Figure 48

Figure 49 Like Fibonacci before him, Piero wrote the Treatise on the Abacus Treatise on the Abacus mainly to provide the merchants of his day with arithmetic recipes and geometrical rules. In a commercial world that had neither a unique system of weights and measures nor even agreed-upon shapes or sizes of containers, the ability to calculate volumes of figures was an absolute must. However, Piero's mathematical curiosity carried him well beyond the subjects that had simple everyday applications. Accordingly, we find in his books "useless" problems, such as calculating the side of an octahedron inscribed inside a cube or calculating the diameter of the five small circles inscribed inside a circle of a known diameter ( mainly to provide the merchants of his day with arithmetic recipes and geometrical rules. In a commercial world that had neither a unique system of weights and measures nor even agreed-upon shapes or sizes of containers, the ability to calculate volumes of figures was an absolute must. However, Piero's mathematical curiosity carried him well beyond the subjects that had simple everyday applications. Accordingly, we find in his books "useless" problems, such as calculating the side of an octahedron inscribed inside a cube or calculating the diameter of the five small circles inscribed inside a circle of a known diameter (Figure 49). The solution of the latter problem involves a pentagon and, therefore, the Golden Ratio.

Much of Piero's algebraic work was incorporated into a book published by Luca Pacioli (14451517), ent.i.tled Summa de arithmetica, geometria, proportioni et proportionalita Summa de arithmetica, geometria, proportioni et proportionalita (The collected knowledge of arithmetic, geometry, proportion and proportionality). Most of Piero's work on solids, which appeared in Latin, was translated into Italian by the same Luca Pacioli and again incorporated (or, many less tactfully say, simply plagiarized) into his famous book on the Golden Ratio: (The collected knowledge of arithmetic, geometry, proportion and proportionality). Most of Piero's work on solids, which appeared in Latin, was translated into Italian by the same Luca Pacioli and again incorporated (or, many less tactfully say, simply plagiarized) into his famous book on the Golden Ratio: Divina Proportione Divina Proportione (The divine proportion). (The divine proportion).

Who was this highly controversial mathematician Luca Pacioli? Was he the greatest mathematical plagiarist of all times or rather a great communicator of mathematics?

UNSUNG HERO OF THE RENAISSANCE?.

Luca Pacioli was born in 1445 in Borgo San Sepolcro (the same Tuscan town in which Piero della Francesca was born and where he had his workshop). In fact, Pacioli had his early education in Piero's workshop. However, unlike other students who displayed skill in the art of painting, and some, like Pietro Perugino, who were destined to become great painters themselves, he showed greater promise in mathematics. Piero and Pacioli were closely a.s.sociated later in life, as manifested by the fact that Piero included a portrait of Pacioli, as St. Peter Martyr, in a painting of "Madonna and Child with Saints and Angels." Pacioli moved to Venice at a relatively young age and became the tutor of the three sons of a wealthy merchant. In Venice he continued his mathematical education (under the mathematician Domenico Bragadino) and wrote his first textbook on arithmetic.

In the 1470s, Pacioli studied theology and was ordained as a Franciscan friar. Consequently, he is customarily referred to as Fra Luca Pacioli. In the following years, he traveled extensively, teaching mathematics at the universities of Perugia, Zara, Naples, and Rome. During this period he may have also tutored for some time Guidobaldo of Montefeltro, who was to become the Duke of Urbino in 1482.

In what may be the best portrait of a mathematician ever produced, Jacopo de' Barbari (14401515) depicts Luca Pacioli giving a lesson in

Figure 50 geometry to a pupil (Figure 50; the painting is currently in the Galleria n.a.z.ionale di Capodimonte in Naples). One of the Platonic solids, a dodecahedron, is seen on the right resting on top of Pacioli's book Summa. Summa. Pacioli himself, dressed in his friar robes and almost resembling a geometrical solid, is shown copying a diagram from volume XIII of Euclid's Pacioli himself, dressed in his friar robes and almost resembling a geometrical solid, is shown copying a diagram from volume XIII of Euclid's Elements. Elements. A transparent polyhedron known as a rhombicuboctahedron (one of the Archimedean Solids, with twenty-six faces of which eighteen are squares and eight equilateral triangles), half filled with water and hanging in mid-air, symbolizes the purity and timelessness of mathematics. The artist has captured the reflections and refractions from this gla.s.s polyhedron with extraordinary skill. The ident.i.ty of the second person in the painting has been the subject of some debate. One of the suggestions is that the student is Duke Guidobaldo. British mathematician Nick MacKinnon raised an interesting possibility in 1993. In a well-researched article ent.i.tled "The Portrait of Fra Luca Pacioli," which appeared in the A transparent polyhedron known as a rhombicuboctahedron (one of the Archimedean Solids, with twenty-six faces of which eighteen are squares and eight equilateral triangles), half filled with water and hanging in mid-air, symbolizes the purity and timelessness of mathematics. The artist has captured the reflections and refractions from this gla.s.s polyhedron with extraordinary skill. The ident.i.ty of the second person in the painting has been the subject of some debate. One of the suggestions is that the student is Duke Guidobaldo. British mathematician Nick MacKinnon raised an interesting possibility in 1993. In a well-researched article ent.i.tled "The Portrait of Fra Luca Pacioli," which appeared in the Mathematical Gazette Mathematical Gazette, MacKinnon suggests that the figure is that of the famous German painter Albrecht Durer, who had great interest in geometry and perspective (and to whose relationship with Pacioli we shall return later in this chapter). The face of the student does in fact bear a striking resemblance to Durer s self-portrait.

Pacioli returned to Borgo San Sepolcro in 1489, after having been granted some special privileges by the Pope, only to encounter jealousy from the existing religious establishment. For about two years he was even banned from teaching. In 1494, Pacioli went to Venice to publish his Summa Summa, which he dedicated to Duke Guidobaldo. Encyclopedic in nature and scope (some 600 pages), the Summa Summa compiled the mathematical knowledge of the time in arithmetic, algebra, geometry, and trigonometry. In this book, Pacioli borrows freely (usually with an appropriate acknowledgment) problems on the icosahedron and dodecahedron from Piero's compiled the mathematical knowledge of the time in arithmetic, algebra, geometry, and trigonometry. In this book, Pacioli borrows freely (usually with an appropriate acknowledgment) problems on the icosahedron and dodecahedron from Piero's Trattato Trattato and problems in algebra and geometry from Fibonacci and others. Identifying Fibonacci as his main source, Pacioli states that when no other is quoted, the work belongs to Leonardus Pisa.n.u.s. An interesting part of the and problems in algebra and geometry from Fibonacci and others. Identifying Fibonacci as his main source, Pacioli states that when no other is quoted, the work belongs to Leonardus Pisa.n.u.s. An interesting part of the Summa Summa is on double-entry accounting, a method of record keeping that lets you track where money comes from and where it goes. While Pacioli did not invent this system but merely summarized the practices of Venetian merchants during the Renaissance, this is considered to be the first published book on accounting. Pacioli's desire to "give the trader without delay information as to his a.s.sets and liabilities" thus gained him the t.i.tle "Father of Accounting," and accountants from all over the world celebrated in 1994 (in Sansepolcro, as the town is now known) the 500 is on double-entry accounting, a method of record keeping that lets you track where money comes from and where it goes. While Pacioli did not invent this system but merely summarized the practices of Venetian merchants during the Renaissance, this is considered to be the first published book on accounting. Pacioli's desire to "give the trader without delay information as to his a.s.sets and liabilities" thus gained him the t.i.tle "Father of Accounting," and accountants from all over the world celebrated in 1994 (in Sansepolcro, as the town is now known) the 500th anniversary of the anniversary of the Summa. Summa.

