Fractals

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Fractals

The term "fractal" was coined by Benoit Mandelbrot to describe a "self-similar" geometrical object that looks much the same on many different scales of measurement. This property contrasts with the property of a circle, for example, which loses its structure when viewed on a different scale and becomes almost a straight line when any arc is greatly magnified.

Fractals are representations of objects with an infinite amount of detail. When magnified, fractals do not become simpler, but instead remain as complex as they were without magnification. This is why fractals seem to describe natural objects in a better way than simple geometric figures like triangles, rectangles, or circles.

A coastline is a classical example of self-similarity in nature. From the air, a sea coast looks irregular by virtue of its bays and headlands. A closer look will reveal the same structure yet on a different scale. Each bay has its own bays and headlands. An even closer look will show even more bays and promontories within the larger bays. Even a beach will have small bays, capes, and peninsulas. On a much smaller scale in nature, a microscope will reveal a self-similar structure even within a grain of sand, which will have indentations and extrusions.

Constructing Geometric Fractals

Any mathematically created fractal can be made by the iteration, or repetition, of a certain rule. There are three basic types of iteration:

generator iteration, which is repeatedly substituting certain geometric shapes with other shapes;
IFS (Iterated Function System) iteration, which is repeatedly applying geometric transformations (such as rotation and reflection) to points; and
formula iteration, which is repeating a certain mathematical formula or several formulas.

The property of self-similarity holds true for the majority of mathematically created fractals.

The figure below illustrates the geometric construction of the Koch Curve, named after Helge von Koch, a Swedish mathematician who introduced this curve in a 1904 paper. First, begin with a straight line, as shown in (a). This initial object can be called the initiator. Partition this into three equal parts, then replace the middle third by an equilateral triangle and take away its base, as shown in (b). These steps are repeated with each resulting segment, as shown in (c). The repetition of steps is known as iteration. The curve shown in (d) is the result after three iterations, and the curve in (e) is after four iterations. The actual Koch Curve cannot be shown because it is theoretical, resulting from an infinite number of iterations.

Other geometric fractals can be created using the same method. Using a triangle as the initiator, the Sierpinski Gasket* is constructed as shown below. With each iteration, the figure becomes more complex as scaled copies build upon identical scaled copies, as shown by the small triangles on the left. The large image on the far right shows the results after six iterations.

*The Sierpinski Gasket is named after Polish mathematician Waclaw Sierpinski (1882–1969).

Characteristics of Fractals

When showing images of fractal figures, approximations given by a finite number of steps are displayed because these approximations—as in the case of the Koch Curve—will yield a curve with finite length. The actual fractal curve will have infinite length.

Because actual fractal figures like the Koch Curve have infinite length, they have interesting properties uncommon in simpler geometric figures. For example, a fractal closed curve such as the Koch Snowflake can enclose a figure with infinite perimeter and finite area. Although not shown here, the Koch Snowflake is constructed from an equilateral triangle, using a Koch Curve to initiate each of its sides. The snowflake has an infinite perimeter because the geometric pattern in (d) of the previous unnumbered figure comprises the snowflake's outer border and can be repeated an infinite number of times at increasingly smaller scales.

Fractal Dimensions. Because of their complexity, fractal objects cannot be assigned a dimension as can a line or a square. For example, the Koch Curve cannot have dimension 1, as a line, nor can it have dimension 2, as a square. So there must be other ways of calculating its fractal dimension.

The calculation of fractal dimensions is related to (1) the number of pieces into which a structure can be divided and (2) the reduction factor. The Koch Curve, in general, has 4^k pieces with a reduction factor of . So when it has four pieces, the reduction factor is ⅓, and when it has sixteen pieces the reduction factor is . Similarly, the Sierpinski Gasket, in general, has 3^k pieces with a reduction factor pieces the reduction factor of ½, and with nine pieces the reduction factor is ¼.

This idea was used in 1919 by the German mathematician Felix Hausdorff to define a fractal dimension that agrees with the usual dimension on the usual spaces. Although it is too complicated to be presented here, it is interesting to know that the dimension of the Koch Curve is approximately 1.2619 (or ) and the Sierpinski Gasket has a dimension close to 1.585 (or ).

For shapes that are not as regular as the Koch Curve or the Sierpinski Gasket, such as clouds or coastlines, this method of determining the fractal dimension does not work. Fractals that are not composed of a certain number of identical versions of itself require other methods for determining the fractal dimension.

