fractality

Fractality is a property of an object or pattern that shows self-similarity across different scales, often appearing in mathematical objects called fractals. A fractal is typically generated by an iterative process or recursive rule, and its detail persists regardless of magnification.

The concept was popularized by Benoit Mandelbrot in the 20th century, who coined the term fractal. Fractals commonly have non-integer, or fractional, dimensions, measured by fractal dimension, such as the Hausdorff dimension or box-counting dimension. Self-similarity can be exact, where each part is a reduced copy of the whole, or statistical, where smaller parts resemble the whole in aggregate.

Classic examples include the Cantor set, Koch snowflake, and Sierpinski triangle, which exhibit exact self-similarity. The Mandelbrot set and related Julia sets arise from complex quadratic mappings and display intricate, infinitely detailed boundaries. Many natural phenomena resemble fractals through statistical self-similarity, including coastlines, mountain profiles, clouds, and plant growth.

Applications span mathematics and computer science as tools for modeling irregular shapes, generating natural-looking graphics, compressing data, and optimizing antenna designs. Fractal analysis provides a framework for describing complexity in systems that do not fit traditional Euclidean geometry, capturing patterns that repeat at multiple scales.