In 1480, Ludovico Sforza became effectively the Duke of Milan. In fact, he was only the regent of the real seven-year-old duke, following an episode of political intrigue and murder. Determined to make his court a home for scholars and artists, Ludovico invited Leonardo da Vinci in 1482 as a "painter and engineer of the duke." Leonardo had considerable interest in geometry, especially for its practical applications in mechanics. In his words: "Mechanics is the paradise of the mathematical sciences, because by means of it one comes to the fruits of mathematics." Consequently, Leonardo was probably the one who induced the duke to invite Pacioli to join the court, as a teacher of mathematics, in 1496. Undoubtedly, Leonardo learned some of his geometry from Pacioli, while he infused in the latter a greater appreciation for art.

During his stay in Milan, Pacioli completed work on his three-volume treatise Divina Proportione Divina Proportione (The divine proportion), which was eventually published in Venice in 1509. The first volume, (The divine proportion), which was eventually published in Venice in 1509. The first volume, Compendio de Divina Proportione Compendio de Divina Proportione (Compendium of the divine proportion), contains a detailed summary of the properties of the Golden Ratio (which Pacioli refers to as the "Divine Proportion") and a study of Platonic solids and other polyhedra. On the first page of (Compendium of the divine proportion), contains a detailed summary of the properties of the Golden Ratio (which Pacioli refers to as the "Divine Proportion") and a study of Platonic solids and other polyhedra. On the first page of The Divine Proportion The Divine Proportion Pacioli declares somewhat grandiloquently that this is: "A work necessary for all the clear-sighted and inquiring human minds, in which everyone who loves to study philosophy, perspective, painting, sculpture, architecture, music and other mathematical disciplines will find a very delicate, subtle and admirable teaching and will delight in diverse questions touching upon a very secret science." Pacioli declares somewhat grandiloquently that this is: "A work necessary for all the clear-sighted and inquiring human minds, in which everyone who loves to study philosophy, perspective, painting, sculpture, architecture, music and other mathematical disciplines will find a very delicate, subtle and admirable teaching and will delight in diverse questions touching upon a very secret science."

Pacioli dedicated the first volume of The Divine Proportion The Divine Proportion to Ludovico Sforza, and in the fifth chapter he lists five reasons why he believes that the appropriate name for the Golden Ratio should be to Ludovico Sforza, and in the fifth chapter he lists five reasons why he believes that the appropriate name for the Golden Ratio should be The Divine Proportion. The Divine Proportion.

1. "That it is one only and not more." Pacioli compares the unique value of the Golden Ratio to the fact that unity "is the supreme epithet of G.o.d himself."

2. Pacioli finds a similarity between the fact that the definition of the Golden Ratio involves precisely three lengths (AC, CB (AC, CB, and AB AB in in Figure 24 Figure 24) and the existence of a Holy Trinity, of Father, Son, and Holy Ghost.

3. To Pacioli, the incomprehensibility of G.o.d and the fact that the Golden Ratio is an irrational number are equivalent. In his own words: Just like G.o.d cannot be properly defined, nor can be understood through words, likewise our proportion cannot be ever designated by intelligible numbers, nor can it be expressed by any rational quant.i.ty, but always remains concealed and secret, and is called irrational by the mathematicians."

4. Pacioli compares the omnipresence and invariability of G.o.d to the self-similarity a.s.sociated with the Golden Ratio-that its value is always the same and does not depend on the length of the line being divided or the size of the pentagon in which ratios of lengths are calculated.

5. The fifth reason reveals an even more Platonic view of existence than Plato himself expressed. Pacioli states that just as G.o.d conferred being to the entire cosmos through the fifth essence, represented by the dodecahedron, so does the Golden Ratio confer being to the dodecahedron, since one cannot construct the dodecahedron without the Golden Ratio. He adds that it is impossible to compare the other four Platonic solids (representing earth, water, air, and fire) to each other without the Golden Ratio.

In the book itself, Pacioli raves ceaselessly about the properties of the Golden Ratio. He a.n.a.lyzes in succession what he calls the thirteen different "effects" of the "divine proportion" and attaches to each one of these "effects" adjectives like "essential," "singular," "wonderful," "supreme," and so on. For example, he regards the "effect" that Golden Rectangles can be inscribed in the icosahedron (Figure 22) as "incomprehensible." Pacioli stops at thirteen "effects," concluding that, "for the sake of salvation, the list must end," because thirteen men were present at the table at the Last Supper.

Figure 51 There is no doubt that Pacioli had a great interest in the arts and that his intention in The Divine Proportion The Divine Proportion was partly to perfect their mathematical basis. His opening statement on the book's first page expresses his desire to reveal to artists, through the Golden Ratio, the "secret" of harmonic forms. To ensure its attractiveness, Pacioli secured for was partly to perfect their mathematical basis. His opening statement on the book's first page expresses his desire to reveal to artists, through the Golden Ratio, the "secret" of harmonic forms. To ensure its attractiveness, Pacioli secured for The Divine Proportion The Divine Proportion the services of the dream ill.u.s.trator of any author-Leonardo da Vinci himself provided sixty ill.u.s.trations of solids, depicted in both skeletal ( the services of the dream ill.u.s.trator of any author-Leonardo da Vinci himself provided sixty ill.u.s.trations of solids, depicted in both skeletal (Figure 51) and solid forms (Figure 52). Pacioli was quick to express his grat.i.tude; he wrote about Leonardo's contribution: "the most excellent painter in perspective, architect, musician, the man endowed with all virtues, Leonardo da Vinci, who deduced and elaborated a series of diagrams of regular solids." The text itself, however, falls somewhat short of its declared high goals.

While the book starts with a sensational flourish, it continues with a rather conventional set of mathematical formulae loosely wrapped up in philosophical definitions.

