What is a fractal?

Q2: What is a fractal? What are some examples of fractals?
A2: A fractal is a rough or fragmented geometric shape that can be subdivided
in parts, each of which is (at least approximately) a reduced-size copy of the
whole.  Fractals are generally self-similar and independent of scale.

There are many mathematical structures that are fractals; e.g.  Sierpinski
triangle, Koch snowflake, Peano curve, Mandelbrot set, and Lorenz attractor.
Fractals also describe many real-world objects, such as clouds, mountains,
turbulence, and coastlines, that do not correspond to simple geometric shapes.

Benoit Mandelbrot gives a mathematical definition of a fractal as a set for
which the Hausdorff Besicovich dimension strictly exceeds the topological di-
mension.  However, he is not satisfied with this definition as it excludes
sets one would consider fractals.