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IV. The Principle of Self-Similarity
Euclidean geometry is the science that deals with regular one-dimensional
lines, two-dimensional planes and three-dimensional solids.
An example of an artificial fractal is the Koch curve. In the early
A shape
is self-similar if any portion is similar to the whole. This means
that the shape has a basic scaling property, i.e., that it
has symmetry throughout the scale. (Symmetry
means having good proportion and order. For example, a shape has symmetry
when the two sides on either side of a dividing line or plane are
similar.)
Mandelbrot was studying artificial fractals that are self-similar
under linear transformation Poincaré's is a visual mathematics of relationships and patterns in geometric figures. It is called topology, the science that studies those properties that are unchanged when the figures are distorted. With a simple nonlinear mathematical transformation, the lengths, angles, and areas of geometric figures can be deformed so that a triangle, for instance, is transformed into a square and then into a circle and a cube is changed into a cylinder, then into a cone and this into a sphere. Nevertheless, some things remain unchanged throughout the transformation. Intersections of lines, for example, do not change and a hole is likewise unaltered. Topology is the study of these unchanging features. Poincaré used topology to investigate the "three-body problem" in celestial mechanics. The laws of motion formulated by Isaac Newton in the 17th century were derived from calculations of the relative motion generated by reciprocal gravitational attraction of only two celestial bodies. The laws imply a universe that is deterministic and predictable. Nevertheless, neither Newton nor the mathematicians that followed were able to calculate the relative motion of three bodies with the mathematical tools at their disposal. More than two hundred years later, however, Poincaré applied topology to its solution. The mathematics he developed allowed him to visualize the individual motion of the bodies. He discovered that minute differences in the initial conditions of each body were amplified exponentially, becoming increasingly larger as the process continued and thus making the motions impossible to predict. Poincaré also discovered that the motion of the whole, although unpredictable, was driven toward an attractor by friction. (An attractor is a pattern or geometric form that describes long-term behavior. In general terms, it is what the system settles down to or is attracted to.) This behavior went contrary to planetary motion. Unknown to him, Poincaré had discovered what later came to be called chaotic or strange attractors. Although he was able to visualize the general shape of the paths, he found it too complex to draw. Only during the past thirty years have developments in mathematics and computer technology enabled scientists to depict them. With this and other topological applications, Poincaré established the foundations dynamical systems theory, the new science of turbulence, commonly known as chaos theory. (Deterministic chaos is the term given to the paradoxical nature of the process, chaotic and unpredictable in its unfolding but orderly when unfolded.)
Strange attractor In order to help visualize the behavior of a dynamical system, scientists have created state space, an abstract space where the coordinates are the degrees of freedom of a system. These fall into three categories, each with its own attractor: 1. Without freedom, corresponding to systems at equilibrium as in the case of a pendulum; 2. With limited freedom, corresponding to systems with periodic oscillations near equilibrium; 3. With unlimited freedom, corresponding to systems far from equilibrium. Chaotic or strange attractors belong to the last group. The flow of water in a stream is an example that illustrates both the changing and unchanging characteristics of a system that goes from equilibrium to near equilibrium to far from equilibrium. If a piece of wood is cast into the current, it can easily be followed as it smoothly rides on top of uniformly moving water. When the water passes over smaller rocks, the waves produced make it bob up and down but it holds its course. However, when the water passes over large boulders and becomes turbulent, it disappears from view. The turbulence of the water makes its motion chaotic and impossible to follow. Despite the turbulence, however, the shape of the waves is self-similar, each wavelet being a miniature representation of the whole. The attractor is the lake where the water collects and the sea into which it empties. In the
early 1960s, the American meteorologist Edward N. Lorenz
Lorenz attractor During the 1960s and '70s, building on the mathematics of nonlinear transformations of Poincaré, Mandelbrot developed fractal geometry, the new mathematics involving nonlinear iterations. (An iteration is a mathematical feedback loop consisting of the following steps: start with a number, square it, add the constant, square the result, add the constant...) When an equation is iterated instead of solved it turns from a description into a process, and from static to dynamic. Mandelbrot also found that the figures he created exhibited self-similarity. Mandelbrot discovered his strange attractor by accident while iterating a nonlinear equation he had devised in which one number was real and the other imaginary. (An imaginary number is the square root of a negative number.) His chaotic attractor is thought to be the most complex object in mathematics discovered thus far. It is a set of points of extreme complexity in which the picture becomes more detailed as the area of magnification gets smaller. Some chaologists (scientists who study chaos) believe that he set existed before Mandelbrot discovered it, formed by natural laws that started repeating themselves when the Universe began. That is why it has been called the island molecule.
The Mandelbrot set
Dissipative structures, hypercycles, and fractals are natural systems that maintain stability as they change and grow in diversity. Evidence shows that their behavior is also purposeful.
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Nevertheless,
about the same time the theory of self-organization was being developed
but independently of it, the French mathematician Benoit Mandelbrot
formulated a geometry that portrays the irregular shapes
of clouds, mountains, coastlines, leaves, flowers, trees, and the
countless other irregular shapes found in Nature. He named it fractal
geometry.
(The word fractal comes from the Latin "fractus" which means "broken."
Fractus is also at the core of the words "fracture" and "fraction".)
1900s, the Swedish mathematician Helge von Koch constructed
an object made up of a sequence of identical steps. He started with
an equilateral triangle. On each side he added equilateral triangles
with sides half the size of the original, making a Star of David.
The operation was repeated which led to a shape similar to a snowflake.
Koch
discovered that by continuing the process endlessly on an increasingly
smaller scale it became a continuous curve containing an infinite
number of minute bends. Through this process of repetition the triangle
became a fractal. The Koch curve is of infinite length yet finite
space; it is more than a line with one dimension but less than a plane
with two dimensions; it is an endless loop where lines never cross;
and at any level of magnification a portion resembles the whole. Because
the repetitions are identical, the Koch curve is a fractal that is
self-similar under linear transformation.
was
the first to generate a strange attractor. He had designed a simple
model of weather conditions consisting of three coupled nonlinear
equations. Although the deterministic nature of the laws of motion
should have given precise solutions, he found that the solutions were
extremely sensitive to initial conditions. Two trajectories (curved
paths) would develop in totally different ways from practically the
same starting point so that long-range predictions were impossible.
The attractor with the form of butterfly wings, the figure generated
by the trajectories of the equations, was named after him. 
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