Figure 52 The second book in the Divina Proportione Divina Proportione is a treatise on proportion and its application to architecture and the structure of the human body. Pacioli's treatment was largely based on the work of the eclectic Roman architect Marcus Vitruvius Pollio (ca. 7025 is a treatise on proportion and its application to architecture and the structure of the human body. Pacioli's treatment was largely based on the work of the eclectic Roman architect Marcus Vitruvius Pollio (ca. 7025 B.C. B.C.). Vitruvius wrote: ... in the human body the central point is naturally the navel. For if a man be placed flat on his back, with his hands and feet extended, and a pair of compa.s.ses centered at his navel, the fingers and toes of his two hands and feet will touch the circ.u.mference of a circle described therefrom. And just as the human body yields a circular outline, so too a square figure may be found from it. For if we measure the distance from the soles of the feet to the top of the head, and then apply that measure to the outstretched arms, the breadth will be found to be the same as the height, as in the case of plane surfaces which are perfectly square.

This pa.s.sage was taken by the Renaissance scholars as yet another demonstration of the link between the organic and geometrical basis of beauty, and it led to the concept of the "Vitruvian man," drawn beautifully by Leonardo (Figure 53; currently in the Galleria dell'Accademia, Venice). Accordingly, Pacioli's book also starts with a discussion of proportions in the human body, "since in the human body every sort of pro portion and proportionality can be found, produced at the beck of the all-Highest through the inner mysteries of nature." However, contrary to frequent claims in the literature, Pacioli does not insist on the Golden Ratio as determining the pro portions of all works of art. Rather, when dealing with de sign and proportion, he specifically advocates the Vitruvian system, which is based on simple (rational) ratios. Author Roger Herz-Fischler traced the fallacy of the Golden Ratio as Pacioli's canon for proportion to a false statement made in the 1799 edition of Histoire de Mathematiques Histoire de Mathematiques (History of mathematics) by the French mathematicians Jean Etienne Montucla and Jerome de Lalande. (History of mathematics) by the French mathematicians Jean Etienne Montucla and Jerome de Lalande.

Figure 53 The third volume of the Divina Divina (A short book divided into three partial tracts on the five regular bodies) is essentially an Italian word-by-word translation of Piero's (A short book divided into three partial tracts on the five regular bodies) is essentially an Italian word-by-word translation of Piero's Five Regular Solids Five Regular Solids composed in Latin. The fact that nowhere in the text does Pacioli acknowledge that he was merely the translator of the book provoked a violent denunciation from art historian Giorgio Vasari. Vasari writes about Piero della Francesca that he composed in Latin. The fact that nowhere in the text does Pacioli acknowledge that he was merely the translator of the book provoked a violent denunciation from art historian Giorgio Vasari. Vasari writes about Piero della Francesca that he ... was regarded as a great master of the problems of regular solids, both arithmetical and geometrical, but he was prevented by the blindness that overtook him in his old age, and then by death, from making known his brilliant researches and the many books he had written. The man who should have done his utmost to enhance Piero's reputation and fame, since Piero taught him all he knew, shamefully and wickedly tried to obliterate his teacher's name and to usurp for himself the honor which belonged entirely to Piero; for he published under his own name, which was Fra Luca dal Borgo [Pacioli], all the researches done by that admirable old man, who was a great painter as well as an expert in the sciences.

So, was Pacioli a plagiarist? Quite possibly, although in Summa Summa he did render homage to Piero, whom he regarded as "the monarch of our times in painting" and one who "is familiar to you in that copious work which he composed on the art of painting and on the force of the line in perspective." he did render homage to Piero, whom he regarded as "the monarch of our times in painting" and one who "is familiar to you in that copious work which he composed on the art of painting and on the force of the line in perspective."

R. Emmett Taylor (18891956) published in 1942 a book ent.i.tled No Royal Road: Luca Pacioli and His Times. No Royal Road: Luca Pacioli and His Times. In this book, Taylor adopts a very sympathetic att.i.tude toward Pacioli, and he argues that, on the basis of style, Pacioli may have had nothing to do with the third book of the In this book, Taylor adopts a very sympathetic att.i.tude toward Pacioli, and he argues that, on the basis of style, Pacioli may have had nothing to do with the third book of the Divina Divina and it was just appended to Pacioli's work. and it was just appended to Pacioli's work.

Be that as it may, there is no question that if not for Pacioli's printed printed books, Piero's ideas and mathematical constructions (which were not published in printed form) would not have reached the wide circulation that they eventually achieved. Furthermore, up until Pacioli's time, the Golden Ratio had been known only by rather intimidating names, such as "extreme and mean ratio" or "proportion having a mean and two extremes," and the concept itself was familiar only to mathematicians. The publication of books, Piero's ideas and mathematical constructions (which were not published in printed form) would not have reached the wide circulation that they eventually achieved. Furthermore, up until Pacioli's time, the Golden Ratio had been known only by rather intimidating names, such as "extreme and mean ratio" or "proportion having a mean and two extremes," and the concept itself was familiar only to mathematicians. The publication of The Divine Proportion The Divine Proportion in 1509 gave a new topical interest to the Golden Ratio. The concept could now be considered with fresh attention, because its publication in book form identified it as worthy of respect. The infusion of theological/philosophical meaning into the in 1509 gave a new topical interest to the Golden Ratio. The concept could now be considered with fresh attention, because its publication in book form identified it as worthy of respect. The infusion of theological/philosophical meaning into the name name ("Divine Proportion") also singled out the Golden Ratio as a mathematical topic into which an increasingly eclectic group of intellectuals could delve. Finally, with Pacioli's book, the Golden Ratio started to become available to artists in theoretical treatises that were not overly mathematical, that they could actually use. ("Divine Proportion") also singled out the Golden Ratio as a mathematical topic into which an increasingly eclectic group of intellectuals could delve. Finally, with Pacioli's book, the Golden Ratio started to become available to artists in theoretical treatises that were not overly mathematical, that they could actually use.

Leonardo da Vinci's drawings of polyhedra for The Divine Proportion The Divine Proportion, drawn (in Pacioli's words) with his "ineffable left hand," had their own impact. These were probably the first ill.u.s.trations of skeletal solids, which allowed for an easy visual distinction between front and back. Leonardo may have drawn the polyhedra from a series of wooden models, since records of the Council Hall in Florence indicate that a set of Pacioli's wooden models was purchased by the city for public display. In addition to the diagrams for Pacioli's book, we can find sketches of many solids scattered throughout Leonardo's notebooks. In one place he presents an approximate geometrical construction of the pentagon. This fusion of art and mathematics reaches its climax in Leonardo's Trattato della pittura Trattato della pittura (Treatise on painting; organized by Francesco Melzi, who inherited Leonardo's ma.n.u.scripts), which opens with the admonition: "Let no one who is not a mathematician read my works"-hardly a likely statement to be found in any contemporary art handbook! (Treatise on painting; organized by Francesco Melzi, who inherited Leonardo's ma.n.u.scripts), which opens with the admonition: "Let no one who is not a mathematician read my works"-hardly a likely statement to be found in any contemporary art handbook!

