Figure 66

Figure 67 Interestingly, two other men who played significant roles in the history of the Golden Ratio before Kepler (and whose work was described in previous chapters) also showed interest in tiling-the tenth-century mathematician Abu'l-Wafa and the painter Albrecht Durer. Both of them presented designs containing figures with fivefold symmetry. (An example of Durer's work is shown in Figure 67 Figure 67.) The fifth book of Harmony of the World Harmony of the World contains Kepler's most significant result in astronomy-Kepler's Third Law of planetary motion. This represents the culmination of all of his agonizing over the sizes of the orbits of the planets and their periods of revolution around the Sun. Twenty-five years of work have been condensed into one incredibly simple law: The ratio of the period squared to the semimajor axis cubed is the same for all the planets (the semimajor axis is half the long axis of the ellipse; contains Kepler's most significant result in astronomy-Kepler's Third Law of planetary motion. This represents the culmination of all of his agonizing over the sizes of the orbits of the planets and their periods of revolution around the Sun. Twenty-five years of work have been condensed into one incredibly simple law: The ratio of the period squared to the semimajor axis cubed is the same for all the planets (the semimajor axis is half the long axis of the ellipse; Figure 62 Figure 62). Kepler discovered this seminal law, which served as the basis for Newton's formulation of the law of universal gravitation, only when Harmony of the World Harmony of the World was already in press. Unable to control his exhilaration he announced: "I have stolen the golden vessels of the Egyptians to build a tabernacle for my G.o.d from them, far away from the borders of Egypt." The essence of the law follows naturally from the law of gravity: The force is stronger the closer the planet is to the Sun, so inner planets must move faster to avoid falling toward the Sun. was already in press. Unable to control his exhilaration he announced: "I have stolen the golden vessels of the Egyptians to build a tabernacle for my G.o.d from them, far away from the borders of Egypt." The essence of the law follows naturally from the law of gravity: The force is stronger the closer the planet is to the Sun, so inner planets must move faster to avoid falling toward the Sun.

In 1626, Kepler moved to Ulm and completed the Rudolphine Tables Rudolphine Tables, the most extensive and accurate astronomical tables produced until that time. While I was visiting the University of Vienna in June 2001, my hosts showed me in the observatory's library a first edition of the tables (147 copies are known to exist today). The frontispiece of the book (Figure 68), a symbolic representation of the history of astronomy, contains at the lower left corner what may be Kepler's only self-portrait (Figure 69). It shows Kepler working by candlelight, under a banner listing his important publications.

Kepler died at noon on November 15, 1630, and was buried in Regensburg. Befitting his turbulent life, wars have totally destroyed his tomb, without a trace. Luckily, a sketch of the gravestone made by a friend survived, and it contains Kepler's epitaph:

Figure 68

Figure 69 I used to measure the heavens, Now the Earth's shadows I measure My mind was in the heavens, Now the shadow of my body rests here.

Today, Kepler's originality and productivity are almost incomprehensible. We should realize that this was a man who endured unimaginable personal hardships, including the loss of three of his children in less than six months during 1617 and 1618. The English poet John Donne (15721631) perhaps described him best when he said that Kepler "hath received it into his care, that no new thing should be done in heaven without his knowledge."

Painting isn't an aesthetic operation; it's a form of magic designed as a mediator between this strange hostile world and us.-PABLO P PICa.s.sO (18811973) (18811973) 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 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? (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 a.s.sertions concerning the employment of the Golden Ratio in painting are directly a.s.sociated 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 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 (12671337). 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).

Figure 70

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. 12551319), and the "Santa Trinita Madonna" by the Florentine painter Cenni di Pepo, known as Cimabue (ca. 12401302). 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 . 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) Fascinating Fibonaccis) with rather thick lines, making any measurement rather uncertain, but, in fact, the upper horizontal side is placed somewhat arbitrarily. 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 The Divine Proportion brought the ratio to wider attention. 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. 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 a.s.serts that 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 a.s.serts that The Divine Proportion 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 doc.u.mented 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., 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 doc.u.mented 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 Giotto by Francesca Flores D'Arcais; by Francesca Flores D'Arcais; Cimabue 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. 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 doc.u.mented) 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."

