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


Understanding is, after all, what science is all about—and science
is a great deal more than mere mindless computation.


The tangled tale of the Golden Ratio has taken us from the sixth century B.C. to contemporary times. Two intertwined trends thread these twenty-six centuries of history. On one hand, the Pythagorean motto “all is number” has materialized spectacularly, in the role that the Golden Ratio plays in natural phenomena ranging from phyllotaxis to the shape of galaxies. On the other, the Pythagorean obsession with the symbolic meaning of the pentagon has metamorphosed into what I believe is the false notion that the Golden Ratio provides a universal canon of ideal beauty. After all of this, you may wonder whether there still is room left for any further exploration of this seemingly simple division of a line.


The Dutch painter Johannes Vermeer (1632–1675) is best known for his fantastically alluring genre paintings, which typically show one or


Figure 92


Figure 93

two figures engaged in some domestic task. In many of these paintings, a window on the viewer's left softly lights the room, and the way the light reflects off the tiled floor is purely magical. If you examine some of these paintings closely, you will find that quite a few, such as “The Concert,” “A Lady Writing a Letter with Her Maid,” “Love Letter” (Figure 92; located in the Rijksmuseum, Amsterdam), and “The Art of Painting” (Figure 93; located in the Kunsthistorisches Museum, Vienna), have identical floor tiling patterns, composed of black and white squares.


Figure 94

Squares, equilateral triangles, and hexagons are particularly easy to tile with, if one wants to cover the entire plane and achieve a pattern that repeats itself at regular intervals—known as periodic tiling (Figure 94). Simple, undecorated square tiles and the patterns they form have a fourfold symmetry—when rotated through a quarter of a circle (90 degrees), they remain the same. Similarly, equilateral, triangular tiles have a threefold symmetry (they remain the same when rotated by a third of a circle or 120 degrees), and hexagonal tiles have a sixfold symmetry (they remain the same when rotated by 60 degrees).

Periodic tilings also can be generated with more complex shapes. One of the most astounding monuments of Islamic architecture, the citadel-palace Alhambra in Granada, Spain, contains numerous examples of intricate tilings (Figure 95). Some of those patterns inspired the famous Dutch graphic artist M. C. Escher (1898–1972), who produced many imaginative examples of tilings (e.g., Figure 96), to which he referred as “divisions of the plane.”


Figure 95

The geometrical plane figure most directly related to the Golden Ratio is, of course, the regular pentagon, which has a fivefold symmetry. Pentagons, however, cannot be used to fill the plane entirely and form a periodic tiling pattern. No matter how hard you try, unfilled gaps will remain. Consequently, it has long been thought that no tiling pattern with long-range order can also exhibit a fivefold symmetry. However, in 1974, Roger Penrose discovered two basic sets of tiles that can fit together to fill the entire plane and exhibit the “forbidden” five-fold rotational symmetry. The resulting patterns are not strictly periodic, even though they display a long-range order.


Figure 96


Figure 97


Figure 98

The Penrose tilings have the Golden Ratio written all over them. One pair of tiles that Penrose considered consists of two shapes known as a “dart” and a “kite” (Figure 97; a and b, respectively). Note that the two shapes are composed of the isosceles triangles that appear in the pentagon (Figure 25). The triangle in which the ratio of side to base is ö (Figure 97b) is the one known as a Golden Triangle, and the one in which the ratio of side to base is 1/ ö(Figure 97a) is the one known as a Golden Gnomon. The two shapes can be obtained by cutting a diamond shape or rhombus with angles of 72 degrees and 108 degrees in a way that divides the long diagonal in a Golden Ratio (Figure 98).

Penrose and Princeton mathematician John Horton Conway showed that in order to cover the whole plane with darts and kites in a nonperiodic way (as in Figure 99), certain matching rules must be obeyed. The latter can be ensured by adding “keys” in the form of notches and protrusions on the sides of the figures, like in the pieces of a jigsaw puzzle (Figure 100). Penrose and Conway further proved that darts and kites can fill the plane in infinitely many nonperiodic ways, with every pattern that can be discerned being surrounded by every other pattern. One of the most startling properties of any Pen-rose kite-dart tiling design is that the number of kites is about 1.618 times the number of darts. That is, if we denote by Nkites the number of kites and Ndarts the number of darts, then Nkjtes/Ndarts approaches ö the larger the area we take in.


Figure 99


Figure 100


Figure 101


Figure 102


Figure 103


Figure 104

Another pair of Penrose tiles that can fill the entire plane (nonperiodically) is composed of two diamonds (rhombi), one fat (obtuse) and one thin (acute; Figure 101). As in the kite-dart pair, each of the rhombi is composed to two Golden Triangles or Golden Gnomons (Figure 102), and special matching rules have to be obeyed (in this case described by decorating the appropriate sides or angles of the rhombi; Figure 103) to obtain a plane-filling pattern (as in Figure 104). Again, in large areas there are 1.618 times more fat rhombi than thin ones, Nrfat/Ntin = ö.

