Fractals: A Self-Similar World?

The geometry fractal and chaos theory have set a real revolution in mathematics and science, bringing to light a new way of looking at reality. Science and mathematics have always sought an order in the chaos of the universe. It is interesting first of all to explain the meaning of an expression as “science of chaos” that can initially sound absurd. This science has been able to find an order in phenomena that until now have always been seen as absolutely chaotic, but such phenomena remain unpredictable and uncontrollable. This revolution was born from the discovery of a new basic tool to understand the universe, i.e. geometry fractal. Among the things that fractals can best represent can be counted plants, fluid flow, geological activity, planetary orbits, human physiological rhythms, behaviour of animal groups, socio-economic trends, etc.

The thinkers of ancient times noted that there were two fundamental forms: the line and the curve. The lines have always been the favorite daughters of the intellectuals, while the curves have always passionate artists. The geometry as we know it was invented around 300 BCE by Euclide of Alexandria, and its conceptions have since dominated Western thought. Starting with intuitive axioms such as “a straight has an infinite length”, Euclide developed a coherent group of logical rules to describe simple points, lines and shapes.

A few centuries later, at the beginning of 1600, Cartesio dissected physical space by proposing the possibility to measure the universe using three perpendicular straights intersected and divided into perfectly regular intervals, which allows to assign to any existing object a precise position in space. All creation could in this way be seen as a giant stack of perfectly cubic small boxes. This idea became the basis of the vision of the world of modern science. Newton and Leibnitz considered the Cartesian vision of the world to bring it to its logical conclusion a century later inventing differential calculation. The fundamental idea underlying this calculation is to transform curves into straight lines so that you can apply linear concepts. Leibnitz asserted that all curves consist of infinitely small segments (called tangent or differential lines). Tangent lines are the core of almost all modern sciences and mathematics. Nowadays all, from architects to economists, use techniques of differentiation and its reverse, i.e. integration, to formulate a system of interpretation of the universe. Everyone is based on the assumption that wants each curve consisting of an infinity number of segments.

In 1875 the German mathematician K. Weierstrass described a continuous curve that could not be differentiated, and did not seem to have tangent lines. In the same period, the mathematics was shaken by a tsunami caused by an ever increasing number of this strange type of curves suddenly appeared on the scene. Perhaps, the most famous of these phenomena ancestors of fractals is the triangle of Sierpinski. To draw this form, the Polish mathematician left a triangle and divided it into four equal portions. Figure 1 exhibition as the three external pieces are divided in the same way as the initial triangle is divided, continuing in the infinity procedure. Having the possibility to continue the design indefinitely, you would get the shape defined Sierpinski. Try now to consider another method of construction of the same form. Imagine taking a full triangle and washing it instead of filling it with lines. First of all, you can extract the central piece, then a smaller piece from each of the corners and so on. In the end, you would get the same triangle (figure 2). Sierpinski encountered the first problems when he attempted to calculate the surface of this figure. On the one hand, an infinity number of holes empties each particle of the full area and therefore the surface is zero. For another direction, in each phase of the procedure only a quarter of the remaining surface is taken, thus leaving three quarters (i.e. most) unchanged. Regardless of the level of iteration of the procedure, what remains is more than what is taken and the surface should never reach zero. So the surface is or is not equal to zero? The intuition suggests a negative response, but a verification as rigorous as simple demonstrates the opposite. The problems of this kind have complicated the life of Leibnitz and even that of ancient Greeks.

All these problematic curves formed a sort of “gallery of monsters” (the term used by H. Poincarè at the threshold of the twentieth century) that could not be solved with the mathematical knowledge of the nineteenth century.

If we measure the length of an irregular coastal line, the calculation results depend on the length of the ruler you use; there is no “exact” answer. Figure 3 is a “linea” philosophy. This form was invented by Swedish mathematician H. Von Koch in 1904, during the great crisis of mathematical sciences (as many of his contemporaries, Koch was obsessed with infinity). In addition to the fractal coasts, mathematicians must identify the instruments for the solution of the infinite matrixes of linear equations. To build its coast, Koch departed from a line and designed a triangular extrusion. The same extrusion is then drawn on each line segment that composes the resulting figure and so on. The subsequent approximations of the Koch coastline have increasing lengths (1- 1,333- 1,778- 2,370......7,492). These approximations can continue indefinitely and the total length would seem infinite. By placing three coastal lines of Koch around a triangle, you can create another form that is even more known, the snowflake of Koch (figure 4).

