Dimension for Irregular Fractals

Student: The irregular fractals I made using Flake Maker and Fractured Pictures are much more complicated than the regular fractals that I made earlier with tools like the Koch Snowflake and Hilbert Curve . How do I calculate the fractal dimension for these? What is the scale factor (S) and the number of pieces (N)?

Mentor: Unfortunately, I don't have a good answer for you on the dimension of irregular fractals. The basic formula must be generalized using some fairly complicated mathematics, and in many cases -- especially with some of the Julia Sets we'll look at later -- the fractional dimension is unknown. There are even situations in which we can show that the dimension is fractional, but not what the fraction actually is.

Student: So if the number of pieces in the replacement and the scale factor both stay the same at each stage, I can use

to calculate the dimension D, but if something changes, I can't?

Mentor: Yes. It's almost always the scale factor that messes us up.