Part 7 (1/2)
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 a.s.semble 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 irresistible and I kept coming back to it.” and I kept coming back to it.”
FRACTALS.
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 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. 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 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 gla.s.s 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 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 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. as, fractals. The name ”fractal” (from the Latin The name ”fractal” (from the Latin fractus 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. 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 (17741840; currently in the Gemaldegalerie 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- 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 (18701924) 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 a.s.sociated 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 Figure 113) takes up more s.p.a.ce than a one-dimensional line but less s.p.a.ce 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 (18681942), 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 f = ). When we divide a (two-dimensional) square into subsquares with half the side length (again a reduction factor ( = ). When we divide a (two-dimensional) square into subsquares with half the side length (again a reduction factor (f=) we get 4 = 22 squares. For a side length of one-third ( squares. For a side length of one-third (f=), there are 9=32 subsquares ( subsquares (Figure 114). For a (three-dimensional) cube, a division into cubes of half the edge-length (f=) produces 8 = 23 cubes, and one-third the length ( cubes, and one-third the length (f=) produces 27=33 cubes ( cubes (Figure 114). If you examine all of these examples, you find that there is a relation between the number of subobjects, n n, the length reduction factor, f f, and the dimension, D. D. The relation is simply The relation is simply n = (1/f) 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 (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. Hammond Atlas of the World. Using computers to do as much as possible una.s.sisted, 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. Using computers to do as much as possible una.s.sisted, 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 at 120 (Figure 115). Each branch further divides in a similar fas.h.i.+on, and the process goes on without bound.
Figure 115
Figure 116 If instead of a length reduction factor of we had chosen a somewhat larger number (e.g., 0.6), the s.p.a.ces 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 Figure 116. Surprisingly (or maybe not, by now), this happens for a reduction factor that is equal precisely to one over the Golden Ratio one over the Golden Ratio, 1/ = 0.618.... (A short proof is given in Appendix 8.) This is known as a Golden Tree 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 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/ . The resulting image, a Golden Tree composed of lunes, is shown in Figure 118 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 . At each of the free corners of the second-generation triangles, attach a triangle with a side length of , and so on (Figure 119). Again, you may wonder at what reduction factor do the three boughs start to touch, as in Figure 120 Figure 120, and again the answer turns out to be 1/ . Precisely the same situation occurs if you build a similar fractal using a square (Figure 121)-overlapping occurs when the reduction factor is 1/ = 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 a.s.sociated 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 s.p.a.ce 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 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 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 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 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 ill.u.s.trated schematically as in Figure 124 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 Figure 124, each one of them was actually the size of the whole first bar. (This is not shown in the Figure because of s.p.a.ce 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 s.p.a.ce 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 ent.i.tled ”Blake and Fractals” in the Mathematics Magazine. Mathematics Magazine. Referring to the mystic poet William Blake's line ”To see a World in a Grain of Sand,” Memory wrote: 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.