Fraktal

Fraktal, or fractal in English, is a geometric object or set that exhibits self-similarity across scales and often possesses a non-integer dimension. The term was introduced by Benoit Mandelbrot in 1975 to describe shapes and sets that repeat patterns at progressively smaller scales.

As a rule, fractals arise from simple iterative rules or recursive processes. They may show exact self-similarity, where a miniature copy resembles the whole, or statistical self-similarity, where the pattern repeats in a probabilistic sense. Fractals are frequently generated by iterative function systems, L-systems, or complex dynamics, such as the Mandelbrot and Julia sets.

Classic examples include the Cantor set, obtained by removing middle thirds from a line segment; the Koch snowflake, formed by replacing each line segment with a three-segment bump; and the Sierpinski triangle, created by repeatedly removing central triangles. The Mandelbrot set visualizes the behavior of a simple quadratic family in the complex plane and has become a symbol of fractal geometry. Julia sets provide a related family of complex shapes.

Fractals are characterized by non-integer (fractal) dimensions, which can be described by Hausdorff or box-counting dimensions. They often exhibit infinite detail and non-differentiability, and their production can be exact or stochastic.

Applications span computer graphics, modeling of natural phenomena (coastlines, terrain, clouds), data compression, antenna design, and the study of irregular structures in physics and biology. Fraktal geometry provides a framework for describing complex patterns that traditional geometry cannot capture.