The drawings of solids in the Divina Divina have also inspired some of the have also inspired some of the intarsia intarsia constructed by Fra Giovanni da Verona around 1520. Intarsia represent a special art form, in which elaborate flat mosaics are constructed of pieces of inlaid wood. Fra Giovanni's intarsia panels include an icosahedron, which almost certainly used Leonardo's skeletal drawing as a template. constructed by Fra Giovanni da Verona around 1520. Intarsia represent a special art form, in which elaborate flat mosaics are constructed of pieces of inlaid wood. Fra Giovanni's intarsia panels include an icosahedron, which almost certainly used Leonardo's skeletal drawing as a template.

The lives of Leonardo and Pacioli continued to be somewhat intertwined even after the completion of The Divine Proportion. The Divine Proportion. In October of 1499 the two men fled Milan, after the French army, led by King Louis XII, captured that city. After spending brief periods of time in Mantua and Venice, both settled for some time in Florence. During the period of their friendship, Pacioli's name became a.s.sociated with two other major mathematical works-a translation into Latin of Euclid's In October of 1499 the two men fled Milan, after the French army, led by King Louis XII, captured that city. After spending brief periods of time in Mantua and Venice, both settled for some time in Florence. During the period of their friendship, Pacioli's name became a.s.sociated with two other major mathematical works-a translation into Latin of Euclid's Elements Elements and an unpublished work on recreational mathematics. Pacioli's translation of the and an unpublished work on recreational mathematics. Pacioli's translation of the Elements Elements was an annotated version, based on an earlier translation by Campa.n.u.s of Novara (12201296), which appeared in printed form in Venice in 1482 (and which was the first was an annotated version, based on an earlier translation by Campa.n.u.s of Novara (12201296), which appeared in printed form in Venice in 1482 (and which was the first printed printed version). Pacioli did not manage to publish his compilation of problems in recreational mathematics and proverbs version). Pacioli did not manage to publish his compilation of problems in recreational mathematics and proverbs De Viribus Quant.i.tatis De Viribus Quant.i.tatis (The powers of numbers) before his death in 1517. This work was a collaborative project between Pacioli and Leonardo, and Leonardo's own notes contain many of the problems in (The powers of numbers) before his death in 1517. This work was a collaborative project between Pacioli and Leonardo, and Leonardo's own notes contain many of the problems in De Viribus. De Viribus.

Fra Luca Pacioli certainly cannot be remembered for originality, but his influence on the development of mathematics in general, and on the history of the Golden Ratio in particular, cannot be denied.

MELANCHOLY.

Another major Renaissance figure who entertained an intriguing combination of artistic and mathematical interests is the German painter Albrecht Durer.

Durer is considered by many to be the greatest German artist of the Renaissance. He was born on May 21, 1471, in the Imperial Free City of Nurnberg, to a hardworking jeweler. At age nineteen, he already demonstrated talents and ability as a painter and woodcut designer that surpa.s.sed those of his teacher, the leading painter and book ill.u.s.trator in Nurnberg, Michael Wolgemut. Durer therefore embarked on four years of travel, during which he became convinced that mathematics, "the most precise, logical, and graphically constructive of the sciences," has to be an important ingredient of art.

Consequently, after a short stay in Nurnberg, during which he married Agnes Frey, the daughter of a successful craftsman, he left again for Italy, with the goal of expanding both his artistic and mathematical horizons. His visit to Venice in 14941495 seems to have accomplished precisely that. Durer's meeting with the founder of the Venetian School of painting, Giovanni Bellini (ca. 14261516), left a great impression on the young artist, and his admiration for Bellini persisted throughout his life. At the same time, Durer's encounter with Jacopo de' Barbari, who painted the wonderful portrait of Luca Pacioli (Figure 50), acquainted him with Pacioli's mathematical work and its relevance for art. In particular, de' Barbari showed Durer two figures, male and female, that were constructed by geometrical methods, and the experience motivated Durer to investigate human movement and proportions. Durer probably met with Pacioli himself in Bologna, during a second visit to Italy in 1505 to 1507. In a letter from that period, he describes his visit to Bologna as being "for art's sake, for there is one there who will instruct me in the secret art of perspective." The mysterious "one" in Bologna has been interpreted by many as referring to Pacioli, although other names, such as those of the outstanding architect Donato di Angelo Bramante (14441514) and the architectural theorist Sebastiano Serlio (14751554), have also been suggested. During the same Italian trip Durer also met again with Jacopo de' Barbari. This second visit, though, was marked by Durer's somewhat paranoiac nervousness about harm that might be done to him by artists envious of his fame. For example, he refused invitations to dinner for fear that someone might try to poison him.

Starting in 1495, Durer showed a serious interest in mathematics. He spent much time studying the Elements Elements (a Latin translation of which he obtained in Venice, although he spoke little Latin), Pacioli's works on mathematics and art, and the important works on architecture, proportion, and perspective by the Roman architect Vitruvius and by the Italian architect and theorist Leon Baptista Alberti (14041472). (a Latin translation of which he obtained in Venice, although he spoke little Latin), Pacioli's works on mathematics and art, and the important works on architecture, proportion, and perspective by the Roman architect Vitruvius and by the Italian architect and theorist Leon Baptista Alberti (14041472).

Durer's contributions to the history of the Golden Ratio come both in the form of written work and through his art. His major treatise, Unterweisung der Messung mit dem Zirkel und Richtscheit Unterweisung der Messung mit dem Zirkel und Richtscheit (Treatise on measurement with compa.s.s and ruler), was published in 1525 and was one of the first books on mathematics published in German. In it Durer complains that too many artists are ignorant of geometry, "without which no one can either be or become an absolute artist." The first of the four books of the (Treatise on measurement with compa.s.s and ruler), was published in 1525 and was one of the first books on mathematics published in German. In it Durer complains that too many artists are ignorant of geometry, "without which no one can either be or become an absolute artist." The first of the four books of the Treatise Treatise gives detailed descriptions of the construction of various curves, including the logarithmic (or equiangular) spiral, which is, as we have seen, closely related to the Golden Ratio. The second book contains precise and approximate methods for the construction of many polygons, including two constructions of the pentagon (one exact and one approximate). The Platonic solids, as well as other solids, some of Durer's own invention, together with the theory of perspective and of shadows, are discussed in the fourth book. Durer's book was not intended to be used as a textbook of geometry-for example, he gives only one example of a proof. Rather, Durer always starts with a practical application and then continues with an exposition of the very basic theoretical aspects. The book contains some of the earliest presentations of nets of polyhedra. These are plane sheets on which the surfaces of the polyhedra are drawn in such a way that the figures can be cut out (as single pieces) and folded to form the three-dimensional solids. gives detailed descriptions of the construction of various curves, including the logarithmic (or equiangular) spiral, which is, as we have seen, closely related to the Golden Ratio. The second book contains precise and approximate methods for the construction of many polygons, including two constructions of the pentagon (one exact and one approximate). The Platonic solids, as well as other solids, some of Durer's own invention, together with the theory of perspective and of shadows, are discussed in the fourth book. Durer's book was not intended to be used as a textbook of geometry-for example, he gives only one example of a proof. Rather, Durer always starts with a practical application and then continues with an exposition of the very basic theoretical aspects. The book contains some of the earliest presentations of nets of polyhedra. These are plane sheets on which the surfaces of the polyhedra are drawn in such a way that the figures can be cut out (as single pieces) and folded to form the three-dimensional solids.

