Mathematicians have studied objects with noninteger
dimension since the early 20'th century. Pioneers were Russian and French
Mathematicians. The topic became fashion in the 80'ies, when people
started to generate fractals on computers. Evenso, like any fashion, the
hype has faded, it is a beautiful corner of Mathematics.
To define fractals, the notion of dimension is extended. Here is a
definition of which illustrates this:
Define the svolume of accuracy r of a set X as
h_{s,r}(X) = n r^{s}, where n is the smallest
number of cubes of side length r needed to cover X.
The svolume is then defined as the limit h_{s}(X) of
h_{s,r}(X), when r goes to zero.
The dimension is defined as the limiting value s, where
h_{s}(X) jumps from 0 to infinity.