The fat and thin rhombi are intimately related to the darts and kites and both, through the Golden Ratio, to the pentagon-pentagram system.

Recall that the Pythagorean interest in the Golden Ratio started with the infinite series of nested pentagons and pentagrams in Figure 105. All four of the Penrose tiles are hidden in this figure. Points B and D mark the opposite far corners of the kite DCBA, while points A and C mark the “wings” of the dart EABC Similarly, you can find the fat rhombus AECD and the thin one (not to scale) ABCF.


Figure 105

Penrose s work on tiling has been expanded to three dimensions. In the same way that two-dimensional tiles can be used to fill the plane, three-dimensional “blocks” can be used to fill up space. In 1976, mathematician Robert Ammann discovered a pair of “cubes” (Figure 106), one “squashed” and one “stretched,” known as rhombohedra, that can fill up space with no gaps. Ammann was further able to show that given a set of face-matching rules, the pattern that emerges is nonperiodic and has the symmetry properties of the icosahedron (Figure 20e; this is the equivalent of fivefold symmetry in three dimensions, since five symmetric edges meet at every vertex). Not surprisingly, the two rhombohedra are Golden Rhombohedra—their faces actually are identical to the rhombi of the Penrose tiles (Figure 101).


Figure 106

Penrose s tilings might have remained in the relative obscurity of recreational mathematics were it not for a dramatic discovery in 1984. Israeli materials engineer Dany Schectman and his collaborators found that the crystals of an aluminum manganese alloy exhibit both long-range order and fivefold symmetry. This was just about as shocking to crystallographers as the discovery of a herd of five-legged cows would be to zoologists. For decades, solid-state physicists and crystallographers were convinced that solids can come in only two basic forms: Either they are highly ordered and fully periodic crystals, or they are totally amorphous. In ordered crystals, like those of ordinary table salt, atoms or groups of atoms appear in precisely recurring motifs, called unit cells, which form periodic structures. For example, in salt, the unit cell is a cube, and each chlorine atom is surrounded by sodium neighbors and vice versa (Figure 107). Just as in a perfectly tiled floor, the position and orientation of each unit cell determines uniquely the entire pattern. In amorphous materials, such as glasses, on the other hand, the atoms are totally disordered. In the same way that only shapes like squares (with a fourfold symmetry), triangles (threefold symmetry), and hexagons (sixfold symmetry) can fill the entire plane with a periodic tiling, only crystals with two-, three-, four-, and six fold symmetry were thought to exist. Schectman's crystals caused complete bewilderment because they appeared both to be highly ordered (like periodic crystals) and to exhibit fivefold (or icosahedral) symmetry. Before this discovery, few people suspected that another state of matter could exist, sharing important aspects with both crystalline and amorphous substances.


Figure 107

These new kinds of crystals (since the original discovery, other alloys of aluminum have been found) are now known as quasi-crystals— they are neither amorphous like glass nor precisely periodic like salt. In other words, these unusual materials appear to have precisely the properties of Penrose tilings! But this realization by itself is of little use to physicists, who want to understand why and how the quasi-crystals form. Penrose's and Ammann's matching rules are in this case little more than a clever mathematical exercise that does not explain the behavior of real atoms or atom clusters. In particular, it is difficult to imagine energetics that permit precisely the existence of two types of clusters (like the two Ammann rhombohedra) in just the required proportion in terms of density.

A clue toward a possible explanation came in 1991, when mathematician Sergei E. Burkov of the Landau Institute of Theoretical Physics in Moscow realized that two shapes of tiles are not needed to achieve quasi-periodic tiling in the plane. Burkov showed that quasi-periodicity could be generated even using a single, decagonal (ten-sided) unit, provided that the tiles are allowed to overlap—a property forbidden in previous tiling attempts. Five years later, German mathematician Petra Gummelt of the Ernst Moritz Arndt University in Greifswald proved rigorously that Penrose tiling can be obtained by using a single “decorated” decagon combined with a specific overlapping rule. Two decagons may overlap only if shaded areas in the decoration overlap (Figure 108). The decagon is also closely related to the Golden Ratio—the radius of the circle that circumscribes a decagon with a side length of 1 unit is equal to ö.

Based on Gummelt's work, mathematics finally could be turned into physics. Physicists Paul Steinhardt of Princeton University and Hyeong-Chai Jeong of Sejong University in Seoul showed that the purely mathematical rules of overlapping units could be transformed into a physical picture in which “quasi-unit cells,” which are really clusters of atoms, simply share atoms. Steinhardt and Jeong suggested that quasi-crystals are structures in which identical clusters of atoms (quasi-unit cells) share atoms with their neighbors, in a pat tern that is designed to maximize the cluster density. In other words, quasi-periodic packing produces a system that is more stable (higher density and lower energy) than otherwise. Steinhardt, Jeong, and collaborators also attempted to verify this model experimentally in 1998. They bombarded a quasi-crystal alloy of aluminum, nickel, and cobalt with X-ray and electron beams. The images of the structure obtained from the scattered beams were in remarkable agreement with the picture of overlapping decagons. This is shown in Figure 109, where a decagon tiling pattern is superimposed on the experimental result. More recent experiments gave results that were somewhat more ambiguous. Nevertheless, the general impression remains that quasi-crystals can be explained by the Steinhardt-Jeong model.


