Part 3 (2/2)
In a paper that appeared in 1984 in Journal de Physique Journal de Physique, a team of scientists led by N. Rivier from the Universite de Provence in Ma.r.s.eille, France, used a simple mathematical algorithm to show that when a growth angle equal to the Golden Angle is used, structures that closely resemble real sunflowers are obtained. (See Figure 36 Figure 36.) Rivier and his collaborators suggested that this provided an answer to the question that had been posed in the cla.s.sical work of biologist Sir D'Arcy Wentworth Thompson. In his monumental book On Growth and Form On Growth and Form (first published in 1917 and revised in 1942), Thompson wonders: ”... and not the least curious feature of the case [phyllotaxis] is the limited, even the small number of possible arrangements which we observe and recognize.” Rivier's team found that the requirements of (first published in 1917 and revised in 1942), Thompson wonders: ”... and not the least curious feature of the case [phyllotaxis] is the limited, even the small number of possible arrangements which we observe and recognize.” Rivier's team found that the requirements of h.o.m.ogeneity h.o.m.ogeneity (that the structure is the same everywhere) and of (that the structure is the same everywhere) and of self-similarity self-similarity (that when one examines the structure on different scales from small to large, it looks precisely the same) limit drastically the number of possible structures. These two properties may be sufficient to explain the preponderance of Fibonacci numbers and the Golden Ratio in phyllotaxis, but they still do not offer any physical cause. (that when one examines the structure on different scales from small to large, it looks precisely the same) limit drastically the number of possible structures. These two properties may be sufficient to explain the preponderance of Fibonacci numbers and the Golden Ratio in phyllotaxis, but they still do not offer any physical cause.
Figure 36 The best clues for a possible dynamical cause of phyllotaxis came not from botany but from experiments in physics by L. S. Levitov (in 1991) and by Stephane Douady and Yves Couder (in 1992 to 1996). The experiment by Douady and Couder is particularly fascinating. They held a dish full of silicone oil in a magnetic field that was stronger near the dish's edge than at the center. Drops of a magnetic fluid, which act like tiny bar magnets, were dropped periodically at the center of the dish. The tiny magnets repelled each other and were pushed radially by the magnetic field gradient. Douady and Couder found patterns that oscillated about, but generally converged to, a spiral on which the Golden Angle separated successive drops. Physical systems usually settle into states that minimize the energy. The suggestion is therefore that phyllotaxis simply represents a state of minimal energy for a system of mutually repelling buds. Other models, in which leaves appear at the points of the highest concentration of some nutrient, also tend to produce separations equal to the Golden Angle.
I hope that the next time you eat a pineapple, send a red rose to a loved one, or admire van Gogh's sunflower paintings, you will remember that the growth pattern of these plants embodies this wonderful number we call the Golden Ratio. Realize, however, that plant growth also depends on factors other than optimal s.p.a.cing. Consequently, the phyllotaxis rules I have described cannot be taken as applying to all circ.u.mstances, like a law of nature. Rather, in the words of the famous Canadian mathematician c.o.xeter, they are ”only a fascinatingly prevalent tendency.” tendency.”
Botany is not the only place in nature where the Golden Ratio and Fibonacci numbers can be found. They appear in phenomena covering a range in sizes from the microscopic to that of giant galaxies. Often that appearance takes the form of a magnificent spiral.
ALTHOUGH CHANGED, I RISE AGAIN THE SAME.
No family in the history of mathematics has produced as many celebrated mathematicians (thirteen in total!) as did the Bernoulli family. Disconcerted by the Spanish Fury (the ravaging riot in the Netherlands by Spanish soldiers), the family fled to Basel, Switzerland, from the Catholic Spanish Netherlands. Three members of the family, the brothers Jacques (16541705) and Jean (16671748), and the latter's second son, Daniel (17001782), stood out head and shoulders above the rest. Strangely, the Bernoullis were almost equally famous for their bitter interfamilial rivalries as they were for their numerous mathematical achievements. In one case, the exchanges between Jacques and Jean became particularly acrimonious. The feud was sparked by a dispute over a solution to a famous problem in mechanics. The problem, known as the brachistochrone (from the Greek brachistos brachistos, ”shortest,” and chronos chronos, ”time”), was to find the curve along which a particle acted on by the force of gravity will pa.s.s in the shortest time from one point to another. The two brothers proposed the same solution independently, but Jean's derivation was incorrect, and he later attempted to present Jacques' derivation as his own. The sad consequence of this chain of events was that Jeanne became a professor in Groningen and did not set foot in Basel until his brother's death.
