Fraktaali

Fraktaali, or fractal in English, is a geometric object or set characterized by self-similarity across different scales and often generated by repeating a simple rule or process. In Finnish usage, fraktaali is the standard term for this concept. Fractals can arise from deterministic rules, stochastic processes, or a combination of both, and they typically exhibit intricate detail at arbitrarily small scales.

Key properties of fractals include self-similarity, where a part of the object resembles the whole, and a fractional or non-integer dimension, known as the fractal dimension, which measures complexity beyond ordinary geometric shapes. Many fractals are created by iterative procedures, such as iterated function systems, L-systems, or recursive equations, and can produce endless detail with relatively simple definitions.

Common examples include the Cantor set, Koch snowflake, and Sierpinski triangle, as well as more famous complex sets like the Mandelbrot set and Julia sets. These illustrate how simple rules can generate highly intricate boundaries and structures.

Historically, the concept traces back to 19th- and 20th-century mathematics, but the term fraktaali was popularized by Benoit Mandelbrot in the 1970s and 1980s with the development of fractal geometry. The field connects geometry, analysis, and probability to study dimension, measure, and structure in both abstract sets and natural phenomena.

Applications range from modeling coastlines, clouds, and biological forms to computer graphics, data compression, and antenna design, where fractal geometries enable efficient representation and realistic rendering of complex shapes.