Question:
I would like to be able to explain to my geometry
students why fractals such as Sierpinski's triangle and the Mandelbrot
Set are both fractals, yet they appear so different and although they
are both generated by an iterative process, the processes seem to be
different.
Can you give me a formal unifying definition of fractals?

Replies:
There are two ideas you have to get to gain a sense of what fractals are
all about: "self similarity", and "fractal dimension." Self similarity
is pretty apparent: if you see some feature in a fractal, you can always
look more closely and see that same feature--or something very like it--
on a smaller scale. Fractal dimension is a harder idea to get. Here are
two explanations from the web:

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