Jacques Bernoulli's a.s.sociation with the Golden Ratio comes through another famous curve. He devoted a treatise ent.i.tled Spira Mirabilis Spira Mirabilis (Wonderful spiral) to a particular type of spiral shape. Jacques was so impressed with the beauty of the curve known as a logarithmic spiral ( (Wonderful spiral) to a particular type of spiral shape. Jacques was so impressed with the beauty of the curve known as a logarithmic spiral (Figure 37; the name was derived from the way in which the radius grows as we move around the curve clockwise) that he asked that this shape, and the motto he a.s.signed to it: ”Eadem mutato resurgo” (although changed, I rise again the same), be engraved on his tombstone.
Figure 37 The motto describes a fundamental property unique to the logarithmic spiral-it does not alter its shape as its size increases. This feature is known as self-similarity Fascinated by this property, Jacques wrote that the logarithmic spiral ”may be used as a symbol, either of fort.i.tude and constancy in adversity, or of the human body, which after all its changes, even after death, will be restored to its exact and perfect self” f you think about it for a moment, this is precisely the property required for many growth phenomena in nature. For example, as the mollusk inside the sh.e.l.l of the chambered nautilus (Figure 4) grows in size, it constructs larger and larger chambers, sealing off the smaller unused ones. Each increment in the length of the sh.e.l.l is accompanied by a proportional increase in its radius, so that the shape remains unchanged. Consequently, the nautilus sees an identical ”home” throughout its lifetime, and it does not need, for example, to adjust its balance as it matures. The latter property applies also to rams, the horns of which are also in the shape of logarithmic spirals (although they do not lie in a plane), and to the curve of elephants' tusks. Increasing by acc.u.mulation from within itself, the logarithmic spiral grows wider, with the distance between its ”coils” increasing, as it moves away from the source, known as the pole. Specifically, turning by equal angles increases the distance from the pole by equal ratios. If we were, with the aid of a microscope, to enlarge the coils that are invisible to the naked eye to the size of Figure 37 Figure 37, they would fit precisely on the larger spiral. This property distinguishes the logarithmic spiral from another common spiral known as the Archimedean spiral, after the famous Greek mathematician Archimedes (ca. 287212 B.C. B.C.), who described it extensively in his book On Spirals. On Spirals. We can see an Archimedean spiral in the side of a roll of paper towels or a rope coiled on the floor. In this type of spiral, the distance between successive coils remains always the same. As a result of a mistake that surely would have caused Jacques Bernoulli much grief, the mason who prepared Bernoulli's tombstone engraved on it an Archimedean rather than a logarithmic spiral. We can see an Archimedean spiral in the side of a roll of paper towels or a rope coiled on the floor. In this type of spiral, the distance between successive coils remains always the same. As a result of a mistake that surely would have caused Jacques Bernoulli much grief, the mason who prepared Bernoulli's tombstone engraved on it an Archimedean rather than a logarithmic spiral.
Nature loves logarithmic spirals. From sunflowers, seash.e.l.ls, and whirlpools, to hurricanes and giant spiral galaxies, it seems that nature chose this marvelous shape as its favorite ”ornament.” The constant shape of the logarithmic spiral on all size scales reveals itself beautifully in nature in the shapes of minuscule fossils or unicellular organisms known as foraminifera. Although the spiral sh.e.l.ls in this case are composite structures (and not one continuous tube), X-ray images of the internal structure of these fossils show that the shape of the logarithmic spiral remained essentially unchanged for millions of years.