Durer's ill.u.s.tration for the net of a dodecahedron (related as we know to the Golden Ratio) is shown in Figure 54

Figure 54 Durer mingled his virtuosity in woodcuts and engravings with his interest in mathematics in the enigmatic allegory "Melencolia I" (Figure 55). This is one of the trio of master engravings (the other two being "Knight, Death and Devil," and "St. Jerome in His Study"). It has been suggested that Durer created the picture in a fit of melancholy after the death of his mother. The central figure in "Melencolia" is a winged female seated listless and dispirited on a stone ledge. In her right hand she holds a compa.s.s, opened for measuring. Most of the objects in the engraving have multiple symbolic meanings, and entire articles have been devoted to their interpretation. The pot on the fire in the middle left and the scale at the top are thought to represent alchemy. The "magic square" on the upper right (in which every row, column, diagonal, the four central numbers, and the numbers in the four corners add up to 34; incidentally, a Fibonacci number) is thought to represent mathematics (Figure 56).

The middle entries in the bottom row make 1514, the date of the engraving. The inclusion of the magic square probably represents Pacioli's influence, since Pacioli's De Viribus De Viribus included a collection of magic squares. The main purport of the engraving, with its geometrical figures, keys, bat, seascape, and so on, seems to be the representation of the melancholy that engulfs the artist or thinker, amid doubts in the success of her endeavors, while time, represented by the hourgla.s.s at the top, goes on. included a collection of magic squares. The main purport of the engraving, with its geometrical figures, keys, bat, seascape, and so on, seems to be the representation of the melancholy that engulfs the artist or thinker, amid doubts in the success of her endeavors, while time, represented by the hourgla.s.s at the top, goes on.

Figure 55 The strange solid in the middle left of the engraving has been the topic of serious discussion and various reconstruction attempts. At first sight it looks like a cube from which two opposite corners have been sliced off (which inspired some Freudian interpretations), but this appears not to be the case. Most researchers conclude that the figure is what is known as a rhombohedron (a six-sided solid with each side shaped as a rhombus; Figure 57 Figure 57), which has been truncated so that it can be circ.u.mscribed by a sphere. When resting on one of its triangular faces, its front fits precisely into the magic square. The angles in the face of the solid have also been a matter of some debate. While many suggest 72 , which would relate the figure to the Golden Ratio (see Figure 25 Figure 25), Dutch crystallographer C. H. MacGillavry concluded on the basis of perspectival a.n.a.lysis that the angles are of 80 . The perplexing qualities of this solid are summarized beautifully in an article by T. Lynch that appeared in 1982 in the Journal of the Warburg and Courtauld Inst.i.tutes. the Journal of the Warburg and Courtauld Inst.i.tutes. The author concludes: "As a representation of polyhedra was seen as one of the main problems of perspective geometry, what better way could Durer prove his ability in this field, than to include in an engraving a shape that was so new and perhaps even unique, and to leave the question of what it was, and where it came from, for other geometricians to solve?" The author concludes: "As a representation of polyhedra was seen as one of the main problems of perspective geometry, what better way could Durer prove his ability in this field, than to include in an engraving a shape that was so new and perhaps even unique, and to leave the question of what it was, and where it came from, for other geometricians to solve?"

Figure 56

Figure 57 With the exception of the influential work of Pacioli and the mathematical/artistic interpretations of the painters Leonardo and Durer, the sixteenth century brought about no other surprising developments in the story of the Golden Ratio. While a few mathematicians, including the Italian Rafael Bombelli (15261572) and the Spanish Franciscus Flussates Candalla (15021594) used the Golden Ratio in a variety of problems involving the pentagon and the Platonic solids, the more exciting applications had to await the very end of the century.

However, the works of Pacioli, Durer, and others revived the interest in Platonism and Pythagoreanism. Suddenly the Renaissance intellectuals saw a real opportunity to relate mathematics and rational logic to the universe around them, in the spirit of the Platonic worldview. Concepts like the "Divine Proportion" built, on one hand, a bridge between mathematics and the workings of the cosmos and, on the other, a relation among physics, theology, and metaphysics. The person who, in his ideas and works, exemplifies more than any other this fascinating blending of mathematics and mysticism is Johannes Kepler.

MYSTERIUM COSMOGRAPHIc.u.m.

Johannes Kepler is best remembered as an outstanding astronomer responsible (among other things) for the three laws of planetary motion that bear his name. But Kepler was also a talented mathematician, a speculative metaphysician, and a prolific author. Born at a time of great political upheaval and religious chaos, Kepler's education, life, and thinking were critically shaped by the events around him. Kepler was born on December 27, 1571, in the Imperial Free City of Weil der Stadt, Germany, in his grandfather Sebald's house. His father, Heinrich, a mercenary soldier, was absent from home throughout most of Kepler's childhood, and during his short visits he was (in Kepler's words): "a wrongdoer, abrupt and quarrelsome." The father left home when Kepler was about sixteen, never to be seen again. He is supposed to have partic.i.p.ated in a naval war for the Kingdom of Naples and to have died on his way home. Consequently, Kepler was raised mostly by his mother, Katharina, who worked in her father's inn. Katharina herself was a rather strange and unpleasant woman, who gathered herbs and believed in their magical healing powers. A series of events involving personal grudges, unfortunate gossip, and greed eventually led to her arrest at old age in 1620, and to an indictment of witchcraft. Such accusations were not uncommon at that time-no fewer than thirty-eight women were executed for witchcraft in Weil der Stadt in the years between 1615 and 1629. Kepler, who was already well known at the time of her arrest, reacted to the news of his mother's trial "with unutterable distress." He effectively took charge of her defense, enlisting the help of the legal faculty at the University of Tubingen. The charges against Katharina Kepler were eventually dismissed after a long ordeal, mainly in light of her own testimony under the threat of great pain and torture. This story conveys the atmosphere and the intellectual confusion that prevailed during the period of Kepler's scientific work. Kepler was born into a society that experienced (only fifty years earlier) Martin Luther's breaking with the Catholic church, proclaiming that humans' sole justification before G.o.d was faith. That society was also about to embark on the b.l.o.o.d.y and insane conflict known as the Thirty Years' War. We can only be astonished how, with this background and with the violent ups and downs of his tumultuous life, Kepler was able to produce a discovery that is regarded by many as the true birth of modern science.