Figure 108


Figure 109


Figure 110

Images of the surfaces of quasi-crystals (taken in 1994 and 2001) reveal another fascinating relation to the Golden Ratio. Using a technique known as scanning tunneling microscopy (STM), scientists from the University of Basel, Switzerland, and from the Ames Laboratory at Iowa State University were able to obtain high-resolution images of the surfaces of an aluminum-copper-iron alloy and an aluminum-palladium-manganese alloy, both of which are quasi-crystals. The images show flat “terraces” (Figure 110) terminating in steps that come primarily in two heights, “high” and “low” (both measuring only a few hundred-millionths of an inch). The ratio of the two heights was found to be equal to the Golden Ratio!

Quasi-crystals are a magnificent example of a concept that started out as a purely mathematical entity (based on the Golden Ratio) but that eventually provided an explanation of a real, natural phenomenon. What is even more amazing about this particular development is that the concept emerged out of recreational mathematics. How could mathematicians have “anticipated” later discoveries by physicists? The question becomes more intriguing yet when we recall that Dürer and Kepler showed interest in tilings with fivefold symmetric shapes already in the sixteenth and seventeenth centuries. Can even the most esoteric topics in mathematics eventually find applications in either natural or human-inspired phenomena? We shall return to this question in Chapter 9.

Another fascinating aspect of the quasi-crystals story is related to two of the main theorists involved. Both Penrose and Steinhardt spent much of their scientific careers on topics related to cosmology—the study of the universe as a whole. Penrose is the person who discovered that Einstein's theory of general relativity predicts its own defects, points in which the strength of gravity becomes infinite. These mathematical singularities correspond to the objects we call black holes, which are masses that have collapsed to such densities that their gravity is sufficiently strong to prevent any light, mass, or energy to escape from them. Observations during the past quarter century have revealed that black holes are not just imaginary theoretical concepts but actual objects that exist in the universe. Recent observations with the two large space observatories, the Hubble Space Telescope and the Chandra X-ray Observatory, have shown that black holes are not even very rare. Rather, the centers of most galaxies harbor monstrous black holes with masses between a few million and a few billion times the mass of our Sun. The presence of the black holes is revealed by the gravitational pull they exert on stars and gas in their neighborhood. According to the standard “big bang” model that describes the origin of our entire universe, the cosmos as a whole started its expansion from such a singularity—an extremely hot and dense state.

Paul Steinhardt was one of the key figures in the development of what is known as the inflationary model of the universe. According to this model, originally proposed by physicist Alan Guth of MIT, when the universe was only a tiny fraction of a second old (0.000 … 1; with the “1” at the 35th decimal place), it underwent a fantastically rapid expansion, increasing in size by a factor of more than 1030 (1 followed by 30 zeros) within a fraction of a second. This model explains a few otherwise puzzling properties of our universe, such as the fact that it looks almost precisely the same in every direction—it is exquisitely isotropic. In 2001, Steinhardt and collaborators proposed a new version of the universe's very beginnings, known as the Ekpyrotic Universe (from the Greek word for “conflagration,” or a sudden burst of fire). In this still very speculative model, the big bang occurred when two three-dimensional universes moving along a hidden extra dimension collided.

The intriguing question is: Why did these two outstanding cosmologists decide to get involved in recreational mathematics and quasi-crystals?

I have known Penrose and Steinhardt for many years, being in the same business of theoretical astrophysics and cosmology. In fact, Penrose was an invited speaker in the first large conference that I organized on relativistic astrophysics in 1984, and Steinhardt was an invited speaker in the latest one in 2001. Still, I did not know what motivated them to delve into recreational mathematics, which appears to be rather remote from their professional interests in astrophysics, so I asked them.

Roger Penrose replied: “I am not sure I have a deep answer for that. As you know, mathematics is something most mathematicians do for enjoyment.” After some reflection he added: “I used to play with shapes fitting together since I was a child; some of my work on tiles therefore predated my work in cosmology. At the particular time, however, my recreational mathematics work was at least partially motivated by my cosmological research. I was thinking about the large-scale structure of the universe and was looking for toy models with simple basic rules, which could nevertheless generate complicated structures on large scales.”

“But,” I asked, “what was it that induced you to continue to work on that problem for quite a while?”

Penrose laughed and said, “As you know, I have always been interested in geometry; that problem simply intrigued me. Furthermore, while I had a hunch that such structures could occur in nature, I just couldn't see how nature could assemble them through the normal process of crystal growth, which is local. To some extent I am still puzzled by that.”

