A fractal as a geometric object generally has the following features:
- fine structure at arbitrarily small scales
- is too irregular to be easily described in traditional Euclidean geometric language.
- is self-similar (at least approximatively or stochastically)
- has a Hausdorff dimension that is greater than its topological dimension (although this requirement is not met by space-filling curves such as the Hilbert curve)
- has a simple and recursive definition.
Due to them appearing similar at all levels of magnification, fractals are often considered to be ‘infinitely complex’. Obvious examples include clouds, mountain ranges and lightning bolts. However, not all self-similar objects are fractals — for example, the real line (a straight Euclidean line) is formally self-similar but fails to have other fractal characteristics.
Personally, I could just look at them all day long….