In his cla.s.sic book The Curves of Life The Curves of Life (1914), English author and editor Theodore Andrea Cook gives numerous examples of the appearance of spirals (not just logarithmic) in nature and art. He discusses spirals in things as diverse as climbing plants, the human body, staircases, and Maori tattoos. In explaining his motivation for writing the book, Cook writes: ”for the existence of these chapters upon spiral formations no other apology is needed than the interest and beauty of an investigation.” (1914), English author and editor Theodore Andrea Cook gives numerous examples of the appearance of spirals (not just logarithmic) in nature and art. He discusses spirals in things as diverse as climbing plants, the human body, staircases, and Maori tattoos. In explaining his motivation for writing the book, Cook writes: ”for the existence of these chapters upon spiral formations no other apology is needed than the interest and beauty of an investigation.”
Artists have also not failed to see the beauty of logarithmic spirals. In Leonardo da Vinci's study for the mythological subject ”Leda and the Swan,” for example, he draws the hair arranged in the shape of a nearly logarithmic spiral (Figure 38). Leonardo repeats this shape many times in his study of spirals in clouds and water in the impressive series of sketches for the ”Deluge.” In that work, he combined his scientific observations of frightening floods with the allegorical aspects of destructive forces descended from heaven. Describing the violent flow of water Leonardo wrote: ”The sudden waters rush into the pond that contains them, striking the various obstacles with swirling eddies.... The momentum of the circular waves flying from the point of impact hurls them in the way of other circular waves moving in the opposite direction.”
Figure 38
Figure 39 Twentieth-century designer and ill.u.s.trator Edward B. Edwards developed hundreds of decorative designs based on the logarithmic spiral; many can be seen in his book Pattern and Design with Dynamic Symmetry Pattern and Design with Dynamic Symmetry (an example is shown in (an example is shown in Figure 39 Figure 39).
The logarithmic spiral and the Golden Ratio go hand in hand. Examine again the series of nested Golden Rectangles obtained when you snip off squares from a Golden Rectangle (Figure 40; we encountered this property already in Chapter 4). If you connect the successive points where these ”whirling squares” divide the sides in Golden Ratios, you obtain a logarithmic spiral that coils inward toward the pole (the point given by the intersection of the diagonals in Figure 25 Figure 25, which was called fancifully ”the eye of G.o.d”).
Figure 40
Figure 41 You can also obtain a logarithmic spiral from a Golden Triangle. We saw in Chapter 4 that if you start from a Golden Triangle (an isosceles triangle in which the side is in Golden Ratio to the base) and bisect a base angle, you get a smaller Golden Triangle. If you continue the process of bisecting the base angle ad infinitum, you will generate a series of whirling triangles. Connecting the vertices of the Golden Triangles in the progression will trace a logarithmic spiral (Figure 41).
The logarithmic spiral is also known as the equiangular spiral. equiangular spiral. This name was coined in 1638 by the French mathematician and philosopher Rene Descartes (15961650), after whom we name the numbers used to locate a point in the plane (with respect to two axes)-Cartesian coordinates. The name ”equiangular” reflects another unique property of the logarithmic spiral. If you draw a straight line from the pole to any point on the This name was coined in 1638 by the French mathematician and philosopher Rene Descartes (15961650), after whom we name the numbers used to locate a point in the plane (with respect to two axes)-Cartesian coordinates. The name ”equiangular” reflects another unique property of the logarithmic spiral. If you draw a straight line from the pole to any point on the
Figure 42 curve, it cuts the curve at precisely the same angle (Figure 42). Falcons use this property when attacking their prey. Peregrine falcons are some of the fastest birds on Earth, plummeting toward their targets at speeds of up to two hundred miles per hour. But they could fly even faster if they would just fly straight instead of following a spiral trajectory to their victims. Biologist Vance A. Tucker of Duke University in North Carolina wondered for years why peregrines don't take the shortest distance to their prey. He then realized that because falcons' eyes are on either side of their heads, to take advantage of their razor-sharp vision, they must c.o.c.k their heads 40 to one side or the other. Tucker found in wind-tunnel experiments that such a head tip would slow them considerably. The results of his research, which were published in the November 2000 issue of the Journal of Experimental Biology the Journal of Experimental Biology, show that falcons keep their head straight and follow a logarithmic spiral. Because of the spiral's equiangular property, this path allows them to keep their target in view while maximizing speeds.