The elementary geometry teaches us that an isolated point, or a finite number of points, constitute a figure of dimension 0; that a straight is a figure of size 1; that a plane, and every other ordinary surface, constitute of the figures of size 2; that a cube has size 3. To these well-known facts the mathematicians, starting from Hausdorff 1919, added that it can be said, of certain idealized figures, that their size is not a full-price admission: it can be a fraction, for example1/2, 3/5, but it is often an irrational number, like log 4/log 3 =1,2618, or even the solution of a complicated equation. The fractional size of the triangular coastline of Koch is approximately 1,26 while the fractional size of the Koch square coastline is about 1,46. More specifically, the fractal size is defined as the ratio between the logarithm of the number of copies and that of the size of the original form compared to each copy. In fact, the idea of a fractional dimension has revealed a potential that goes far beyond what its creators had imagined. Since in nature abounds in self-like forms such as coastal lines, there is now the possibility of characterizing most of our environment through this new index. The mountains, clouds, trees and flowers have dimensions between two and three, and it is now possible to decipher the characteristics of an object by observing its size.

Research in the field of complex curves was abruptly interrupted at the beginning of the twentieth century by an insurmountable barrier: the difficulty of calculations. Mathematicians spent months doing complex calculations to produce a approximation even vague with non-linear curves of infinity detail level. From 1925 to 1960, the limits of manual calculation prevented significant progress in the geometry of complexity and infinity. After that, the first computers appeared that became the trusty mates of the mathematicians.

He was a biologist the first to base a theory on computer simulations. Lindenmayer advanced the idea of cell automatons to define a model of growth of living organisms in 1968. In particular, he was interested in cell development and plant branching schemes. The theory itself, however, can be applied to any number of complex scenarios.

A mobile automaton is an artificial universe governed by simple natural laws. You can choose the structure of this universe and the laws it obeys and then let it develop according to this system. By programming laws in a computer, you can simulate an automated universe and observe its evolution (see Figure 5 and 6).

Figure 7 shows the development of another famous graphic automaton. In this case, the universal consists of geometric “cells” that appear as lines. There is only one rule of propagation: each line is divided into two derived lines of halved length; the first starts from the same point from which the line of origin is separated and proceeds with a inclination of 45 degrees, while the second continues to the final point of the original line. The first two parts of figure 7 represent the geometric pattern of this rule. The horizontal line is that of origin while the two inclined lines are those derived.

By the 1970s, these researches had achieved sufficient development to lay the conditions for a revolution in the field of geometry. The scenario included the experiments of Koch and Peano, the new dimensions of Hausdorff, the fascinating bizarres of Heighway and the self-plication models defined by Lindenmayer. The foundations of this revolution were the new computer technology that allowed us to overcome the problems of calculation and visualization that had until then kept all scholars in check. From the early 1950s onwards, the Mandelbrot mathematician devoted himself to the study of the unpredictable, and at the same time almost cyclical, high and low consumer goods market. The price of cotton was the privileged object of its attention because reliable data relating to centuries of trade were available. The cost of cotton behaves with a strange type of appeal: its variations have approximately the same entity over the centuries as decades or even a few years. In fact, if you enlarge a chart on the trend of the price of cotton over time, each part has approximately the same trend of the lightning-price admission. Mandelbrot called this statistical similarity of a part to the lightning-price admission invariance of scale (see figure 8). Recordings of other unpredictable phenomena, such as floods of rivers or stock market trends, present the same structure intimately cyclical.

What makes fluctuations so complex is the fact that they contain cycles within larger cycles. To simulate this trend, Mandelbrot used iteration techniques such as those of Sierpinski and Koch by introducing a random factor at each stage, so that the curve was not periodic and artificially regular. In 1977 Mandelbrot published the book “Fractal objects: form, case and size” where he demonstrated how many everyday phenomena in the field of physics, biology and mathematics originated in fractals. All the fractals he had taken into account were, like the Koch curve, self-like for changes in scale and translation (in mathematical language, invariants for linear transformation). At the end of 1979 Mandelbrot concluded that it was worth examining, using a computer, the behavior of the particular x^2+c function, in which both the x variable and the constant parameter there are complex numbers. In 1982, Mandelbrot published the milestone of his production “The fractal geometry of Nature”. He coined the term fractal (from the Latin fringe, which means subdividing into irregular fragments).

To adhere to the fractal category, a form must have a Hausdorff/Besicovitch dimension greater than its traditional topological dimension. Basically, fractals are those strange things that occupy a space that the mathematicians have abandoned considering them incomprehensibly complex. Mandelbrot also noted that “algebra” and “fractal” are etymologically opposite. Once explored the self-absorbing “natural” fractals, Mandelbrot discovered iterative procedures that were used to produce abstract mathematical constructions, such as the famous Mandelbrot and Julia sets. Like the other fractals, these sets had been discovered long before the Mandelbrot era, but they were so complex that it would have been impossible to view them and study them without using computers.

