As mathematical equations, fractals are usually nowhere differentiable, which means that they cannot be measured in traditional ways. An infinite fractal curve can be perceived of as winding through space differently from an ordinary line, still being a 1-dimensional line yet having a fractal dimension indicating it also resembles a surface.

The mathematical roots of the idea of fractals have been traced through a formal path of published works, starting in the 17th century with notions of recursion, then moving through increasingly rigorous mathematical treatment of the concept to the study of continuous but not differentiable functions in the 19th century, and on to the coining of the word fractal in the 20th century with a subsequent burgeoning of interest in fractals and computer-based modelling in the 21st century. The term "fractal" was first used by mathematician Benoît Mandelbrot in 1975. Mandelbrot based it on the Latin frāctus meaning "broken" or "fractured", and used it to extend the concept of theoretical fractional dimensions to geometric patterns in nature.

There is some disagreement amongst authorities about how the concept of a fractal should be formally defined. The general consensus is that theoretical fractals are infinitely self-similar, iterated, and detailed mathematical constructs having fractal dimensions, of which many examples have been formulated and studied in great depth. Fractals are not limited to geometric patterns, but can also describe processes in time. Fractal patterns with various degrees of self-similarity have been rendered or studied in images, structures and sounds and found in nature, technology, and art.

Pronunciation

• IPA: /fʁaktal/

Noun

fractal m. (plural fractals)

1. fractal

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• Chaos (39)
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• Software (50)

See also:

• Science: Math: Applications: Complex Systems (81)
• Science: Math: Differential Equations: Dynamical Systems (64)

• The Dynamical Systems and Technology Project
  math.bu.edu
  Educational resource from the Boston University. Includes mathematical framework and formulation, animated illustrations and calculation spreadsheets.
• Fractal Geometry
  classes.yale.edu
  An educational resource on the mathematical framework and formalism from the Yale University, covering the concept of self similarity. Includes topical examples, images, algorithms and software.
• Mathematical Figures Using Mathematica
  mathforum.org
  Collection of sets and attractors. Includes Mathematica source code, mathematical formulations and illustrations.
• Wikipedia Category: Fractals - Collection of encyclopedia articles about self-similar geometric objects with both aesthetical and scientific uses.
• Wikipedia: Fractal - Free encyclopedia article covering historical aspects and mathematical formulations. Includes two and three dimensional illustration sets.

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