Fractal

TL;DR

• Patterns that repeat across scales and exhibit self-similarity.
• Appear in natural phenomena such as snowflakes, mountains, and coastlines.
• Used in computer graphics, art, music, and in modeling or analyzing chaotic and complex systems.

Definition

A fractal is a mathematical concept that describes a pattern that repeats itself at different scales. It is a self-similar structure that can be found in a wide range of natural phenomena, such as snowflakes, mountains, and coastlines.

Explanation

Fractals show the same pattern or structure at different magnifications: the pattern at a small scale resembles the pattern at a larger scale. This self-similarity is the defining property of fractals. They arise in both simple geometric constructions and in more complex mathematical objects generated by iterative processes. Fractals are applied in fields such as computer graphics, art, music, and the study of chaos and complexity, and they can serve as mathematical models of certain natural phenomena.

Examples

Mandelbrot set

The Mandelbrot set is a mathematical object generated using complex numbers and plotted on the complex plane. It is named after mathematician Benoit Mandelbrot and is known for intricate patterns that emerge from its complex mathematical calculations.

Sierpinski triangle

The Sierpinski triangle is created by starting with a triangle, dividing it into four smaller triangles, and repeating this process at each smaller scale. This produces a pattern described as infinitely complex, with no clear beginning or end. The Sierpinski triangle is often used as a simple, visual example of a fractal.

Use cases

• Computer graphics
• Art
• Music
• Mathematical models of natural phenomena
• Study of chaos and complexity

Related terms

• Self-similarity
• Mandelbrot set
• Sierpinski triangle
• Complex numbers
• Complex plane
• Chaos
• Complexity