Fractal notesÉÉ A modern mathematical theory that radically departs from traditional EUCLIDEAN GEOMETRYÉÉ ¥fractal geometry describes objects that are self-similar, or scale symmetric. This means that when such objects are magnified, their parts are seen to bear an exact resemblance to the whole, the likeness continuing with the parts of the parts and so on to infinity. Fractals, as these shapes are called, also must be devoid of translational symmetry--that is, the smoothness associated with Euclidean lines, planes, and spheres. Instead a rough, jagged quality is maintained at every scale at which an object can be examined. ¥The nature of fractals is reflected in the word itself, coined by mathematician Benoit B. Mandelbrot from the Latin verb frangere, "to break," and the related adjective fractus, "irregular and fragmented." ¥The simplest fractal is the Cantor bar (named after the 19th- century German mathematician Georg CANTOR). One may be constructed by dividing a line in 3 parts and removing the middle part. The procedure is repeated indefinitely, first on the 2 remaining parts, then on on 4 parts produced by that operation, and so on, until the object has an infinitely large number of parts each of which is infinitely small. ¥Fractals are not relegated exclusively to the realm of mathematics. If the definition is broadened a bit, such objects can be found virtually everywhere in the natural world. The difference is that "natural" fractals are randomly, statistically, or stochastically rather than exactly scale symmetric. The rough shape revealed at one length scale bears only an approximate resemblance to that at another, but the length scale being used is not apparent just by looking at the shape. Moreover, there are both upper and lower limits to the size range over which the fractals in nature are indeed fractal. Above and below that range, the shapes are either rough (but not self-similar) or smooth--in other words, conventionally Euclidean. ¥Whether natural or mathematical, all fractals have particular fractal dimensions. These are not the same as the familiar Euclidean dimensions, measured in discrete whole integers--1, 2, or 3--but a different kind of quantity. Usually noninteger, a fractal dimension indicates the extent to which the fractal object fills the Euclidean dimension in which it is embedded. ¥A natural fractal of fractal dimension 2.8, for example, would be a sponge-like shape nearly 3-dimensional in appearance. A natural fractal of fractal dimension 2.2 would be a much smoother object that just misses being flat. Background ¥The roots of fractal geometry can be traced to the late 19th century, when mathematicians started to challenge Euclid's principles. ¥Fractional dimensions were not discussed until 1919, however, when the German mathematician Felix Hausdorff put forward the idea in connection with the small-scale structure of mathematical shapes. As completed by the Russian mathematician A. S. Besicovitch, Hausdorff's dimensionality was a forerunner of fractal dimensionality. Other mathematicians of the time, however, considered such strange shapes as "pathologies" that had no significance. ¥This attitude persisted until the mid-20th century and the work of Mandelbrot, a Polish-born French mathematician who moved to the United States in 1958. His 1961 study of similarities in large- and small-scale fluctuations of the stock market was followed by work on phenomena involving nonstandard scaling, including the turbulent motion of fluids and the distribution of galaxies in the universe. A 1967 paper on the length of the English coast showed that irregular shorelines are fractals whose lengths increase with increasing degree of measurable detail. By 1975, Mandelbrot had developed a theory of fractals, and publications by him and others made fractal geometry accessible to a wider audience. The subject began to gain importance in the sciences. ¥Mandelbrot later also investigated another fractal terrain, that of shapes distorted in some way from one length to another. These fractals are now called nonlinear, since the relationships between their parts is subject to change. They retain some degree of self-similarity, but it is a local rather than global characteristic in them. The general definition of the word fractal may thus need further refinement, to indicate more precisely which shapes should be included and which excluded by the term. ¥The most intriguing of the nonlinear fractals thus far has been the mathematical set named after Mandelbrot by the American mathematicians John Hubbard and Adrien Douady. The more the set is magnified, the more its unpredictability increases, until unpredictability comes to dominate the bud-like shape that is the set's major element of stability. The set has become the source of stunning color COMPUTER GRAPHICS images. It is also important in mathematics because of its centrality to dynamical system theory. An entire Mandelbrot set is actually a catalog of dynamical mathematical objects--that is, objects generated through an iterative process called Julia sets. These