The amazing thing is that the same spiral shape that is found in the unicellular foraminifera and in the sunflower and that guides the flight of a falcon can also be found in those ”systems of stars gathered together in a common plane, like those of the Milky Way” which philosopher Immanuel Kant (17241804) speculated about long before they were actually observed (Figure 43). These became known as island universes-giant galaxies containing hundreds of billions of stars like our Sun. Observations conducted with the Hubble s.p.a.ce
Figure 43 Telescope revealed that there are some one hundred billion galaxies in our observable universe, many of which are spiral galaxies. You can hardly think of a better manifestation of the grand vision expressed by English poet, painter, and mystic William Blake (17571827), when he wrote: To see a World in a Grain of Sand, And a Heaven in a Wild Flower, Hold Infinity in the Palm of your hand, And Eternity in an hour.
Why do so many galaxies exhibit a spiral pattern? Spiral galaxies like our own Milky Way have a relatively thin disk (like a pancake) composed of gas, dust (miniature grains), and stars. The entire galactic disk is rotating about the galactic center. In the vicinity of the Sun, for example, the orbital speed around the Milky Way's center is about 140 miles per second, and it takes material about 225 million years to complete one revolution. At other distances from the center the speed is different-higher closer to the center, lower at greater distances-that is, galactic disks do not rotate like a solid compact disk but rather rotate differentially. Seen face on, spiral galaxies show spiral arms originating close to the galactic center and extending outward throughout much of the disk (as in the ”Whirlpool Galaxy,” Figure 43 Figure 43). The spiral arms are the part of the galactic disk where many young stars are being born. Since young stars are also the brightest, we can see the spiral structure of other galaxies from afar. The basic question that astrophysicists had to answer is: How do the spiral arms retain their shape over long periods of time? Because the inner parts of the disk rotate faster than the outer parts, any large-scale pattern that is somehow ”attached” to the disk material (e.g., the stars) cannot survive for long. A spiral structure tied always to the same collection of stars and gas clouds would inevitably wind up, contrary to observations. The explanation for the longevity of the spiral arms relies on density waves- density waves-waves of gas compression sweeping through the galactic disk-squeezing gas clouds along the way and triggering the formation of new stars. The spiral pattern that we observe simply marks the denser-than-average parts of the disk and its newborn stars. The pattern is therefore created repeatedly without winding up. The situation is similar to that observed near a lane closed for repairs by a work crew on a major highway. The density of cars in the vicinity of the closed stretch is higher because cars have to slow down there. If you take a long-exposure photograph of the highway from above, you will record the high-traffic density near the place where repairs are being undertaken. Just as the traffic density wave is not a.s.sociated with any particular set of cars, the spiral-arms pattern is not tied to any particular piece of disk material. Another similarity is in the fact that the density wave itself moves through the disk more slowly than the motion of the stars and the gas, just as the speed at which the repair work proceeds along the highway is typically much slower than the unperturbed speed of individual cars.
The agent that deflects the motion of the stars and the gas clouds and generates the spiral density wave (a.n.a.logous to the repair crew that deflects the moving cars to fewer lanes) is the gravitational force resulting from the fact that the distribution of matter in the galaxy is not perfectly symmetric. For example, a set of oval orbits around the center (Figure 44a) in which each orbit is perturbed (rotated) slightly by an amount that changes with distance from the center results in a spiral pattern (Figure 44b).
