Fractal
Introduction
A Fractal is a complex geometric shape that can be split into parts, each of which is a reduced-scale copy of the whole. They are self-similar and independent of scale. Fractals are found in nature and are used in computer models.
History
The term "fractal" was first used by mathematician Benoit Mandelbrot in 1975. He derived it from the Latin word "fractus", meaning "broken" or "fractured". Mandelbrot was fascinated by irregularities in nature and believed that these could be modelled mathematically.
Mathematical Definition
In mathematics, a fractal is a subset of a Euclidean space for which the Hausdorff dimension strictly exceeds the topological dimension. Fractals can be defined in terms of a recursive or iterative process that can be carried out indefinitely, creating self-similar patterns at arbitrarily small scales.
Types of Fractals
There are several types of fractals, including:
- Mandelbrot set: This is a set of complex numbers defined by a simple iterative algorithm. It is named after Benoit Mandelbrot, who studied and popularized it.
- Julia set: This is a set of complex numbers that is defined by a recursive algorithm. It is named after the French mathematician Gaston Julia.
- Cantor set: This is a set of points lying on a single line segment that has a number of remarkable and deep properties.
- Sierpinski triangle: This is a fractal that is a self-similar set of equilateral triangles.
- Koch snowflake: This is a fractal curve, also known as the Koch island, which was first described by Helge von Koch.
Fractals in Nature
Fractals are not just mathematical constructs, but they also appear in nature. The branching patterns of trees and rivers, the structure of snowflakes, and the formation of clouds are all examples of natural fractals. The study of fractals in nature is a part of the field of chaos theory.
Fractals in Computer Graphics
Fractals have found extensive application in computer graphics, both for creating interesting shapes and for textures. They are used in the generation of certain types of digital images and animations, and in the design of computer game landscapes.