Field:Fractals

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Fractals

A fractal is often defined as a geometric shape that is self-similar, that is, whose magnified parts look like a smaller copy of the whole. The term "fractal" was coined by Benoit Mandelbrot in 1975 from the latin term fractus meaning "fragmented" or "irregular".

1 Fractals

The Mandelbrot Set

The basic concept

This concept can be explained in a commonly used analogy involving the coastline of an island.

Suppose you wanted to measure the total perimeter of an island. You could begin by roughly estimating the perimeter of the island by measuring the border of the island from a high vantage point like an airplane and using miles as units. Next, to be more accurate, you could walk along the island's borders and measure around its various coves and bays using a measuring tape and foot as units. Then, if you wanted to be really accurate, you could carefully measure around every single protruding rock and detail of the island with foot-long ruler and use inches as a measuring unit.

The perimeter of the island would grow as you decrease the size of your measuring device and increase the accuracy of your measurements. Also, the island would more or less self-similar (in terms of becoming more and more jagged and complex) as you continued to shorten your measuring device.

More Information

In addition to self-similarity, there are other properties exhibited by fractals:

Fine or complex structure at small scales
Too irregular to be described by traditional geometric dimension
Defined recursively

Self-Similarity

Self-Similiarity of Sierpinkisi's Triangle

Although all fractals exhibit self-similarity, they do not necessarily have to possess exact self-similarity, which would mean that the parts look exactly like the whole. The coastline fractal explained above does not have exact self-similarity, but its parts are very similar to the whole, while fractals made by iterated function systems (such as Sierpinski's Triangle, shown at the right) have exact-similarity.^[1]

Fractal (Non Integer) Dimension

Fractals are too irregular to be defined by traditional or Euclidean geometry language. Objects that can be described by Euclidean geometric dimensions include a line (1 dimension), an square (2 dimension), and a cube (3 dimension). Fractals are instead described by what is called Hausdorff or fractal dimension that measures how fully a fractal seems to fill space. For example, going back to the coastline example above, the coastline of Norway has an estimated fractal dimension of about 1.52 so it is not quite a line, but not quite an area either.

Recursive

A fractal must have a recursive definition, meaning that the fractal is defined in terms of itself. Fractals can be described by a single equation or by a system of equations, and created by taking an initial starting value and applying the recursive equation(s) to that value over and over again (a process called iteration). This iteration takes the output calculated from the previous iteration as the input for the next statement. Similarly, if the recursive definition of a fractal is a process, that process is first applied to the starting geometric shape and then continuously iterated to the segments resulting from the previous iteration. Recursive can be seen as a kind of positive feedback loop, where the same definition is applied infinitely by using the results from the previous iteration to start the next iteration.

Click here to learn more about Iterated Functions.

Types of Fractals

There are four main types of fractals that are categorized by how they are generated. In addition, numerous fractals occur naturally in lightening, broccoli, blood vessels, landscapes, and other phenomena.

Iterated function systems (IFS)

A IFS fractal consists of one of more recursive equations or processes that describe the behavior of the fractal and are iterated (or applied continually). These fractals are always exactly self-similar and are made up of an infinite number of self-copies that are transformed by a function or set of functions.

Examples include: Koch Snowflake, Cantor Set, Harter-Heighway Dragon, Barnsley’s Fern (Blue Fern), and Sierpinski's Triangle.

Strange attractors

Fractals that are considered strange attractors are generated from a set of functions called attractor maps or systems. These systems are chaotic and dynamic. Initially, the functions appear to map points in a seemingly random order, but the points are in fact over time evolving towards a recognizable structure called an attractor (because it "attracts" the points into a certain shape).

Examples include: Lorenz Attractor, Henon Attractor, and Rössler Attractor.

Random fractals

These fractals are created through stochastic methods, meaning that the behavior of these fractals depend on a random factor and usually probability restraints. One way to differentiate between chaotic and random fractals is to observe that chaotic fractals have errors (the difference between one plotted value to the next) that grow exponentially, while random fractal errors are simply random.

Examples include: Levy Flights, Brownian Motion, Brownian Tree, and fractal landscapes.

Escape-time (orbit) fractals

Escape-time fractals are created in the complex plane with a single function, some $f(z)$ , where z is a complex number. On a computer, each pixel corresponds to a complex number value. Each complex number value is applied recursively to the function until it reaches infinity or until it is clear that value will converge to zero. A color is assigned to each complex number value or pixel: the pixel is either colored black if the value converges to zero or the pixel is given a color based on the number of iterations (aka. escape time) it took for the value to reach infinity. The intermediary numbers that arise from the iterations are referred to as their orbit. The boundary between black and color pixels is infinite and increasingly complex.