Kepler started his studies at the higher seminary at Maulbronn and then won a scholarship from the Duke of Wurttemberg to attend the Lutheran seminary at the University of Tubingen in 1589. The two topics that attracted him most, and which in his mind were closely related, were theology and mathematics. At that time astronomy was considered a part of mathematics, and Kepler's teacher of astronomy was the prominent astronomer Michael Mastlin (15501631), with whom he continued to maintain contact even after leaving Tubingen.

In his formal lessons, Mastlin must have taught only the traditional Ptolemaic or geocentric system, in which the Moon, Mercury, Venus, the Sun, Mars, Jupiter, and Saturn all revolved around the stationary Earth. Mastlin, however, was fully aware of Nicolaus Copernicus' heliocentric system, which was published in 1543, and in private he did discuss the merits of such a system with his favorite student, Kepler. In the Copernican system, six planets (including Earth, but not including the Moon, which was no longer considered a planet but rather a "satellite ") revolved around the Sun. In the same way that from a moving car you can observe only the relative motions of the other cars, in the Copernican system, much of what appears to be the motion of the planets simply reflects the motion of Earth itself.

Kepler seems to have taken an immediate liking to the Copernican system. The fundamental idea of this cosmology, that of a central Sun surrounded by a sphere of the fixed stars with a s.p.a.ce between the sphere and the Sun, fit perfectly into his view of the cosmos. Being a profoundly religious person, Kepler believed that the universe represents a reflection of its Creator. The unity of the Sun, the stars, and the intervening s.p.a.ce symbolized to him an equivalence to the Holy Trinity of the Father, Son, and the Holy Spirit.

While Kepler graduated with distinction from the faculty of arts and was close to finishing his theological studies, something happened to change his profession from that of a pastor to that of a mathematics teacher. The Protestant seminary in Graz, Austria, asked the University of Tubingen to recommend a replacement for one of their math teachers who had pa.s.sed away, and the university selected Kepler. In March of 1594 Kepler therefore began, unwillingly, a month-long trip to Graz, in the Austrian province of Styria.

Realizing that fate had forced upon him the career of a mathematician, Kepler became determined to fulfill what he regarded as his Christian duty-to understand G.o.d's creation, the universe. Accordingly, he delved into the translations of the Elements Elements and the works of the Alexandrian geometers Apollonius and Pappus. Accepting the general principle of the Copernican heliocentric system, he set out to search for answers to the following two major questions: Why were there precisely six planets? and What was it that determined that the planetary orbits would be s.p.a.ced as they are? These "why" and "what" questions were entirely new in the astronomical vocabulary. Unlike the astronomers before him, who satisfied themselves with simply recording the observed positions of the planets, Kepler was seeking a theory that would explain it all. He expressed this new approach to human inquiry beautifully: and the works of the Alexandrian geometers Apollonius and Pappus. Accepting the general principle of the Copernican heliocentric system, he set out to search for answers to the following two major questions: Why were there precisely six planets? and What was it that determined that the planetary orbits would be s.p.a.ced as they are? These "why" and "what" questions were entirely new in the astronomical vocabulary. Unlike the astronomers before him, who satisfied themselves with simply recording the observed positions of the planets, Kepler was seeking a theory that would explain it all. He expressed this new approach to human inquiry beautifully: In all acquisition of knowledge it happens that, starting out from those things which impinge on the senses, we are carried by the operation of the mind to higher things which cannot be grasped by any sharpness of the senses. The same thing happens also in the business of astronomy, in which we first of all perceive with our eyes the various positions of the planets at different times, and reasoning then imposes itself on these observations and leads the mind to recognition of the form of the universe.

But, wondered Kepler, what tool would G.o.d use to design His universe?

The first glimpse of what was to become his preposterously fantastic explanation to these cosmic questions dawned on Kepler on July 19, 1595, as he was trying to explain the conjunctions of the outer planets, Jupiter and Saturn (when the two bodies have the same celestial coordinate). Basically, he realized that if he in scribed an equilateral triangle within a circle (with its vertices lying on the circle) and another circle inside the triangle (touching the midpoints of the sides; Figure 58 Figure 58), then the ratio of the radius of the larger circle to that of the smaller one was about the same as the ratio of the sizes of Saturn's...o...b..t to Jupiter's...o...b..t. Continuing with this line of thought, he decided that to get to the orbit of Mars (the next planet closer to the Sun), he would need to use the next geometrical figure-a square-inscribed inside the small circle. Doing this, however, did not produce the right size. Kepler did not give up, and being already along a path inspired by the Platonic view, that "G.o.d ever geometrizes," it was only natural for him to take the next geometrical step and try three-dimensional figures. The latter exercise resulted in Kepler's first use of geometrical objects related to the Golden Ratio.

Figure 58 Kepler gave the answer to the two questions that intrigued him in his first treatise, known as Mysterium Cosmographic.u.m Mysterium Cosmographic.u.m (The cosmic mystery), which was published in 1597. The full t.i.tle, given on the t.i.tle page of the book ( (The cosmic mystery), which was published in 1597. The full t.i.tle, given on the t.i.tle page of the book (Figure 59; although the publication date reads 1596, the book was published the following year) reads: "A precursor to cosmographical dissertations, containing the cosmic mystery of the admirable proportions of the Celestial Spheres, and of the True and Proper Causes of their Numbers, Sizes, and Periodic Motions of the Heavens, Demonstrated by the Five Regular Geometric Solids."

Kepler's answer to the question of why there were six planets was simple: because there are precisely five regular Platonic solids. Taken as boundaries, the solids determine six s.p.a.cings (with an outer spherical boundary corresponding to the heaven of the fixed stars). Furthermore, Kepler's model was designed so as to answer at the same time the question of the sizes of the orbits as well. In his words:

Figure 59 The Earth's sphere is the measure of all other orbits. Circ.u.mscribe a dodecahedron around it. The sphere surrounding it will be that of Mars. Circ.u.mscribe a tetrahedron around Mars. The sphere surrounding it will be that of Jupiter. Circ.u.mscribe a cube around Jupiter. The surrounding sphere will be that of Saturn. Now, inscribe an icosahedron inside the orbit of the Earth. The sphere inscribed in it will be that of Venus. Inscribe an octahedron inside Venus. The sphere inscribed in it will be that of Mercury. There you have the basis for the number of the planets.