derive from the work done by a French mathematician, Gaston Julia, on the iteration of nonlinear transformations in a complex plane. Impact on the Sciences ¥Scientists have begun to investigate the fractal character of a wide range of phenomena. Researchers are interested in doing so for the practical reason that behavior on a fractal shape may differ markedly from that on a Euclidean shape. Physics is by far the discipline most affected by fractal geometry. In condensed-matter, or solid-state physics, for example, the so-called "percolation cluster" model used to describe critical phenomena involved in phase transitions and in mixture of atoms with opposing properties is clearly fractal. This has implications, as well, for a host of attributes, including electrical conductivity. The percolation cluster model may also apply to the atomic structure of glasses, gels, and other amorphous materials, and their fractal nature may give them unique heat-transport properties that could be exploited technologically. ¥Another major area of condensed-matter physics to invoke the concept of self-similarity is that of kinetic growth, in which particles are gradually added to a structure in such a way that once they stick, they neither come off nor rearrange themselves. In the case of the simplest model of kinetic growth, the most important physical phenomenon to which it applies appears to be the fingering of a less-viscous fluid (water) through a more viscous fluid (oil) lodged in a porous substance (limestone and other kinds of rock). A more complex model explains the growth of colloidal agglomerates. ¥Mathematical physics, for its part, has a particular interest in nonlinear fractals. When dynamical systems--those that change their behavior over time--become chaotic, or totally unpredictable, physicists describe the route they take with such fractals . Called strange attractors, these objects are not real physical entities but abstractions that exist in "phase space," an expanse with as many dimensions as physicists need to describe dynamical physical behavior. ¥One point in phase space represents a single measurement of the state of a dynamical system as it evolves over time. When all such points are connected, they form a trajectory that lies on the surface of a strange attractor. Most physicists who study chaos do so with carefully controlled laboratory setups of turbulent fluid flow. Individual strange attractors have been identified for different kinds of turbulent fluid flow, suggesting the existence of numerous routes to chaos. ¥Although not concerned with fractals to the same extent as physics, other sciences have discovered them. In biology, the anomolous thermal relaxation rate of iron-containing proteins has been explained as resulting from the fractal shape of the linear polymer chain that comprises all proteins. The distribution pattern of atoms on the protein surface, a different aspect of protein structure, also appears to be fractal. ¥Many more fractals have been detected in geology, including both random exterior surfaces--ragged mountains and valleys--and interior fractal surfaces in the brittle crust, such as California's famous San Andreas fault. Earthquake processes for small tremors--those of magnitude 6 or less--appear to be fractal in time as well as space, since these quakes occur in self-similar clusters rather than at regular intervals. ¥Meteorology provides a different kind of space-time fractalÉ the contour of the area over which tropical rain falls is self-similar, and the amount of rain that falls varies in a self-similar fashion over time. ¥Finally, on the interface of science and art, computer-graphics specialists, using a recursive splitting technique, have produced striking new fractal images of great statistical complexity. Landscapes made this way have been used as backgrounds in many motion pictures; trees and other branching structures have been used in still lifes and animations. Still more FractalsÉ ¥Fractal geometry which, like topology, includes structures that are non-Euclidean, has been applied in Chaos Theory. ¥On the software side, graphics programs employ a wide variety of special display ALGORITHMS, or internal data-processing procedures, to create realistic images. For example, FRACTAL algorithms produce random images that can reveal an infinite amount of texture and are ideal for illustrating mountains and oceans. Fuzzy sets, or algorithms based on statistical probabilities, are useful for generating images of fire, smoke, and other natural phenomena. An iterative algorithm can reproduce the same image countless times, making tiny changes in each one. Hidden-surface removal algorithms can erase lines and surfaces from unseen portions of an image, such as the underside of a cube. ¥Pseudocoloring, or "colorizing," can endow a colorless image with a wide range of hues. Modeling some complicated images, such as a human face, is still difficult to achieve, due to the complex algorithms involve Modern refinements of logical procedure have long since shown that Euclid's work needs