Figure 72

Figure 73 The case of the two versions of "Madonna of the Rocks" (one in the Louvre in Paris, Figure 72 Figure 72, and the other in the National Gallery in London, Figure 73 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 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 a.s.sumed 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 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. (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 partic.i.p.ation 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) 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. 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 hors.e.m.e.n in red chalk, which are a.s.sociated with Leonardo's "Battle of Anghiari," were added to the same page around 15031504.

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-

Figure 74

Figure 75 teresting manifestation. In an article that appeared in 1995 in the Scientific American 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 a.n.a.lysts 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 ill.u.s.trator for the Oivina 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 (18591891) 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 a.n.a.logy 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 Figure 76; currently in The Metropolitan Museum of Art, New York). Note in particular the right angle formed by the bal.u.s.trade 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 h ha philosophie des Beaux-Arts appliquee a la peinture Beaux-Arts appliquee a 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." (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."

Figure 76 Golden Ratio aficionados often present a.n.a.lyses of "The Parade" (as well as other paintings, such as "The Circus") to "prove" the use of . Even in the beautiful book Mathematics Mathematics, by Bergamini and the editors of Life Magazine Life Magazine, we find: "La Parade "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 (18811965), who was also the "art expert" quoted by Bergamini. Ghyka published two influential books, Esthetique des proportions dans la nature et dans les arts Esthetique des proportions dans la nature et dans les arts (Aesthetics of proportions in nature and in the arts; 1927) and (Aesthetics of proportions in nature and in the arts; 1927) and Le Nombre d'Or: Rites et rythmes pytagoriciens dans le developpement de la civilisation occidentale Le Nombre d'Or: Rites et rythmes pytagoriciens dans le developpement 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. (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 a.n.a.lysis of the entire composition of this and other paintings by Seurat, as well as paintings by the Symbolist painter Pierre Puvis de Chavannes (18241898), 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 a.n.a.lysis 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 (18591926) 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 Serusier (18641927). Serusier was born in Paris, and after studying philosophy he entered the famous art school Academie Julian. A meeting with the painters Paul Gaugin and emile Bernard converted him to their expressive use of color and symbolist views. Together with the post-Impressionist painters Pierre Bon-nard, edouard 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 a.s.sociated with the group. Serusier 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 (18681946). 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 ent.i.ties such as the circle, equilateral triangle, and hexagon. Serusier 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 (18701943) wrote biographical notes on Serusier, from which we learn that those "measures" employed by Father Lenz included the Golden Ratio. Even though Serusier admits that his initial studies of the mathematics of Beuron were "not all plain sailing," the Golden Ratio and the story of its potential a.s.sociation with the Great Pyramid and Greek artworks made it also into Serusier's important art theory book L'ABC de la Peinture L'ABC de la Peinture (The ABC of painting). (The ABC of painting).

While Serusier'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 Serusier, 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 Pica.s.so's painting "Les Demoiselles d Avignon" and Braque's "Nude." In revolt against the pa.s.sionate use of color and form in Expressionism, Pica.s.so and Braque developed an austere, almost monochrome style that deliberately rejected any subject matter that was likely to evoke emotional a.s.sociations. Objects like musical instruments and even human figures were dissected into faceted geometrical planes, which were then combined in shifting perspectives. This a.n.a.lysis 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 ent.i.tled "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 (18871927) and the Lithuanian-born sculptor Jacques (Chaim Jacob) Lipchitz (18911973) 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 Figure 77), in which the two artists used Kepler's triangle (which is based on the Golden Ratio; see Figure 61 Figure 61) for the production of the desired proportions.

Figure 77 Another artist who used the Golden Ratio in the early 1920s was the Italian painter Gino Severini (18831966). 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 Figure 78).

Figure 78 Russian Cubist painter Maria Vorobeva, 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 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 Pica.s.so, 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 Pica.s.so, 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 (18671924). In a series of articles and books, Hambidge defined two types of symmetry in cla.s.sical 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-edouard Jeanneret, 18871965).

