e.g. a sierpinski triangle, one copy of size n is the same as 3 copies of size n/2, giving the recursive "bound" f(n) = 3f(n/2)

which expands to a tree of height log_2 n, with weight tripling at each level, so:
f(n) = 3^(log_2 n) = n^(log 3 / log 2)

which you would see in most algorithms classes as a runtime bound, but geometrically this means the "dimension" of the sierpinski triangle is log 3 / log 2