Figure 44 Actually, we should be quite happy that the force of gravity behaves in our universe the way it does. According to Newton's universal law of gravitation, every ma.s.s attracts every other ma.s.s, and the force of attraction decreases as the ma.s.ses get farther apart. In particular, doubling the distance weakens the force by a factor of four (the force decreases as the inverse of the square of the distance). Newton's laws of motion show that as a result of this dependence on the distance, the orbits of the planets around the Sun are in the shapes of ellipses. Imagine what would have happened had we lived in a universe in which gravity had decreased by a factor of eight (instead of four) upon doubling of the distance (a force decreasing as the inverse of the cube of the distance). In such a universe, Newton's laws predict that one possible orbit of the planets is a logarithmic spiral. In other words, Earth would have spiraled into the Sun or rushed off into s.p.a.ce.
Leonardo Fibonacci, who initiated all of this frenzy of mathematical activity, is far from forgotten today. In today's Pisa, a statue of Fibonacci constructed in the nineteenth century stands in the Scotto Garden on the grounds of the Sangallo Fortress, next to a street named after Fibonacci, which runs along the south side of the Arno River.
Since 1963 the Fibonacci a.s.sociation has published a journal ent.i.tled the Fibonacci Quarterly. Fibonacci Quarterly. The group was formed by mathematicians Verner Emil Hoggatt (19211981) and Brother Alfred Brousseau (19071988) ”in order to exchange ideas and stimulate research in Fibonacci numbers and related topics.” Perhaps against the odds, the The group was formed by mathematicians Verner Emil Hoggatt (19211981) and Brother Alfred Brousseau (19071988) ”in order to exchange ideas and stimulate research in Fibonacci numbers and related topics.” Perhaps against the odds, the Fibonacci Quarterly Fibonacci Quarterly has since grown into a well-recognized journal in number theory. As Brother Brousseau humorously put it: ”We got a group of people together in 1963, and just like a bunch of nuts, we started a mathematics magazine.” The Tenth International Conference on Fibonacci Numbers and Their Applications is planned for June 2428, 2002, at Northern Arizona University in Flagstaff, Arizona. has since grown into a well-recognized journal in number theory. As Brother Brousseau humorously put it: ”We got a group of people together in 1963, and just like a bunch of nuts, we started a mathematics magazine.” The Tenth International Conference on Fibonacci Numbers and Their Applications is planned for June 2428, 2002, at Northern Arizona University in Flagstaff, Arizona.
All of this is but a small tribute to the man who used rabbits to discover a world-embracing mathematical concept. As important as Fibonacci's contribution was, however, the story of the Golden Ratio did not end in the thirteenth century; fascinating developments were still to come in Renaissance Europe.
The quest for our origin is that sweet fruit's juice which maintains satisfaction in the minds of the philosophers.-LUCA P PACIOLI (14451517) (14451517) Few famous painters in history have also been gifted mathematicians. However, when we speak of a ”Renaissance man,” we mean a person who exemplifies the Renaissance ideal of wide-ranging culture and learning. Accordingly, three of the best-known Renaissance painters, the Italians Piero della Francesca (ca. 14121492) and Leonardo da Vinci and the German Albrecht Durer, also made interesting contributions to mathematics. Not surprisingly perhaps, the mathematical investigations of all three painters were related to the Golden Ratio.
The most active mathematician of this ill.u.s.trious trio of artists was Piero della Francesca. The writings of Antonio Maria Graziani (the brother-in-law of Piero's great-grandchild), who purchased Piero's house, indicate that the artist was born in 1412 in Borgo San Sepolcro (today Sansepolcro) in central Italy. His father, Benedetto, was a prosperous tanner and shoemaker. Little else is known about Piero's very early life, but newly discovered doc.u.ments show that he spent some time before 1431 as an apprentice in the workshop of the painter Antonio D'Anghiari (by whom no works have survived). By the late 1430s Piero had moved to Florence, where he started to work with the artist Domenico Veneziano. In Florence, the young painter was exposed to the works of such early Renaissance painters as Fra Angelico and Masaccio and to the sculptures of Donatello. He was particularly impressed with the serenity of the religious works of Fra Angelico, and his own style, in terms of application of color and light, reflected this influence. Later in life, every phase in Piero's work was characterized by a burst of activity, in a variety of places including Rimini, Arezzo, and Rome. The figures that Piero painted either have an architectural solidity about them, as in the ”Flagellation of Christ” (currently in the Galleria Nationale delle Marche in Urbino; Figure 45 Figure 45), or they seem like natural extensions
Figure 45
Figure 46 of the background, as in ”The Baptism” (currently in the National Gallery, London; Figure 46 Figure 46).