modification. The recently developed FRACTAL geometry, for example, requires a more abstract, general definition of dimension than Euclid's. Euclidean statements as "a point is that which has no part" and "a line is breadthless length" and they recognize that any logical system must contain some undefined terms. chaos theory ¥Chaos theory, a modern development in mathematics and science, provides a framework for understanding irregular or erratic fluctuations in nature. Chaotic systems are found in many fields of science and engineering. Many bodies in the solar system alone, for example, have already been determined to exhibit chaotic orbits, and evidence of chaotic behavior has also been found in the pulsations of variable stars. Evidence of chaos occurs in models and experiments describing convection and mixing in fluids, in wave motion, in oscillating chemical reactions, and in electrical currents in semiconductors. It is also found in the dynamics of animal populations and of medical disorders such as heart arrhythmia and epileptic seizures. In addition, attempts are being made to apply chaotic dynamics in the social sciences, such as the study of business cycles and the modeling of arms races. ¥A chaotic system is defined as one that shows "sensitivity to initial conditions." That is, any uncertainty in the initial state of the given system, no matter how small, will lead to rapidly growing errors in any effort to predict the future behavior. For example, the motion of a dust particle floating on the surface of a pair of oscillating whirlpools can display chaotic behavior. The particle will move in well-defined circles around the centers of the whirlpools, alternating between the two in an irregular manner. An observer who wants to predict the motion of this particle will have to measure its initial location. If the measurement is not infinitely precise, however, the observer will instead obtain the location of an imaginary particle very close by. The "sensitivity to initial conditions" mentioned above will cause the nearby imaginary particle to follow a path that diverges from the path of the real particle. This makes any long-term prediction of the trajectory of the real particle impossible. In other words, the system is chaotic. Its behavior can be predicted only if the initial conditions are known to an infinite degree of accuracy, which is impossible. ¥The possibility of chaos in a natural, or deterministic, system was first envisaged by the French mathematician Henri Poincare in the late 19th century, in his work on planetary orbits. For many decades thereafter, however, little interest was shown in such possibilities. The modern study of chaotic dynamics may be said to have begun in 1963, when American meteorologist Edward Lorenz demonstrated that a simple, deterministic model of thermal convection in the Earth's atmosphere showed sensitivity to initial conditions--or, in current terms, that it was a chaotic system. ¥Following this observation, scientists and mathematicians began to study the progression from order to chaos in various systems, as the parameters of the systems were varied. In 1971 a Belgian physicist, David Ruelle, and a Dutch mathematician, Floris Takens, together predicted that the transition to chaotic turbulence in a moving fluid would take place at a well-defined critical value of the fluid's velocity (or some other important factor controlling the fluid's behavior). They predicted that this transition to turbulence would occur after the system had developed oscillations with at least three distinct frequencies. Experiments with rotating fluid flows conducted by American physicists Jerry Gollub and Harry Swinney in the mid-1970s supported these predictions. ¥Another American physicist, Mitchell Feigenbaum, then predicted that at the critical point when an ordered system begins to break down into chaos, a consistent sequence of period-doubling transitions would be observed. This so-called "period-doubling route to chaos" was thereafter observed experimentally by various investigators, including the French physicist Albert Libchaber and his coworkers. Feigenbaum went on to calculate a numerical constant that governs the doubling process (Feigenbaum's number) and showed that his results were applicable to a wide range of chaotic systems. In fact, an infinite number of possible routes to chaos can be described, several of which are "universal," or broadly applicable, in the sense of obeying proportionality laws that do not depend on details of the physical system. ¥The term chaotic dynamics refers only to the evolution of a system in time. Chaotic systems, however, also often display spatial disorder--for example, in complicated fluid flows. Incorporating spatial patterns into theories of chaotic dynamics is an active area of current research. Ultimately, scientists and mathematicians hope to extend theories of chaos to the regime of fully developed turbulence, where complete disorder exists in both space and time. This effort is widely viewed as among the greatest challenges of modern physics