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 19161917, Jeanneret moved to Paris, where he met Amedee 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 Galerie Thomas. More precisely, two canvases by Jeanneret were hung alongside many more paintings by Ozenfant. They called themselves "Purists," and ent.i.tled their catalog Apres le Cubisme Apres le Cubisme (After cubism). Purism invoked Piero della Francesca and the Platonic aesthetic theory in its a.s.sertion that "the work of art must not be accidental, exceptional, impressionistic, inorganic, protestatory, picturesque, but on the contrary, generalized, static, expressive of the invariant." (After cubism). Purism invoked Piero della Francesca and the Platonic aesthetic theory in its a.s.sertion 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 a.n.a.lysis 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 Aesthetics of Proportions in Nature and in the Arts, and his Golden Number, Pythagorean Rites and Rhythms Golden Number, Pythagorean Rites and Rhythms (1931) only enhanced the mystical aspects of even further. Le Corbusier's fascination with (1931) only enhanced the mystical aspects of even further. Le Corbusier's fascination with Aesthetics 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." 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. B.C. "Man is the measure of all things." Accordingly, in the spirit of the Vitruvian man ( "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).

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; 75), 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 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").

Figure 80

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 s.p.a.ces. In a world with an increasing need for ma.s.s production, the Modulor was supposed to provide the model for standardization. Le Corbusier's two books, Le Modulor Le Modulor (published in 1948) and (published in 1948) and Modulor II 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 mora.s.s 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." (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 mora.s.s 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 part.i.tioning 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 s.p.a.ces already furnished by the frames."

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 in one of his paintings. 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 ent.i.tled "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 (18721944) is best known for his geometric, non.o.bjective 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.

Figure 83 Not surprisingly, per haps, Mondrian's geometrical compositions attracted quite a bit of Golden Numberist speculation. In Mathematics 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 The Painter's Secret Geometry, a.s.serting 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 a.n.a.lyses 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 the real expert-Yves-Alain Bois of Harvard University, who coauth.o.r.ed the book real expert-Yves-Alain Bois of Harvard University, who coauth.o.r.ed the book Mondrian 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 19181919, 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." 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 19181919, 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 mult.i.tude of psychological experiments and a vast literature.

THE SENSES DELIGHT IN THINGS DULY PROPORTIONED.

With the words in the t.i.tle of this section, Italian scholastic philosopher St. Thomas Aquinas (ca. 12251274) 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 Korpers Neue Lehre von den Proportionen des menschlichen Korpers (The latest theory of proportions in the human body) and culminated in the publication (after Zeising's death) of a ma.s.sive book, (The latest theory of proportions in the human body) and culminated in the publication (after Zeising's death) of a ma.s.sive book, Der Goldne Schnitt 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 part.i.tion 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." (The golden section), in 1884. In these works, Zeising combined his own interpretation of Pythagorean and Vitruvian ideas to argue that "the part.i.tion 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 (18011887), 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 (14971543), 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 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. (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 n.o.body accuses Fechner of altering the results, some speculate that he may have subconsciously produced circ.u.mstances 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 . 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 G.o.dkewitsch of the University of Toronto cast even greater doubts about the notion of the Golden Rectangle being the most pleasing rectangle. G.o.dkewitsch 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 n.o.body will ever buy it! Consequently, first choices provide a more meaningful measure of preference than mean preference rankings. G.o.dkewitsch 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.

G.o.dkewitsch 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 G.o.dkewitsch, 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. G.o.dkewitsch concluded that "the basic question whether there is or is not, in the Western world, a reliable verbally expressed aesthetic 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." 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 G.o.dkewitsch's conclusions. British psychologist Chris McMa.n.u.s 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. McMa.n.u.s 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." McMa.n.u.s admitted, however, that "whether the Golden Section per se per se is important, as opposed to similar ratios (e.g. 1.5, 1.6 or even 1.75), is very unclear." is important, as opposed to similar ratios (e.g. 1.5, 1.6 or even 1.75), is very unclear."

Figure 84 You can test yourself (or your friends) on the question of which rectangle you prefer best. Figure 84 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.) 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. 10643 B.C. 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 (16461716) 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 (16851750) 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 Bacha.n.a.lia Bacha.n.a.lia (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 (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 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 Hitchc.o.c.k 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 (16441737) 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.

Figure 85

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