In the Lives of the Most Eminent Painters, Sculptors, and Architects Lives of the Most Eminent Painters, Sculptors, and Architects, the first art historian, Giorgio Vasari (15111574), writes that Piero demonstrated great mathematical ability since early youth, and he attributes to him ”many” mathematical treatises. Some of these were written when the painter, because of his old age, could no longer practice art. In the dedicatory letter to Duke Guidobaldo of Urbino, Piero says about one of his books that it was composed ”in order that his wits might not go torpid with disuse.” Three of Piero's mathematical works have survived: De Prospectiva pingendi De Prospectiva pingendi (On perspective in painting), (On perspective in painting), Li-bellus de Quinque Corporibus Regularibus Li-bellus de Quinque Corporibus Regularibus (Short book on the five regular solids), and (Short book on the five regular solids), and Trattato d'Abaco Trattato d'Abaco (Treatise on the abacus). (Treatise on the abacus).
Piero's On Perspective On Perspective (written in the mid-1470s to 1480s) contains numerous references to Euclid's (written in the mid-1470s to 1480s) contains numerous references to Euclid's Elements Elements and and Optics Optics, since he was determined to demonstrate that the technique for achieving perspective in a painting relies firmly on the scientific basis for visual perception. In his own paintings, perspective provides a s.p.a.cious container that is in complete consonance with the geometrical properties of the figures within. In fact, to Piero, painting itself was primarily ”the demonstration in a plane of bodies in diminis.h.i.+ng or increasing size.” This att.i.tude is manifested magnificently in the ”Flagellation” (Figures 45 and 47), which is one of the few Renaissance paintings with a very meticulously determined perspectival construction. As modern-day artist David Hockney puts it in his 2001 book Secret Knowledge Secret Knowledge, Piero paints ”the way he knows the figures to be, not the way he sees them.”
Figure 47 With the occasion of the 500th anniversary of Piero's death, researchers Laura Geatti of the University of Rome and Luciano Fortunati of the National Research Council in Pisa performed a detailed, computer-aided a.n.a.lysis of the ”Flagellation.” They digitized the entire image, determined the coordinates of all the points, measured all the distances, and conducted a complete perspectival a.n.a.lysis using algebraic calculations. Doing this allowed them to determine the precise location of the ”vanis.h.i.+ng point,” at which all lines receding directly from the viewer converge ( anniversary of Piero's death, researchers Laura Geatti of the University of Rome and Luciano Fortunati of the National Research Council in Pisa performed a detailed, computer-aided a.n.a.lysis of the ”Flagellation.” They digitized the entire image, determined the coordinates of all the points, measured all the distances, and conducted a complete perspectival a.n.a.lysis using algebraic calculations. Doing this allowed them to determine the precise location of the ”vanis.h.i.+ng point,” at which all lines receding directly from the viewer converge (Figure 47), that Piero used to achieve the painting's impressive ”depth.”
Piero's lucid book on perspective became the standard manual for artists who attempted to paint plane figures and solids, and the less mathematical (and more accessible) parts of the treatise were incorporated into most subsequent works on perspective. Vasari testifies that due to Piero's strong mathematical background, ”he understood better than anyone else all the curves in the regular bodies and we are indebted to him for the light shed on that subject.” An example of Piero's careful a.n.a.lysis of how to draw a pentagon in perspective is shown in Figure 48 Figure 48.
In both the Treatise on the Abacus Treatise on the Abacus and the and the Five Regular Solids Five Regular Solids, Piero presents a wide range of problems (and their solutions) that involve the pentagon and the five Platonic solids. He calculates the lengths of sides and diagonals as well as areas and volumes. Many of the solutions involve the Golden Ratio, and some of Piero's techniques represent innovative thinking and originality.
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