Fractaline - What is a Fractal ?

What is a Fractal ?

"Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth,
nor does lightning travel in a straight line."

Benoit Mandelbrot, 1983

A Fractal is an object or quantity which displays Self-Similarity, in a somewhat technical sense, on all scales. The object need not exhibit exactly the same structure at all scales, but the same "type" of structures must appear on all scales. A plot of the quantity on a log-log graph versus scale then gives a straight line, whose slope is said to be the Fractal Dimension. The prototypical example for a fractal is the length of a coastline measured with different length rulers. The shorter the ruler, the longer the length measured, a paradox known as the Coastline Paradox.

The Modeling Power of Fractals

Many objects in nature are so complicated and irregular that they cannot be modeled well using conic sections, polygons, spheres and the other familiar objects of classical geometry. For example, circulatory systems, clouds, trees, mountains, and coastlines cannot be reduced to combinations of simple shapes from classical geometry. Where classical geometry ends as a tool for analyzing the complexity of natural objects, fractal geometry begins. Today, fractals are used to model a wide range of biological and topographical entities and to produce ultra-realistic special effects for movies and video games.

How Long is the Coastline of Britain?

The question "How long is the coastline of Britain?" posed by Benoit Mandelbrot, the father of modern fractal theory, in his book The Fractal Geometry of Nature is not as simple as it appears. The problem is that one's answer to this question depends on the length of the ruler one uses. Unlike circles and the other shapes from classical geometry, coastlines are very irregular. They're full of inlets, bays, and rocky shores. A shorter measuring stick will fit more snugly in these nooks and crannies and increase the estimated length of the coastline. Hence, if we measure the length of Britain's coastline using a mile-long ruler, we will get one value. If we use a shorter ruler, say a yardstick, we will get a larger value because a yardstick can more closely approximate Britain's convoluted boundary. In fact, as the scale of measurement decreases, the estimated length increases without limit. Thus, as the length of the ruler approaches zero, the estimated length of the coastline approaches infinity. This difficulty in measuring due to the irregularity of the object being measured is characteristic of fractal curves and surfaces.

Self-Similarity

Fractal theory is grounded in geometry and dimension theory. Geometrically, fractals are independent of scale and appear equally detailed at any level of magnification. This property, called self-similarity, means that any portion of a self-similar fractal curve, if blown up in scale, would appear identical to the whole curve. In other words, if we shrink or enlarge a fractal pattern, its appearence remains unchanged. This repetition of a pattern at all scales, no matter how small, is exhibited by many natural objects. For example, imagine that you are in space looking at the coastline of Britain. As you approach the Earth, the coastline still looks like a coastline. No matter how close you get to Britain's shore, the coastline appears equally complex. Even after you land your spacecraft and get down on your hands and knees with a microscope at the water's edge, the coastline still looks jagged and irregular.

Fractal dimension

The term "fractal," introduced in 1975 by Benoit Mandelbrot, is an abbreviation for "fractional dimension." We all learned in high school that in classical geometry a line is an one dimensional object and a plane is two dimensional. Strangely, if we put enough kinks in a line, the resulting fractal curve will have a dimension somewhere between one and two, so that it is neither a line nor a plane but something in between. Similarly, an extremely convoluted surface will have a dimension beween two and three. Such a figure is called a fractal. For objects of classical geomery, the classical dimension of the object and its fractal dimension are the same. A fractal, on the other hand, is an object that has a fractal dimension that is strictly greater than its classical dimension. Although they are continuous, fractal curves are so rough that they are nowhere differentiable. The concept of fractal dimension provides a way to measure how rough fractal curves are. The more jagged and irregular a curve is, the higher its fractal dimension, a value betwen one and two. Fractional dimension is related to self-similarity in that the easiest way to create a figure that has fractional dimension is through self-similarity.

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