To display the Handdelbrot set (see figure 9 ), each point of the computer screen is multiplied by itself repeatedly and added each time to the original point. It is certainly the frontier of the whole that offers the most interesting investigation ground. It has been shown that the Mandelbrot set is closely linked with the behaviour of all dynamic processes; as such it occupies a special place of primary importance in kinship, along with other particular figures such as circle and regular polygons. Despite the simplicity of the formula that generates the Mandelbrot set, the mathematicians who have discovered it have defined it as the most complex object in absolute mathematics (it is worth noting that it was not Mandelbrot to discover the Mandelbrot set). Someone could define the Mandelbrot set as a mathematical monster. Mandelbrot called it “the geometry of nature”.

An enlargement of the Mandelbrot set exhibition the same type of branching that is found in natural phenomena such as lightning, crystals, air growth of plants, liquids that dissolve, spiral forms such as those of shells or sunflowers or, with the addition of a biomorphic coloration, forms that resemble monocellular organisms, etc.

A Julia set is originated by taking a certain point, repeatedly multiplying each other point for the first, and adding each time to the original point. Therefore, each point on the floor has its own Julia set. In figure 10 you can see some sets of Julia. Julia’s equation of iteration is exactly the same as the Mandelbrot set. The only differences are the meaning of c and the initial point of the orbit. When you calculate a Julia set, you use a constant value c for all points on the screen and the orbit begins at the point of the screen that is colored instead of from the origin (0.0). The Mandelbrot and Julia sets obviously have remarkable correlations. In fact, if you enlarge a point of the Mandelbrot set, the image around that point increasingly resembles a Julia set as you approach.

Julia’s sets with strange shapes, bizarre, and therefore more interesting, are those corresponding to points near the frontier of the Mandelbrot set, that is to those that give rise to periodic successions. No one knows with certainty how it happens that the spirals and ramifications of Mandelbrot and Julia's sets can originate from simple non-linear equations, not to mention the reasons why they follow the archetypical patterns of nature so adherent. These topics are the most advanced frontier of current research and scientific theory. Since ancient times, the sharp order of mathematics has opposed the unpredictable chaos of nature.

The turbulent flow of water flowing in a stream is a typical example of those phenomena traditionally excluded from a punctual scientific investigation as chaotic, i.e. unpredictable and non-repeatable behaviour. The science of complexity today represents a real scientific revolution comparable to that, radical, which marked the most classic and famous of scientific revolutions, that of Copernican. If, in fact, the astronomical system proposed by Copernicus demanded the abandonment of the Previous anthropocentric view, which saw man occupy a central place in the lightning-price admission universe, the science of complexity imposes a definitive change of perspective towards an ancient concept as human thought: chaos

Order and chaos. Throughout history, and within the universe, they contend with supremacy: a small change of pressure can transform the regular flow of water from a tap into a complex chaos of vortices; ordered animal communities, including human ones, can be transformed with incredible ease into uncontrollable anarchists. At the opposite pole the order can emerge from chaos, as evidenced by the evolution of life from the formal chaos of the universe, the last step is the human race. The passage from order to chaos, and the Next emerge of order from within that chaos, is clearly revealed by the study of simple retroactive circuits. The essential aspect of the retroaction mechanism is this: there is a certain amount x that varies (in time or in relation to some other variable) so that the value of x at any moment depends with regular trend from its value in the instant Previous (figure 11). Procedures of this type permeate all the exact sciences and most, if not all, of the experimental sciences. A lot of modern thinking has been developed to treat such procedures; for example, the case where the increase between the old x and the new x is infinitesimal led to the development of various techniques for the resolution of differential equations.

The fundamental intuition in Cartesio is due to the fact that the link between physical quantities which can take different values, depending on the circumstances, and for this reason they are called variables – it is expressible with mathematical relationships of equality: algebra equations (so defined because precisely the rules of algebra establish how to make sums and products between such variables). An effective method to represent very small variations of variables is the infinitesimal calculation elaborated, simultaneously but independently, by Newton and Leibnitz. It gives rise to a new type of equations, such differentials, which relate variables and derivatives, the elements expressing the extent of variation. The sum of an infinitely large number of infinitesimal quantities is then allowed by integration (i.e. the operation that allows to calculate an integral). Differential equations consisting of terms that each contain only one variable (or one of its derivatives) are called linear, while non-linear are defined as those differential equations where there are also terms that contain products of variables (or their derivatives). The first are solved by applying the methods of infinitesimal calculation and correspond to mathematical models of physical systems that introduce simplifications and idealizations. To describe the chaotic phenomena in their concreteness, the second ones are indispensable, which in general can be resolved only by assigning away numerical values different to the variables and their derivatives; a method of calculation that is approachable only by having the most powerful electronic processing tools.