Paul Steinhardt's immediate reaction on the phone was: “Good question!” After thinking about it for a few minutes he reminisced: “As an undergraduate student I really wasn't sure what I wanted to do. Then, in graduate school, I looked for some mental relief from my strenuous efforts in particle physics, and I found that in the topic of order and symmetry in solids. Once I stumbled on the problem of quasi-periodic crystals, I found it irresistible and I kept coming back to it.”


The Steinhardt-Jeong model for quasi-crystals has the interesting property that it produces long-range order from neighborly interactions, without resulting in a fully periodic crystal. Amazingly enough, we can also find this general property in the Fibonacci sequence. Consider the following simple algorithm for the creation of a sequence known as the Golden Sequence. Start with the number 1, and then replace 1 by 10. From then on, replace each 1 by 10 and each 0 by 1. You will obtain the following steps:


and so on. Clearly, we started here with a “short-range” law (the simple transformation of 0 → 1 and 1 → 10) and obtained a nonperiodic long-range order. Note that the numbers of 1s in the sequence of lines 1, 1, 2, 3, 5, 8… form a Fibonacci sequence, and so do the numbers of 0s (starting from the second line). Furthermore, the ratio of the number of 1s to the number of 0s approaches the Golden Ratio as the sequence lengthens. In fact, an examination of Figure 27 reveals that if we take 0 to stand for a baby pair of rabbits and 1 to stand for a mature pair, then the sequence just given mirrors precisely the numbers of rabbit pairs. But there is even more to the Golden Sequence than these surprising properties. By starting with 1 (on the first line), followed by 10 (on the second line), and simply appending to each line the line just preceding it, we can also generate the entire sequence. For example, the fourth line, 10110, is obtained by appending the second line, 10, to the third, 101, and so on.

Recall that “self-similarity” means symmetry across size scale. The logarithmic spiral displays self-similarity because it looks precisely the same under any magnification, and so does the series of nested pentagons and pentagrams in Figure 10. Every time you walk into a hair stylist shop, you see an infinite series of self-similar reflections of yourself between two parallel mirrors.

The Golden Sequence is also self-similar on different scales. Take the sequence


and probe it with a magnifying glass in the following sense. Starting from the left, whenever you encounter a 1, mark a group of three symbols, and when you encounter a 0, mark a group of two symbols (with no overlap among the different groups). For example, the first digit is a 1, we therefore mark the group of the first three digits 101 (see below). The second digit from the left is a zero, therefore we mark the group of two digits 10 that follow the first 101. The third digit is 1; therefore we mark the three digits 101 that follow the 10; and so on. The marked sequence now looks like this


Now from every group of three symbols retain the first two, and from every group of two retain the first one (the retained symbols are underlined):


If you now look at the retained sequence


you find that it is identical to the Golden Sequence.

We can do another magnification exercise on the Golden Sequence simply by underlining any pattern or subsequence. For example, suppose we choose “10” as our subsequence, and we underline it whenever it occurs in the Golden Sequence:


If we now treat each 10 as a single symbol and we mark the number of places by which each pattern of 10 needs to be moved to overlap with the next 10, we get the sequence: 2122121… (the first “10” needs to be moved two places to overlap with the second, the third is one place after the second, etc.). If we would now replace each 2 by a 1 and each 1 by a 0 in the new sequence, we recover the Golden Sequence. In other words, if we look at any pattern within the Golden Sequence, we discover that the same pattern is found in the sequence on another scale. Objects with this property, like the Russian Matrioshka dolls that fit one into the other, are known as, fractals. The name “fractal” (from the Latin fractus, meaning “broken, fragmented”) was coined by the famous Polish-French-American mathematician Benoit B. Mandelbrot, and it is a central concept in the geometry of nature and in the theory of highly irregular systems known as chaos.

Fractal geometry represents a brilliant attempt to describe the shapes and objects of the real world. When we look around us, very few forms can be described in terms of the simple figures of Euclidean geometry, such as straight lines, circles, cubes, and spheres. An old mathematical joke tells of a physicist who thought that he could become rich from betting at horse races by solving the exact equations of motion for the horses. After much work, he indeed managed to solve the equations—for spherical horses. Real horses, unfortunately, are not spherical, and neither are clouds, cauliflowers, or lungs. Similarly, lightning, rivers, and drainage systems do not travel in straight lines, and they all remind us of the branching of trees and of the human circulatory system. Examine, for example, the fantastically intricate branching of the “Dolmen in the Snow” (Figure 111), a painting by the German romantic painter Caspar David Friedrich (1774–1840; currently in the Gemäldegalerie Neue Meister in Dresden).


