Part 4 (2/2)

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 scholars.h.i.+p 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 finis.h.i.+ng 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 relations.h.i.+ps 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 fas.h.i.+on 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

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