fractals

Fractals are geometric objects or sets that exhibit self-similarity across scales and often possess non-integer, or fractional, dimensions. They can be generated by simple rules that are repeated recursively, producing complex forms with infinite detail in theory. The concept was popularized by Benoit Mandelbrot in the 1960s and 1970s, though early examples include the Cantor set, the Koch snowflake, and the Sierpinski triangle.

Fractals are categorized by the type of self-similarity: exact self-similarity (smaller copies identical to the whole), statistical self-similarity (patterns repeat in a statistical sense), and multifractals (a spectrum of scaling behaviors). They are often constructed via iterated function systems of contractive mappings, escape-time algorithms for complex plane sets like the Mandelbrot and Julia sets, or L-systems for plant-like figures.

A key feature is a fractal dimension, a measure (such as the Hausdorff or box-counting dimension) that generally lies between the topological dimension and the embedding dimension, reflecting complexity.

Fractals appear in mathematics and computer graphics, modeling natural phenomena such as coastlines, clouds, mountain ranges, and river networks. They also have practical applications in antenna design, image compression, data analysis, and modeling of turbulent flows.

Fractals remain idealizations; real-world objects exhibit approximate self-similarity and finite detail. Still, the study of fractals links geometry, dynamics, and chaos, offering a framework for describing complex, irregular shapes.