What are Fractuals? When Do You Use Them In The Real World?
A fractal is defined as a jagged or fragmented geometric shape which can be split into parts that are considered a reduced copy of the whole. Although the study of fractals have existed as early as the 17th century, but the term fractal was only coined in 1975 by Benoit Mandelbrot. It is derived from the Latin word fractus, which means broken or fractured. While a fractal is strictly a mathematical construct, it is found in various non-mathematical models such as natural systems and artworks.
To understand fractals, it is important to know first what their characteristics are. Its first characteristic is that its structure is defined by fine and small scales and/or substructures. Another characteristic it has is that its shape cannot be defined by Euclidean geometry. The next is that it is recursive and shows iteration to some degree. In addition, fractals are informally considered to be infinitely complex as they appear similar in all levels of magnification. There are a lot of natural phenomena that can be defined and predicted using fractals. Some of these shapes include clouds, vegetables, color patterns, lightning, and snowflakes.
Fractals started to be considered mathematical in nature when Leibniz considered recursive self similarity. However, a graph that can be considered fractal did not exist until 1872, when Karl Weierstrass was able to create a function that is "continuous everywhere yet differentiable nowhere".
Fractals are considered to be important because they define images that are otherwise cannot be defined by Euclidean geometry. Fractals are described using algorithms and deals with objects that don't have integer dimensions. Some of the more prominent examples of fractals are the Cantor set, the Koch curve, the Sierpinski triangle, the Mandelbrot set, and the Lorenz model.
Contrary to its complicated nature, fractals do have a lot of uses in real life applications. First, we start with art. The image created by a fractal is complex yet striking, and has intrigued artists for a long time already. In fact, fractal art is considered to be true art. Artists such as Jackson Pollock and Max Ernst, has used fractal patterns to create seemingly chaotic yet defined forms. Even in African art and architecture fractal shapes and images are prevalent. In addition, some artists are inspired by fractal images when creating their own art forms. Not only that: fractal images are actually being used nowadays to create special effects. Utilized in shows such as Star Trek and Star Wars, fractals are used to create landscapes that are otherwise impossible with conventional technology. On a related note, fractals are also used in creating some computer graphics.
In addition, fractals are a very important part in biological studies. For example, a lot of objects in nature are composed of complex figures that are otherwise not possible to be defined by Euclidean shapes. Most natural objects, such as clouds and organic structures, resemble fractals. As such, fractals can be used to capture images of these complex structures. In addition, fractals are used to predict or analyze various biological processes or phenomena such as the growth pattern of bacteria, the pattern of situations such as nerve dendrites, etc.
And speaking of imaging, one of the most important uses of fractals is with regards to image compressing. A pretty controversial process, it takes an image and expresses it into an iterated system of functions. This image is displayed quickly and is expressed in detail in any magnification.
All in all, studying fractals is both a complicated yet interesting branch of mathematic study. And yet despite all its intricacies, it still proves to be a useful tool.