Field:Fractals

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{{Field Page |Field=Fractals |BasicDesc=A fractal is often defined as a geometry shape that is self-similarity>, or whose magnified parts look like a smaller copy of the whole. It was coined by Benoit Mandelbolt in 1975 from the latin term ‘’fractus’’ meaning ”fragmented” or “irregular”.

This concept can be explained in a commonly used nature 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. Then, if you wanted to be really
 accurate, you could carefully measurement around every single protruding rock and detail of the island
 with a yardstick or even a foot-long ruler.

Clearly, 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 like similar (in terms of becoming more and more jagged and complex) as you continued to decrease your measuring device. |FurtherInfo=In addition to self-similarity, there are other traits exhibited by fractals:

  • Fine or complex structure at small scales
  • Too irregular to be described by traditional geometric dimension
  • Defined by a recursive statement

Self-Similarity

Although all fractals exhibit self-similarity, they do not necessarily have to possess exact self-similarity. 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 (explained later) have exact-similarity.

Irregular Dimension

File:NorwayCoastline.jpg

Fractals are too irregular to be defined by tradition or Euclidean geometry language. Objects that can be described by Euclidean geometric dimensions include a line (1 dimension), a square (2 dimension), and a cube (3 dimension). Fractals are instead described by a specific term 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. Click here to learn more about Fractal Dimension and how it is calculated.

Recursive

Fractals are defined by recursive or iterating statements that can be equations or geometric curves. Basically, a recursive statement is a rule that defines the shape or behavior of a fractal and is applied over and over again, using the output calculated from the previous statement as the input for the next statement. This can be seen as a kind of positive feedback loop, where the same definition or statement is applied infinitely by using the results from the previous iteration to start the next iteration.

Click here to learn more about Iterated Functions and its mathematical implications.

Examples of fractals

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

  • Iterated function systems (IFS)

Fractals created with IFS are always exactly self-similarity, so that these fractals are essential constructed of infinitely many infinitely smaller pieces of itself that overlap. The IFS consists of one of more functions or methods that describe the behavior of the fractal.
Fractals are often generated by IFS governed by the ‘’chaos game’’. This is where the IFS consists of a set of functions that each have a fixed probability. A starting point is picked at random, and then one function is picked (according to probability) and applied to the point for an infinite amount of iterations.
Examples include: Koch’s Snowflake, Harter-Heighway Dragon, Barnsley’s 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, because the functions map points in a seemingly random order. These points are actually not completely random and are in fact evolving towards a structure that can eventually be seen as the form of the attractor.
Examples include: Lorenzo Attractor, Henon Attractor, Cantor Dust , and Rossler Attractor.
  • Random fractals

These fractals are created through stochastic methods, meaning there are no set functions that determine the behavior of the fractals. Although the fractals still must have rules that they follow, they are unlike chaotic IFS fractals because chaotic fractals are governed by a set of functions that are picked according to probability and these follow some sort of a pattern (and grow exponentially), while random fractals are simply random (and grow randomly).
Examples include: Levy Flights, Brownian Motion, Brownian Tree, and fractal landscapes.
  • Escape-time (“orbit”) fractals

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