Field:Fractals

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Fractals

A fractal is a geometric shape that is self-similar and has a fractal dimension. In 1975, the father of fractals, Benoît Mandelbrot, coined the term from the Latin fractus, meaning “‘to break:’ to create irregular fragments,” which also describes a few of the methods used to create fractals. [1]


Contents

The Mandelbrot Set
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. [1]

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

Main Properties

A few important properties pertaining to the nature of a fractal are:

  • self-similarity[2]
  • iteration ad infinitum (to infinity)[2]
  • fractional dimension [2]

In addition, 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
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.[3]

Fractal (Non Integer) Dimension

Fractals are too irregular to be defined by traditional or Euclidean geometry language. Euclidean geometry is constrained to having forms in a “dimension the same as that of the embedding space”[2]. For example, a line has D = 1 because it is described as having only one direction needed to define it, as if on a number line where only one coordinate is needed to find a location in the one dimensional space. A square has D = 2, a square is created by drawing lines that create the top and bottom of the figure in on direction and then drawing two other lines perpendicular to that direction; analogous to the idea of identifying a point in the first dimension, in D = 2, two coordinates are needed to define a location in the two dimensional space. Fractals are instead described by what is called Hausdorff or fractal dimension that measures how fully a fractal seems to fill space. For instance, the Sierpinski triangle “has a dimension intermediate between that of a line and an area" and is present in a fractional dimension.[1]


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

Fractals are categorized by how they are generated.

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.

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).

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.

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.

Two Basic Fractals

These fractals illustrate some of the features described above.

The Sierpinski Triangle

One method of generating the Sierpinski Triangle is with the growth rule, this method is described as a process similar to how a “child might assemble a castle from building blocks.”[2] In the initial state we have a basic unit, a single triangular-shaped tile with a unit mass (M = 1) and unit length (L = 1). The Sierpinski triangle is “defined operationally as an ‘aggregation process’ obtained by a simple iterative process,” simply put, the triangle is made by adding unit tiles on top of unit tiles to infinity.[2] In stage one, two unit triangles are added to the initial triangle in such a way so that the new formation appears to be a large red triangle with an inverted triangular piece missing from its middle (Figure 1). Now this object has a mass M = 3 and a length L = 2.
With density defined as:
p(L)=M(L)/(L)^2,
note that the density has decreased by 3/4 in the first stage, or iteration. In the following stages, the density will continue to decrease in a monotonic fashion by a factor of (3/4)^n. The factor by which the density decreases is a simple power law relating density and length and can be described as:
p(L)=L^a,
Figure 1. The Sierpinski Triangle after 1 iteration.
Figure 1. The Sierpinski Triangle after 1 iteration.
and when plotted on a log-log graph, we can then observe a, the slope of the function. From two points, we get the expression for slope:
a = log1-log(3/4)/log1-log2=-2log3/log2.
With the slope, a, the fractal dimension of the Sierpinski triangle can be concluded by substituting M(L) \equiv AL^D into p(L)\equiv M(L)/(L)^2 yielding:
 p(L) \equiv AL^{D-2}.
And by comparing this exponent D to a we conclude that Sierpinski's triangle is a shape with a fractal dimension,
 D = \frac{\log 3}{\log 2}\approx 1.585.

The Koch Snowflake

With the construction of the Koch snowflake,we start off with the initiator; in this case, the initiator is also an equilateral triangle. An initiator is a figure that precedes an iteration, it can be the initial figure before any iterations are carried out or it can be the figure at the iteration right before the next. Now to create the generator, we take one of the sides of the triangle and break the edge into thirds and attach a triangle onto the middle third of the side. The triangle added sits on the initiator and is a scaled version of the original figure. A generator is the figure that is created after iteration. The figure at first iteration resembles the Star of David, a star hexagon. Now, each of the sides becomes the initiator and each one is broken up into thirds where, once again, we will place a scaled equilateral triangle on the middle third. After this step, the rule is made clear, break the lengths of the triangle into thirds and add a triangle to the middle third section, and as with our previous fractal this is to be done ad infinitum. Figure 2 illustrates iterations of the snowflake.
Figure 2. First 7 iterations.
Figure 2. First 7 iterations.
As the snowflake is iterated ad infinitum, the fractal nature is evident in the “segmenting process” of breaking the sides apart and the “cascading” of the shapes over the initiator.[1] We can look at one-third of the snowflake, at the Koch curve, and see in greater detail, the self-similarity in the shape (Figure 3).
Figure 3. self similarity.
Figure 3. self similarity.

Examples of Fractals

To see all Fractal related pages, head over to the Fractals category.

References

  1. 1.0 1.1 1.2 1.3 Mandelbrot, Benoît B. The Fractal Geometry Of Nature. New York, NY: W. H. Freeman, 1983. Print.
  2. 2.0 2.1 2.2 2.3 2.4 2.5 Bunde, Armin, and Shlomo Havlin. Fractals and Disordered Systems. 2nd ed. Berlin, Germany: Springer-Verlag Berlin Heidelberg, 1996. Print.
  3. http://en.wikipedia.org/wiki/Fractal

Other Links

Cynthia Lanius, Cynthia Lanius' Lessons: A Fractal Lesson

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