. Curchin of the University of Ottawa and Roger Herz-Fischler repeated in 1981 the analysis with Duckworth's data (but using the ratio m/M) and showed that there is no evidence for the Golden Ratio in the 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.

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Figure 91

Sadly, the statement about Vergil and ö 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 assumption 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.