Figure 111

Mandelbrot's gigantic mental leap in formulating fractal geometry has been primarily in the fact that he recognized that all of these complex zigs and zags are not merely a nuisance but often the main mathematical characteristic of the morphology. Mandelbrot's first realization was the importance of self-similarity—the fact that many natural shapes display endless sequences of motifs repeating themselves within motifs, on many scales. The chambered nautilus (Figure 4) exhibits this property magnificently, as does a regular cauliflower—break off smaller and smaller pieces and, up to a point, they continue to look like the whole vegetable. Take a picture of a small piece of rock, and you will have a hard time recognizing that you are not looking at an entire mountain. Even the printed form of the continued fraction that is equal to the Golden Ratio has this property (Figure 112)—magnify the barely resolved symbols and you will see the same continued fraction. In all of these objects, zooming in does not smooth out the degree of roughness. Rather, the same irregularities characterize all scales.

At this point, Mandelbrot asked himself, how do you determine the dimensions of something that has such a fractal structure? In the world of Euclidean geometry, all the objects have dimensions that can be expressed as whole numbers. Points have zero dimensions, straight lines are one-dimensional, plane figures like triangles and pen tagons are two-dimensional, and objects like spheres and the Platonic solids are three-dimensional. Fractal curves like the path of a bolt of lightning, on the other hand, wiggle so aggressively that they fall somewhere between one and two dimensions. If the path is relatively smooth, then we can imagine that the fractal dimension would be close to one, but if it is very complex, then a dimension closer to two can be expected. These musings have turned into the by now-famous question: “How long is the coast of Britain?” Mandelbrot's surprising answer is that the length of the coastline actually depends on the length of your ruler. Suppose you start out with a satellite-generated map of Britain that is one foot on the side. You measure the length and convert it to the actual length by multiplying by the known scale of your map. Clearly this method will skip over any twists in the coastline that are too small to be revealed on the map. Equipped with a one-yard stick, you therefore start the long journey of actually walking along Britain's beaches, painstakingly measuring the length yard by yard. There is no doubt that the number you get now will be much larger than the previous one, since you managed to capture much smaller twists and turns. You immediately realize, however, that you would still be skipping over structures on smaller scales than one yard. The point is that every time you decrease the size of your ruler, you get a larger value for the length, because you always discover that there exists substructure on even smaller scale. This fact suggests that even the concept of length as representing size needs to be revisited when dealing with fractals. The contours of the coastline do not become a straight line upon magnification; rather, the crinkles persist on all scales and the length increases ad infinitum (or at least down to atomic scales).


Figure 112

This situation is exemplified beautifully by what could be thought of as the coastline of some imaginary land. The Koch snowflake is a curve first described by the Swedish mathematician Helge von Koch (1870–1924) in 1904 (Figure 113). Start with an equilateral triangle, one inch long on the side. Now in the middle of each side, construct a smaller triangle, with a side of one-third of an inch. This will give the Star of David in the second figure. Note that the original outline of the triangle was three inches long, while now it is composed of twelve segments, one-third of an inch each, so that the total length is now four inches. Repeat the same procedure consecutively—on each side of a triangle place a new one, with a side length that is one-third that of the previous one. Each time, the length of the outline increases by a factor of 4/3 to infinity, in spite of the fact that it borders a finite area. (We can show that the area converges to eight-fifths that of the original triangle.)


Figure 113

The realization of the existence of fractals raised the question of the dimensions that should be associated with them. The fractal dimension is really a measure of the wrinkliness of the fractal, or of how fast length, surface, or volume increases if we measure it with respect to ever-decreasing scales. For example, we feel intuitively that the Koch curve (bottom of Figure 113) takes up more space than a one-dimensional line but less space than the two-dimensional square. But how can it have an intermediate dimension? There is, after all, no whole number between 1 and 2. This is where Mandelbrot followed a concept first introduced in 1919 by the German mathematician Felix Hausdorff (1868–1942), a concept that at first appears mind boggling—fractional dimensions. In spite of the initial shock we may experience from such a notion, fractional dimensions were precisely the tool needed to characterize the degree of irregularity, or fractal complexity, of objects.

In order to obtain a meaningful definition of the self-similarity dimension or fractal dimension, it helps to use the familiar whole-number dimensions 0, 1, 2, 3 as guides. The idea is to examine how many small objects make up a larger object in any number of dimensions. For example, if we bisect a (one-dimensional) line, we obtain two segments (for a reduction factor of f = ½). When we divide a (two-dimensional) square into subsquares with half the side length (again a reduction factor (f=½) we get 4 = 22 squares. For a side length of one-third (f=⅓), there are 9=32 subsquares (Figure 114). For a (three-dimensional) cube, a division into cubes of half the edge-length (f=⅓) produces 8 = 23 cubes, and one-third the length (f=⅓) produces 27=33 cubes (Figure 114). If you examine all of these examples, you find that there is a relation between the number of subobjects, n, the length reduction factor, f, and the dimension, D. The relation is simply n = (1/f)D. (I give another form of this relation in Appendix 7.) Applying the same relation to the Koch snowflake gives a fractal dimension of about 1.2619. As it turns out, the coastline of Britain also has a fractal dimension of about 1.26. Fractals therefore serve as models for real coastlines. Indeed, pioneering chaos theorist Mitch Feigenbaum, of Rockefeller University in New York, exploited this fact to help produce in 1992 the revolutionary Hammond Atlas of the World. Using computers to do as much as possible unassisted, Feigenbaum examined fractal satellite data to determine which points along coastlines have the greatest significance. The result—a map of South America, for example, that is better than 98 percent accurate, compared to the more conventional 95 percent scored by older atlases.


