fractal

A fractal is a geometric object that exhibits self-similarity across different scales and often possesses a non-integer, or fractional, dimension. The term was coined by Benoit Mandelbrot in the 1960s–1970s to describe sets and curves that display intricate detail regardless of the level of magnification. Fractals can arise from simple recursive rules or iterative processes, producing complex patterns from straightforward constructions.

Key properties include self-similarity, where parts resemble the whole; infinite or very large detail; and a fractal dimension that, in many cases, lies between the familiar integers, capturing how completely the object fills space at progressively finer scales. Not all fractals have exact self-similarity; many natural fractals show statistical or approximate self-similarity.

Well-known mathematical fractals include the Cantor set, the Sierpinski triangle, and the Koch snowflake, which are generated by repeated subdivision and elimination processes. In the complex plane, the Mandelbrot set and related Julia sets illustrate how simple iterative equations can produce infinitely intricate boundaries.

Fractals also appear in nature and science, with coastlines, mountain ranges, clouds, and plant growth often cited as exhibiting fractal-like structure. Fractal analysis provides tools for measuring roughness and scaling behavior in data and images, with fractal dimension (such as Hausdorff or box-counting dimension) serving as a quantitative descriptor.

Applications extend to computer graphics, natural phenomenon modeling, signal and image processing, antenna design, and financial modeling, where self-similar patterns and scaling laws help simulate or analyze complex systems.

Historically, ideas related to fractals trace back to early work by Cantor, Weierstrass, and Julia, but the modern framing and terminology were established by Mandelbrot in the late 20th century.