Figure 60 shows a schematic from shows a schematic from Mysterium Cosmographic.u.m Mysterium Cosmographic.u.m, which ill.u.s.trates Kepler's cosmological model. Kepler explained at some length why he made the particular a.s.sociations between the Platonic solids and the planets, on the basis of their geometrical, astrological, and metaphysical attributes. He ordered the solids based on relationships to the sphere, a.s.suming that the differences between the sphere and the other solids reflected the distinction between the creator and his creations. Similarly, the cube is characterized by a single single angle-the right angle. To Kepler this symbolized the solitude a.s.sociated with Saturn, and so on. More generally, astrology was relevant to Kepler because "man is the goal of the universe and of all creation," and the metaphysical approach was justified by the fact that "the mathematical things are the causes of the physical because G.o.d from the beginning of time carried within himself in simple and divine abstraction the mathematical objects as prototypes for the materially planned quant.i.ties." angle-the right angle. To Kepler this symbolized the solitude a.s.sociated with Saturn, and so on. More generally, astrology was relevant to Kepler because "man is the goal of the universe and of all creation," and the metaphysical approach was justified by the fact that "the mathematical things are the causes of the physical because G.o.d from the beginning of time carried within himself in simple and divine abstraction the mathematical objects as prototypes for the materially planned quant.i.ties."

Earth's position was chosen so as to separate the solids that can stand upright (i.e., cube, tetrahedron, and dodecahedron), from those that "float" (i.e., octahedron and icosahedron).

Figure 60 The s.p.a.cings of the planets resulting from this model agreed reasonably well for some planets but were significantly discrepant for others (although the discrepancies were usually no more than 10 percent). Kepler, absolutely convinced of the correctness of his model, attributed most of the inconsistencies to inaccuracies in the measured orbits. He sent copies of the book to various astronomers for comments, including a copy to one of the foremost figures of the time, the Danish Tycho Brahe (15461601). One copy even made it into the hands of the great Galileo Galilei (15641642), who informed Kepler that he too believed in Copernicus' model but lamented the fact that "among a vast number (for such is the number of fools)" Copernicus "appeared fit to be ridiculed and hissed off the stage."

Needless to say, Kepler's cosmological model, which was based on the Platonic solids, was not only absolutely wrong, but it was crazy even for Kepler's time. The discovery of the planets Ura.n.u.s (next after Saturn in terms of increasing distance from the Sun) in 1781 and Neptune (next after Ura.n.u.s) in 1846 put the final nails into the coffin of an already moribund idea. Nevertheless, the importance of this model in the history of science cannot be overemphasized. As astronomer Owen Gingerich has put it in his biographical article on Kepler: "Seldom in history has so wrong a book been so seminal in directing the future course of science." Kepler took the Pythagorean idea of a cosmos that can be explained by mathematics a huge step forward. He developed an actual mathematical model mathematical model for the universe, which on one hand was based on existing observational measurements and on the other was for the universe, which on one hand was based on existing observational measurements and on the other was falsifiable falsifiable by observations that could be made subsequently. These are precisely the ingredients required by the "scientific method"-the organized approach to explaining observed facts with a model of nature. An idealized scientific method begins with the collection of facts, a model is then proposed, and the model's predictions are tested through experiments or further observations. This process is sometimes summed up by the sequence: induction, deduction, verification. In fact, Kepler was even given a chance to make a successful prediction on the basis of his theory. In 1610, Galileo discovered with his telescope four new celestial bodies in the Solar System. Had these proven to be planets, it would have dealt a fatal blow to Kepler's theory already during his lifetime. However, to Kepler's relief, the new bodies turned out to be satellites (like our Moon) around Jupiter, not new planets revolving around the Sun. by observations that could be made subsequently. These are precisely the ingredients required by the "scientific method"-the organized approach to explaining observed facts with a model of nature. An idealized scientific method begins with the collection of facts, a model is then proposed, and the model's predictions are tested through experiments or further observations. This process is sometimes summed up by the sequence: induction, deduction, verification. In fact, Kepler was even given a chance to make a successful prediction on the basis of his theory. In 1610, Galileo discovered with his telescope four new celestial bodies in the Solar System. Had these proven to be planets, it would have dealt a fatal blow to Kepler's theory already during his lifetime. However, to Kepler's relief, the new bodies turned out to be satellites (like our Moon) around Jupiter, not new planets revolving around the Sun.

Present-day physical theories that aim at explaining the existence of all the elementary (subatomic) particles and the basic interactions among them rely on mathematical symmetries in a very similar fashion to Kepler's theory relying on the symmetry properties of the Platonic solids to explain the number and properties of the planets. Kepler's model had something else in common with today's fundamental theory of the universe: Both theories are by their very nature reductionistic- reductionistic- they attempt to explain many phenomena in terms of a few fundamental laws. For example, Kepler's model deduced both the number of planets and the properties of their orbits from the Platonic solids. Similarly, modern theories known as string theories use basic ent.i.ties (strings) which are extremely tiny (more than a billion billion times smaller than the atomic nucleus) to deduce the properties of all the elementary particles. Like a violin string, the strings can vibrate and produce a variety of "tones," and all the known elementary particles simply represent these different tones. they attempt to explain many phenomena in terms of a few fundamental laws. For example, Kepler's model deduced both the number of planets and the properties of their orbits from the Platonic solids. Similarly, modern theories known as string theories use basic ent.i.ties (strings) which are extremely tiny (more than a billion billion times smaller than the atomic nucleus) to deduce the properties of all the elementary particles. Like a violin string, the strings can vibrate and produce a variety of "tones," and all the known elementary particles simply represent these different tones.

Kepler's continued interest in the Golden Ratio during his stay in Graz produced another interesting result. In October 1597, he wrote to Mastlin, his former professor, about the following theorem: "If on a line which is divided in extreme and mean ratio one constructs a right angled triangle, such that the right angle is on the perpendicular put at the section point, then the smaller leg will equal the larger segment of the divided line." Kepler's statement is represented by Figure 61 Figure 61. Line AB AB is divided in a Golden Ratio by point is divided in a Golden Ratio by point C. C. Kepler constructs a right-angled triangle Kepler constructs a right-angled triangle ADB ADB on on AB AB as a hypotenuse, with the right angle as a hypotenuse, with the right angle D D being on the perpendicular put at the Golden Section point being on the perpendicular put at the Golden Section point C. C. He then proves that He then proves that BD BD (the shorter side of the right angle) is equal to (the shorter side of the right angle) is equal to AC AC (the longer segment of the line divided in Golden Ratio). What makes this particular triangle special (other than the use of the Golden Ratio) is that in 1855 it was used by pyramidologist Friedrich Rober in one of the false theories explaining the appearance of the Golden Ratio in the design of the pyramids. Rober was not aware of Kepler's work, but he used a similar construction to support his view that the "divine proportion" played a crucial role in architecture. (the longer segment of the line divided in Golden Ratio). What makes this particular triangle special (other than the use of the Golden Ratio) is that in 1855 it was used by pyramidologist Friedrich Rober in one of the false theories explaining the appearance of the Golden Ratio in the design of the pyramids. Rober was not aware of Kepler's work, but he used a similar construction to support his view that the "divine proportion" played a crucial role in architecture.