Figure 114

For many fractals in nature, from trees to the growth of crystals, the main characteristic is branching. Let us examine a highly simplified model for this ubiquitous phenomenon. Start with a stem of unit length, which divides into two branches of length ½ at 120 ° (Figure 115). Each branch further divides in a similar fashion, and the process goes on without bound.


Figure 115


Figure 116

If instead of a length reduction factor of ½ we had chosen a somewhat larger number (e.g., 0.6), the spaces among the different branches would have been reduced, and eventually branches would overlap. Clearly, for many systems (e.g., a drainage system or a blood circulatory system), we may be interested in finding out at what reduction factor precisely do the branches just touch and start to overlap, as in Figure 116. Surprisingly (or maybe not, by now), this happens for a reduction factor that is equal precisely to one over the Golden Ratio, 1/ ö = 0.618.… (A short proof is given in Appendix 8.) This is known as a Golden Tree, and its fractal dimension turns out to be about 1.4404. The Golden Tree and similar fractals composed of simple lines cannot be resolved very easily with the naked eye after several iterations. The problem can be partially resolved by using two-dimensional figures like lunes(Figure 117) instead of lines. At each step, you can use a copying machine equipped with an image reduction feature to produce lunes reduced by a factor 1/ ö. The resulting image, a Golden Tree composed of lunes, is shown in Figure 118.


Figure 117


Figure 118


Figure 119


Figure 120


Figure 121


Figure 122

Fractals can be constructed not just from lines but also from simple planar figures such as triangles and squares. For example, you can start with an equilateral triangle with a side of unit length and at each corner attach a new triangle with a side length of ½. At each of the free corners of the second-generation triangles, attach a triangle with a side length of ¼, and so on (Figure 119). Again, you may wonder at what reduction factor do the three boughs start to touch, as in Figure 120, and again the answer turns out to be 1/ ö. Precisely the same situation occurs if you build a similar fractal using a square (Figure 121)—overlapping occurs when the reduction factor is 1/ ö = 0.618… (Figure 122).

Furthermore, all the unfilled white rectangles in the last Figure are Golden Rectangles. We therefore find that while in Euclidean geometry the Golden Ratio originated from the pentagon, in fractal geometry it is associated even with simpler figures like squares and equilateral triangles.

Once you get used to the concept, you realize that the world around us is full of fractals. Objects as diverse as the profiles of the tops of forests on the horizon and the circulatory system in a kidney can be described in terms of fractal geometry. If a particular model of the universe as a whole known as eternal inflation is correct, then even the entire universe is characterized by a fractal pattern. Let me explain this concept very briefly, giving only the broad-brush picture. The inflationary theory, originally advanced by Alan Guth, suggests that when our universe was only a tiny fraction of a second old, an unbridled expansion stretched our region of space to a size that is actually much larger than the reach of our telescopes. The driving force behind this stupendous expansion is a very peculiar state of matter called a false vacuum. A ball on top of a flat hill, as in Figure 123, can symbolically describe the situation. For as long as the universe remained in the false vacuum state (the ball was on the hilltop), it expanded extremely rapidly, doubling in size every tiny fraction of a second. Only when the ball rolled down the hill and into the surrounding, lower-energy “ditch” (representing symbolically the fact that the false vacuum decayed) did the tremendous expansion stop. According to the inflationary model, what we call our universe was caught in the false vacuum state for a very brief period, during which it expanded at a fantastic rate. Eventually the false vacuum decayed, and our universe resumed the much more leisurely expansion we observe today. All the energy and subatomic particles of our universe were generated during oscillations that followed the decay (represented schematically in the third drawing in Figure 123). However, the inflationary model also predicts that the rate of expansion while in the false vacuum state is much faster than the rate of decay. Consequently, the fate of a region of false vacuum can be illustrated schematically as in Figure 124. The universe started with some region of false vacuum. As time progressed, some part (a third in the figure) of the region has decayed to produce a “pocket universe” like our own. At the same time, the regions that stayed in the false vacuum state continued to expand, and by the time represented schematically by the second bar in Figure 124, each one of them was actually the size of the whole first bar. (This is not shown in the Figure because of space constraints.) Moving in time from the second bar to the third, the central pocket universe continued to evolve slowly as in the standard big bang model of our universe. Each of the remaining two regions of false vacuum, however, evolved in precisely the same way as the original region of false vacuum—some part of them decayed, producing a pocket universe to become the same size Figure because of space constraints). An infinite number of pocket universes thus were produced, and a fractal pattern was generated—the same sequence of false vacua and pocket universes is replicated on ever-decreasing scales. If this model truly represents the evolution of the universe as a whole, then our pocket universe is but one out of an infinite number of pocket universes that exist.