It was the fundamental theorem developed in 1892 by the great mathematician H. Poincaré to irreversibly engrave the thermal dream of classical physics. Analyzing problems of celestial mechanics, Poincaré established that, while the orbits traveled from two bodies (e.g. the Sun and its planet) are regular and predictable, the interaction even with a third body (a satellite of the planet) causes disturbances such that the behavior of the global system is intrinsically unstable; that is – as it is possible to assert strictly adhere to the prescriptions of mathematical This means that such equations do not give rise to a single solution when, starting from a certain initial state, it is meant to determine the trajectory accomplished by the three bodies in a time interval.

Through Poincaré's theorem it becomes evident that dynamic systems, in their large majority, are not stable, and therefore have a certain intrinsic degree of chaotity. Even considering gravitational interactions alone, we must note that even the evolution, in a rather long time interval, of a system that appears so regular and predictable as the solar one cannot be known and described analytically with absolute precision. Figure 12 reproduces a computer simulation of the path, clearly chaotic, traced by a “proof planet” orbiting the gravitational field produced by the combined action of its fixed “soli” S’ and S” of equal mass; t1 and t2 represent two moments of later time.

According to Mandelbrot, a hierarchy of increasing complexity appears in the study of flat curves: At the first level there are regular curves such as straight and circumference, which locally confuses with the straight. The classic elementary curves also belong to this level. At the second level we can place the classic fractal curves, in which complication does not change when we approach: they can become more or less complicated, but there is a “invariance” of form compared to distance. We then have a fractal dimension which is between 1 and 2, and this dimension remains the same when we approach the curve. On the third level we find the Mandelbrot set: when we approach more and more we recognize in some details what is observed globally. But we have a constant increase in complexity, and we can say that chaos increases, but it has an ordered structure, because it can be mathematically described. Finally, at the fourth level everything is really chaotic and if we approach it we no longer see in detail what was seen globally, but we observe new and unexpected things. We can conclude that the simplest level was the one studied by the elementary geometry. The second level is of great importance in applications because it is easily found in nature. The third level is the Mandelbrot set, and the fourth corresponds to the most complete and uncontrollable chaos. With this classification we pass from what is simple and regular to what is extremely chaotic, and then emerge the basic categories of scientific thought, that is the local-global relationship and order-caos. Mandelbrot then arrives at the unexpected conclusion that these objects that were considered “mostruosis” are indeed what we observe in nature. These forms of order within chaos can be formalized with the methods of geometry fractal. In the mechanical conception the Universe is considered as a wonderful machine, formed by independent parts, in which everything is the sum of the individual parts. Within this scheme, cosmic evolution is uniquely determined by mechanical laws, once the initial conditions are known. Modern science has surpassed mechanicalism, determinism and reductionism, for various reasons, very different: With Einstein’s “relativity” (1905) it is shown that it is not possible to reduce electromagnetism to mechanics, and then a deep link between space and time appears. With the “Quantum Physics” (1927) the “principle of indetermination” of Heisenberg is introduced, and the dual corpuscular and ondulatory appearance in microphysics. Physical laws then become probable. The “university theory of the physical and biological world” of Fantappié (1942), introduces “sintropic” phenomena. There is therefore a dependence on phenomena from the past (causes) and the future (fini). Therefore in the universe we have a double tendency towards order and disorder. With the “deterministic caos”, discovered by Lorenz (1963), it is demonstrated as deterministic systems also very simple can have a “caotic” behaviour, and then every forecast on the evolution of the system is impossible. If we study a simplified model of weather phenomena, with three degrees of freedom, it behaves chaoticly, i.e. unpredictable. Using the computer is then found the new chaotic attraction of Lorenz, which is a fractal. In fact, in a chaotic system the microscopic disturbances are enormously amplified and interfere with the microscopic behaviour of the system. It then happens that two neighboring orbits become more and more, making it impossible to predict system behaviour. The new “science of chaos” that is achieved is intended to study “complex” and seemingly disordered systems. It is then found that many phenomena of nature are halfway between determinism and indeterminism and between order and disorder. We can conclude that with the discovery of chaos and fractal-type attractors, there is a serious defeat of “reductionism” according to which global properties are uniquely determined by local ones. In fact, the interactions of the components of a system at a given scale can produce a completely different global behaviour, and this leads to a real revolution in physics, comparable to that of relativity and quantum.