The World of Fractals

A fractal is a mathematical set that has a fractal dimension that usually exceeds its topological dimension and may fall between the integers.

Fractals are typically self-similar patterns, where self-similar means they are “the same from near as from far”.Fractals may be exactly the same at every scale, or, as illustrated in Figure 1, they may be nearly the same at different scales.

The definition of fractal goes beyond self-similarity per se to exclude trivial self-similarity and include the idea of a detailed pattern repeating itself.

fractalmandel

Figure 1

Self-similarity, which may be manifested as:

  • Exact self-similarity: identical at all scales;  e.g. Koch snowflake
  • Quasi self-similarity: approximates the same pattern at different scales; may contain small copies of the entire fractal in distorted and degenerate forms;  e.g., the Mandelbrot set’s satellites are approximations of the entire set, but not exact copies, as shown in Figure 1
  • Statistical self-similarity: repeats a pattern stochastically so numerical or statistical measures are preserved across scales;  e.g., randomly generated fractals; the well-known example of the coastline of Britain, for which one would not expect to find a segment scaled and repeated as neatly as the repeated unit that defines, for example, the Koch snowflake
  • Qualitative self-similarity: as in a time series
  • Multifractal scaling: characterized by more than one fractal dimension or scaling rule

As mathematical equations, fractals are usually nowhere differentiable, which means that they cannot be measured in traditional ways. An infinite fractal curve can be perceived of as winding through space differently from an ordinary line, still being a 1-dimensional line yet having a fractal dimension indicating it also resembles a surface.

The mathematical roots of the idea of fractals have been traced through a formal path of published works, starting in the 17th century with notions of recursion, then moving through increasingly rigorous mathematical treatment of the concept to the study of continuous but not differentiable functions in the 19th century, and on to the coining of the word fractal in the 20th century with a subsequent burgeoning of interest in fractals and computer-based modelling in the 21st century.

The term “fractal” was first used by mathematician Benoît Mandelbrot in 1975. Mandelbrot based it on the Latin frāctus meaning “broken” or “fractured”, and used it to extend the concept of theoretical fractional dimensions to geometric patterns in nature.

Mandelbrot set

The Mandelbrot set, named after Benoit Mandelbrot, is a fractal. Fractals are objects that display self-similarity at various scales. Magnifying a fractal reveals small-scale details similar to the large-scale characteristics. Although the Mandelbrot set is self-similar at magnified scales, the small scale details are not identical to the whole.

In fact, the Mandelbrot set is infinitely complex. Yet the process of generating it is based on an extremely simple equation involving complex numbers.

Understanding complex numbers

The Mandelbrot set is a mathematical set, a collection of numbers. These numbers are different than the real numbers that you use in everyday life. They are complex numbers. A complex number consists of a real number plus an imaginary number. The real number is an ordinary number, for example, -2. The imaginary number is a real number times a special number called i, for example, 3i. An example of a complex number would be -2 + 3i.

The number i was invented because no real number can be squared (multiplied by itself) and result in a negative number. This means that you can not take the square root of a negative number and get a real number. By definition, when you take the square root of a number, you find a number that can be squared to get that number. The number i is defined to be the square root of -1. This means that i squared is equal to -1. So when you square an imaginary number you can get a negative number. For example, 3i squared is -9.

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Real numbers can be represented on a one dimensional line called the real number line. Negative numbers like -2 are plotted to the left of zero and positive numbers like 2 are plotted to the right of zero. Any real number can be graphed on the real number line.

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Since complex numbers have two parts, a real one and an imaginary one, we need a second dimension to graph them. We simply add a vertical dimension to the real number line for the imaginary part. Since our graph is now two-dimensional, it is a plane, the complex number plane. We can graph any complex number on this plane. The colored dots on this graph represent the complex numbers [2, + 1i], [-1.5, + 0.5i], [2, – 2i], [-0.5, – 0.5i], [0, + 1i], and [2, + 0i].

Graphing the Mandelbrot set

The Mandelbrot set is a set of complex numbers, so we graph it on the complex number plane. However, first we have to find many numbers that are part of the set. To do this we need a test that will determine if a given number is inside the set or outside the set. The test is based on the equation Z = Z 2 +  C. C represents a constant number, meaning that it does not change during the testing process. C is the number we are testing, the point on the complex plane that will be plotted when testing is complete. Z starts out as zero, but it changes as we repeatedly iterate this equation. With each iteration we create a new Z that is equal to the old Z squared plus the constant C. So the number Z keeps changing throughout the test.

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We’re not really interested in the actual value of Z as it changes, we just look at its magnitude. The magnitude of a number is its distance from zero. For example, the number -9 is a distance of 9 from zero, so it has a magnitude of 9. The magnitude of a complex number is harder to measure. To calculate it, we add the square of the number’s distance from the x-axis (the horizontal real axis) to the square of the number’s distance from the y-axis (the imaginary vertical axis) and take the square root of the result. In this illustration, a is the distance from the y-axis, b is the distance from the x-axis, and d is the magnitude, the distance from zero.

As we iterate our equation, Z changes and the magnitude of Z also changes. The magnitude of Z will do one of two things. It will either stay equal to or below 2 forever, or it will eventually surpass two. Once the magnitude of Z surpasses 2, it will increase forever. In the first case, where the magnitude of Z stays small, the number we are testing is part of the Mandelbrot set. If the magnitude of Z eventually surpasses 2, the number is not part of the Mandelbrot set.

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