Figure 123


Figure 124

In 1990, North Carolina State University professor Jasper Memory published a poem entitled “Blake and Fractals” in the Mathematics Magazine. Referring to the mystic poet William Blake's line “To see a World in a Grain of Sand,” Memory wrote:

William Blake said he could see
Vistas of infinity
In the smallest speck of sand
Held in the hollow of his hand.
Models for this claim we've got
In the work of Mandelbrot:
Fractal diagrams partake
Of the essence sensed by
Blake. Basic forms will still prevail
Independent of the Scale;
Viewed from far or viewed from near
Special signatures are clear.
When you magnify a spot,
What you had before, you've got.
Smaller, smaller, smaller, yet,
Still the same details are set;
Finer than the finest hair
Blake's infinity is there,
Rich in structure all the way—
Just as the mystic poets say.

Some of the modern applications of the Golden Ratio, Fibonacci numbers, and fractals reach into areas that are much more down to earth than the inflationary model of the universe. In fact, some say that the applications can reach even all the way into our pockets.


One of the best-known attempts to use the Fibonacci sequence and the Golden Ratio in the analysis of stock prices is associated with the name of Ralph Nelson Elliott (1871–1948). An accountant by profession, Elliott held various executive positions with railroad companies, primarily in Central America. A serious alimentary tract illness that left him bedridden forced him into retirement in 1929. To occupy his mind, Elliott started to analyze in great detail the rallies and plunges of the Dow Jones Industrial Average. During his lifetime, Elliott witnessed the roaring bull market of the 1920s followed by the Great Depression. His detailed analyses led him to conclude that market fluctuations were not random. In particular, he noted: “the stock market is a creation of man and therefore reflects human idiosyncrasy.” Elliott's main observation was that, ultimately, stock market patterns reflect cycles of human optimism and pessimism.

On February 19, 1935, Elliott mailed a treatise entitled The Wave Principle to a stock market publication in Detroit. In it he claimed to have identified characteristics which “furnish a principle that determines the trend and gives clear warning of reversal.” The treatise eventually developed into a book with the same title, which was published in 1938.


Figure 125


Figure 126

Elliott's basic idea was relatively simple. He claimed that market variations can be characterized by a fundamental pattern consisting of five waves during an upward (“optimistic”) trend (marked by numbers in Figure 125) and three waves during a downward (“pessimistic”) trend (marked by letters in Figure 125). Note that 5, 3, 8 (the total number of waves) are all Fibonacci numbers. Elliott further asserted that an examination of the fluctuation on shorter and shorter time scales reveals that the same pattern repeats itself Figure 126), with all the numbers of the constituent wavelets corresponding to higher Fibonacci numbers. Identifying 144 as “the highest number of practical value,” the breakdown of a complete market cycle, according to Elliott, might look as follows. A generally upward trend consisting of five major waves, twenty-one intermediate waves, and eighty-nine minor waves (Figure 126) is followed by a generally downward phase with three major, thirteen intermediate, and fifty-five minor waves (Figure 126).


Figure 127

Some recent books that attempt to apply Elliott's general ideas to actual trading strategies go even further. They use the Golden Ratio to calculate the extreme points of maximum and minimum that can be expected (although not necessarily reached) in market prices at the end of upward or downward trends (Figure 127). Even more sophisticated algorithms include a logarithmic spiral plotted on top of the daily market fluctuations, in an attempt to represent a relationship between price and time. All of these forecasting efforts assume that the Fibonacci sequence and the Golden Ratio somehow provide the keys to the operation of mass psychology. However, this “wave” approach does suffer from some shortcomings. The Elliott “wave” usually is subjected to various (sometimes arbitrary) stretchings, squeezings, and other alterations by hand to make it “forecast” the real-world market. Investors know, however, that even with the application of all the bells and whistles of modern portfolio theory, which is supposed to maximize the returns for a decided-on level of risk, fortunes can be made or lost in a heartbeat.

