Definition:Fractal
Definition
A fractal is a geometric object whose dimension is not an integer, but is instead fractional.
A fractal is often generated by an infinitely repeated process
It also has the property of self-similarity on all scales.
Formal Definition
A fractal is a set of points which has a similarity dimension or Hausdorff dimension which is not an integer.
Examples
Koch Snowflake
The Koch snowflake is an example of a fractal.
Cantor Set
The Cantor set is an example of a fractal.
Strange Attractor
In many cases, a strange attractor associated with a transformation or a flow is a fractal.
Also see
- Results about fractals can be found here.
Historical Note
Fractals are used in computer graphics to randomly generate images of natural objects, such as landscapes and other geographical features.
Fractals have also been used to study the formation of crystals, the phenomenon of electrical discharge, coagulation of particles, urban growth and many other areas.
Linguistic Note
The term fractal was coined by Benoît B. Mandelbrot in $1975$.
This arises from such objects having a fractional dimension.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): fractal
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): fractal
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): fractal
- Weisstein, Eric W. "Fractal." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Fractal.html