Figure 61

Figure 62 Kepler's Mysterium Cosmographic.u.m Mysterium Cosmographic.u.m led to a meeting between him and Tycho Brahe in Prague-at the time the seat of the Holy Roman Emperor. The meeting took place on February 4, 1600, and was the prelude to Kepler's moving to Prague as Tycho's a.s.sistant in October of the same year (after being forced out of Catholic Graz because of his Lutheran faith). When Brahe died on October 24, 1601, Kepler became the Imperial Mathematician. led to a meeting between him and Tycho Brahe in Prague-at the time the seat of the Holy Roman Emperor. The meeting took place on February 4, 1600, and was the prelude to Kepler's moving to Prague as Tycho's a.s.sistant in October of the same year (after being forced out of Catholic Graz because of his Lutheran faith). When Brahe died on October 24, 1601, Kepler became the Imperial Mathematician.

Tycho left a huge body of observations, in particular of the orbit of Mars, and Kepler used these data to discover the first two laws of planetary motions named after him. Kepler's First Law states that the orbits of the known planets around the Sun are not exact circles but rather ellipses, with the Sun at one focus (Figure 62; the elongation of the ellipse is greatly exaggerated). An ellipse has two points called foci, such that the sum of the distances of any point on the ellipse from the two foci is the same. Kepler's Second Law establishes that the planet moves fastest when it is closest to the Sun (the point known as perihelion) and slowest when it is farthest (aphelion), in such a way that the line joining the planet to the Sun sweeps equal areas in equal time intervals (Figure 62). The question of what causes Kepler's laws to hold true was the outstanding unsolved problem of science for almost seventy years after Kepler published the laws. It took the genius of Isaac Newton (16421727) to deduce that the force holding the planets in their orbits is gravity. Newton explained Kepler's laws by solving together the laws that describe the motion of bodies with the law of universal gravitation. He showed that elliptical orbits with varying speeds (as described by Kepler's laws) represent one possible solution to these equations.

Kepler's heroic efforts in the calculations of Mars' orbit (many hundreds of sheets of arithmetic and their interpretation; dubbed by him as "my warfare with Mars") are considered by many researchers as signifying the birth of modern science. In particular, at one point he found a circular orbit that matched nearly all of Tycho's observations. In two cases, however, this...o...b..t predicted a position that differed from the observations by about a quarter of the angular diameter of a full moon. Kepler wrote about this event: "If I had believed that we could ignore these eight minutes [of arc], I would have patched up my hypothesis in Chapter 16 accordingly. Now, since it was not permissible to disregard, those eight minutes alone pointed the path to a complete reformation in astronomy."

Kepler's years in Prague were extremely productive in both astronomy and mathematics. In 1604, he discovered a "new" star, now known as Kepler's Supernova. A supernova is a powerful stellar explosion, in which a star nearing the end of its life ejects its outer layers at a speed of ten thousand miles per second. In our own Milky Way galaxy, one such explosion is expected to occur on the average every one hundred years. Indeed, Tycho discovered a supernova in 1572 (Tycho's Supernova), and Kepler discovered one in 1604. Since then, however, for unclear reasons, no other supernova has been discovered in the Milky Way (although one exploded apparently unnoticed in the 1660s). Astronomers remark jokingly that maybe this paucity of supernovae simply reflects the fact that there have been no truly great astronomers since Tycho and Kepler.

In June 2001, I visited the house in which Kepler lived in Prague, at 4 Karlova Street. Today, this is a busy shopping street, and it is easy to miss the rusty plaque above the number 4, which states that Kepler lived there from 1605 to 1612. One of the shop owners just below Kepler's apartment did not even know that one of the greatest astronomers of all times had lived there. The rather sad-looking inner courtyard does contain a small sculpture of the armillary sphere with Kepler's name written across it, and another plaque is located near the mailboxes. Kepler's apartment itself, however, is not marked in any special way and is not open to the public, being occupied by one of the many families who live in the residential upper floors.

Kepler's mathematical work produced a few more highlights in the history of the Golden Ratio. In the text of a letter that he wrote in 1608 to a professor in Leipzig, we find that he discovered the relation between Fibonacci numbers and the Golden Ratio. He repeats the contents of that discovery in an essay tracing the reason for the six-cornered shape of snowflakes. Kepler writes: Of the two regular solids, the dodecahedron and the icosahedron... both of these solids, and indeed the structure of the pentagon itself, cannot be formed without the divine proportion as the geometers of today call it. It is so arranged that the two lesser terms of a progressive series together const.i.tute the third, and the two last, when added, make the immediately subsequent term and so on to infinity, as the same proportion continues unbroken... the further we advance from the number one, the more perfect the example becomes. Let the smallest numbers be 1 and 1... add them, and the sum will be 2; add to this the latter of the 1s, result 3; add 2 to this, and get 5; add 3, get 8; 5 to 8, 13; 8 to 13, 21. As 5 is to 8, so 8 is to 13, approximately, and as 8 to 13, so 13 is to 21, approximately.

In other words, Kepler discovered that the ratio of consecutive Fibonacci numbers converges to the Golden Ratio. In fact, he also discovered another interesting property of the Fibonacci numbers: that the square of any term differs by 1 at most from the product of the two adjacent terms in the sequence. For example, since the sequence is: 1, 1, 2, 3, 5, 8, 13, 21, 34,..., if we look at 32 = 9, it is only different by 1 from the product of the two terms that are adjacent to 3, 2 5 = 10. Similarly, 13 = 9, it is only different by 1 from the product of the two terms that are adjacent to 3, 2 5 = 10. Similarly, 132 = 169 is different by 1 from 821 = 168, and so on. = 169 is different by 1 from 821 = 168, and so on.

This particular property of Fibonacci numbers gives rise to a puzzling paradox first presented by the great creator of mathematical puzzles, Sam Loyd (18411911).

Consider the square of eight units on the side (area of 82 = 64) in = 64) in Figure 63 Figure 63. Now dissect it into four parts as indicated. The four pieces can be rea.s.sembled (Figure 64) to form a rectangle of sides 13 and 5 with an area of 65! Where did the extra square unit come from? The solution to the paradox is in the fact that the pieces actually do not fit exactly along the rectangle's long diagonal-there is a narrow s.p.a.ce (a long thin parallelogram hidden under the thick line marking the long diagonal in Figure 64 Figure 64) with an area of one square unit. Of course, 8 is a Fibonacci number, and its square (82 = 64) differs by 1 from the product of its two adjacent Fibonacci numbers (135 = 65)-the property discovered by Kepler. = 64) differs by 1 from the product of its two adjacent Fibonacci numbers (135 = 65)-the property discovered by Kep