fraktala

Fraktala refers to a geometric object or set that exhibits self-similarity across scales and complexity arising from simple iterative rules. Fractals are often built by repeating a transformation or subdivision, producing structures that look similar regardless of magnification.

Core properties include self-similarity, scale invariance, and often a non-integer fractal dimension, which quantifies how detail changes with scale. Fractal dimension can be defined in several ways, such as the Hausdorff or box-counting dimension. Fractals may be deterministic, produced by fixed rules, or random, incorporating stochastic elements.

Historically, the study of fractals traces to the work of Gaston Julia and Pierre Fatou on iterations of complex functions in the early 20th century. The term fractal was coined by Benoit Mandelbrot in 1975, who popularized fractal geometry as a framework for describing irregular shapes in nature and mathematics. Common construction methods include iterated function systems (IFS) and the chaos game, which generate self-similar patterns through repeated maps.

Typical examples include the Cantor set, Koch snowflake, and Sierpinski triangle, as well as the Mandelbrot and Julia sets in the complex plane. In nature, many structures such as coastlines, mountain profiles, river networks, and clouds exhibit fractal-like features, though not always in exact mathematical form.

Applications of fractals span computer graphics, digital image processing, geographic information systems, and the modeling of natural processes. They are also used in data compression, antenna design, and network theory. The field distinguishes between self-similar, self-affine, and multifractal objects, each with distinct scaling properties.