Julia and Mandelbort Sets

Complex numbers are numbers of the form a + bi, where i = . By representing the complex number a + bi with the point (a, b ) in the Cartesian plane , a graphical representation of the complex numbers known as the complex plane is obtained. Complex numbers can be added, multiplied, and divided, just as real numbers . However, it is important to bear in mind that i ² = −1. So functions can be defined using complex numbers as input, and the output of these functions will be, in general, complex numbers.

Gaston Julia (1893–1978) investigated what happens when functions in the complex plane are iterated. Consider, for example, the function f (z ) = z ² + c, where c is a complex number. For real numbers, it is not difficult to evaluate this function. If c = 1 + i, and one wants to evaluate the function for z = 2, then f (2) = 2² + 1 + i = 4 + 1 + i = 5 + i. Squaring complex numbers is just a little bit more difficult, but it is enough to realize that when a function like this takes a complex number as input, it yields another complex number as output. If this function is iterated (that is, if the output becomes the input), and the function is evaluated again and again, one of two things can happen. Either the output numbers will begin to grow and to go farther from the origin, or somehow they will stay close to the origin, even if the function is iterated many times.

For example, select c = −0.125 + 0.75i, and evaluate f for z = 0. Evaluating the function again using as the input the output of f (0), and continuing this repetition of using each output value as the next input value yields a sequence of complex numbers different than the sequence of complex numbers that would result from evaluating the same function with an initial value of z = −0.5 + 0.5i. The difference is that for the initial value z = 0, the resulting sequence of complex numbers remains bounded; that is, the sequence remains close to the origin. On the other hand, the sequence given by z = −0.5 + 0.5i quickly goes far away from the origin.

The collection of complex numbers, represented as points on the complex plane, that lead to sequences that stay always close to the origin is called the prisoner set for c, whereas the collection of points that lead to unbounded sequences is called the escape set for c. The Julia Set is the boundary between the two sets.

Although not shown here, the prisoner set for c = −0.125 + 0.75i and its bordering Julia Set is considered connected because it appears in one piece. On the other hand, the Julia Set for c = −0.75 + 0.125i is disconnected because it consists of pieces that are separated from each other. If all those values c in the complex plane that have connected Julia Sets are colored black, the result is known as the Mandelbrot Set, named in honor of

Benoit Mandelbrot. It is not surprising that this set has a complexity that placed it beyond the reach of mathematicians until computers were used to study it. Mandelbrot studied Julia's work extensively and used computer graphics to render the Julia Sets and the Mandelbrot Set.

Self-similarity in the Mandelbrot Set is of a different nature than in the Koch Curve and Sierpinski Gasket because it arises from iterations of quadratic functions rather than from generator iterations or IFS iterations, as described above. In the Mandelbrot Set, identical pictures cannot be seen right away. But as the four-frame image shows, under increasing magnifications, the borders will reveal hidden complexities and even tiny copies of the Mandelbrot Set.

Fractals in Science and Art

Before Mandelbrot, none of the mathematical pioneers thought that their theoretical speculations about iterative processes and their relation to extremely unusual sets would end up being the best tools to describe nature. And yet fractals have proven to be a rich subject of study. They have been used to describe nature and are used frequently by scientists of different disciplines to explore very diverse phenomena. Fractal structures can be found in the leaves of a tree, in the course of a river, in the shape of a broccoli, in our arterial system, and on the surface of a virus.

The earliest applications of fractals, and perhaps the most widely seen by nonscientists, occur in the arts and in the film industry, where fractal forgery has been used to create landscapes for science fiction movies. Using fractals, convincing simulations of clouds, mountains, and surfaces of alien worlds have been created for our amusement.

In the 1970s, a young scientist, Loren Carpenter, made a computer movie of a flight over a fractal landscape. This brought him to the attention of Lucasfilm Ltd, whose graphic division, Pixar, immediately hired him. His work with fractals was used to create the geography of the moons of Endor and the outline of the Death Star in the movie Return of the Jedi. Fractals were also used to generate the landscape of the Genesis planet in the movie Star Trek II: Wrath of Khan. Carpenter has received awards for his contributions to the film industry, and his work in these two movies triggered the extended use of fractals for special effects and to simulate landscapes and other irregular shapes in three-dimensional (3-D) computer games.

The study of fractals is still a young branch of mathematics, and more applications are yet to be revealed.

see also Mandelbrot, Benoit B.; Numbers, Complex.

Óscar Chávez and

Gay A. Ragan