You may have noticed that Elliott's wave interpretation has as one of its ingredients the concept that each part of the curve is a reduced-scale version of the whole, a concept central to fractal geometry. Indeed, in 1997, Benoit Mandelbrot published a book entitled Fractals and Scaling in Finance: Discontinuity, Concentration, Risk, which introduced well-defined fractal models into market economics. Mandelbrot built on the known fact that fluctuations in the stock market look the same when charts are enlarged or reduced to fit the same price and time scales. If you look at such a chart from a distance that does not allow you to read the scales, you will not be able to tell if it represents daily, weekly, or hourly variations. The main innovation in Mandelbrot's theory, as compared to standard portfolio theory, is in its ability to reproduce tumultuous trading as well as placid markets. Portfolio theory, on the other hand, is able to characterize only relatively tranquil activity. Mandelbrot never claimed that his theory could predict a price drop or rise on a specific day but rather that the model could be used to estimate probabilities of potential outcomes. After Mandelbrot published a simplified description of his model in Scientific American in February 1999, a myriad of responses from readers ensued. Robert Ihnot of Chicago probably expressed the bewilderment of many when he wrote: “If we know that a stock will go from $10 to $15 in a given amount of time, it doesn't matter how we interpose the fractals, or whether the graph looks authentic or not. The important thing is that we could buy at $10 and sell at $15. Everyone should now be rich, so why are they not?”

Elliott's original wave principle represented a bold if somewhat naïve attempt to identify a pattern in what appears otherwise to be a rather random process. More recently, however, Fibonacci numbers and randomness have had an even more intriguing encounter.


The defining property of the Fibonacci sequence—that each new number is the sum of the previous two numbers—was obtained from an unrealistic description of the breeding of rabbits. Nothing in this definition hinted that this imaginary rabbit sequence would find its way into so many natural and cultural phenomena. There was even less, however, to suggest that experimentation with the basic properties of the sequence themselves could provide a gateway to understanding the mathematics of disordered systems. Yet this was precisely what happened in 1999. Computer scientist Divakar Viswanath, then a postdoctoral fellow at the Mathematical Sciences Research Institute in Berkeley, California, was bold enough to ask a “what if?” question that led unexpectedly to the discovery of a new special number: 1.13198824 …. The beauty of Viswanath's discovery lies primarily in the simplicity of its central idea. Viswanath merely asked himself: Suppose you start with the two numbers 1, 1, as in the original Fibonacci sequence, but now instead of adding the two numbers to get the third, you flip a coin to decide whether to add them or to subtract the last number from the previous one. You can decide, for example, that “heads” means to add (giving 2 as the third number) and “tails” means to subtract (giving 0 as the third number). You can continue with the same procedure, each time flipping a coin to decide whether to add or subtract the last number to get a new one. For example, the series of tosses HTTHHTHTTH will produce the sequence 1, 1, 2, –1, 3, 2, 5, –3, 2, –5, 7, 2. On the other hand, the (rather unlikely) series of tosses HHHHHHHHHHHH… will produce the original Fibonacci sequence.

In the Fibonacci sequence, terms increase rapidly, like a power of the Golden Ratio. Recall that we can calculate the seventeenth number in the sequence, for example, by raising the Golden Ratio to the seventeenth power, dividing by the square root of 5, and rounding off the result to the nearest whole number (which gives 1597). Since Viswanath's sequences were generated by a totally random series of coin tosses, however, it was not at all obvious that a smooth growth pattern would be obtained, even if we ignore the minus signs and take only the absolute value of the numbers. To his own surprise, however, Viswanath found that if he ignored the minus signs, the values of the numbers in his random sequences still increased in a clearly defined and predictable rate. Specifically, with essentially 100 percent probability, the one hundredth number in any of the sequences generated in this way was always close to the one hundredth power of the peculiar number 1.13198824…, and the higher the term was in the sequence, the closer it came to the corresponding power of 1.13198824 …. To actually calculate this strange number, Viswanath had to use fractals and to rely on a powerful mathematical theorem that was formulated in the early 1960s by mathematicians Hillel Furstenberg of the Hebrew University in Jerusalem and Harry Kesten of Cornell University. These two mathematicians proved that for an entire class of randomly generated sequences, the absolute value of a number high in the sequence gets closer and closer to the appropriate power of some fixed number. However, Furstenberg and Kesten did not know how to calculate this fixed number; Viswanath discovered how to do just that.

The importance of Viswanath's work lies not only in the discovery of a new mathematical constant, a significant feat in itself, but also in the fact that it illustrates beautifully how what appears to be an entirely random process can lead to a fully deterministic result. Problems of this type are encountered in a variety of natural phenomena and electronic devices. For example, stars like our own Sun produce their energy in nuclear “furnaces” at their centers. However, for us actually to see the stars shining, bundles of radiation, known as photons, have to make their way from the stellar depths to the surface. Photons do not simply fly through the star at the speed of light. Rather, they bounce around, being scattered and absorbed and reemitted by all the electrons and atoms of gas in their way, in a seemingly random fashion. Yet the net result is that after a random walk, which in the case of the Sun takes some 10 million years, the radiation escapes the star. The power emitted by the Sun's surface determined (and continues to determine) the temperature on Earth's surface and allowed life to emerge. Viswanath's work and the research on random Fibonaccis that followed provide additional tools for the mathematical machinery that explains disordered systems.

There is another important lesson to be learned from Viswanath's discovery—even an eight-hundred-year-old, seemingly trivial mathematical problem can still surprise you.