A fractal is a mathematical set that has a fractal dimension that usually exceeds its topological dimension and may fall between the integers.

Fractals are typically self-similar patterns, where self-similar means they are “the same from near as from far”.Fractals may be exactly the same at every scale, or, as illustrated in Figure 1, they may be nearly the same at different scales.

The definition of fractal goes beyond self-similarity per se to exclude trivial self-similarity and include the idea of a detailed pattern repeating itself.

Figure 1

Self-similarity, which may be manifested as:

• Exact self-similarity: identical at all scales;  e.g. Koch snowflake
• Quasi self-similarity: approximates the same pattern at different scales; may contain small copies of the entire fractal in distorted and degenerate forms;  e.g., the Mandelbrot set’s satellites are approximations of the entire set, but not exact copies, as shown in Figure 1
• Statistical self-similarity: repeats a pattern stochastically so numerical or statistical measures are preserved across scales;  e.g., randomly generated fractals; the well-known example of the coastline of Britain, for which one would not expect to find a segment scaled and repeated as neatly as the repeated unit that defines, for example, the Koch snowflake
• Qualitative self-similarity: as in a time series
• Multifractal scaling: characterized by more than one fractal dimension or scaling rule

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.

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