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I'm interested in the topic of fractals, such as those created by the borders of the Mandelbrot and Julia sets.

My question is if there are other, not yet discovered fractal sets, which one could find by experimenting with the generation formulas (using a computer). I mean of course fractals which are substantially different in their appearance and "behaviour" than the ones which are already known.

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Contrary to popular opinion, "fractal" does not have a rigorous mathematical definition. – Qiaochu Yuan Jan 28 '12 at 20:18
Not all fractals are generated by formulas like the Mandelbrot set is. There are a LOT of fractally things. They come up all over the place; consider this interesting example: – tomcuchta Jan 28 '12 at 20:26
@QiaochuYuan Is it possible to give it one? – Pedro Tamaroff Apr 19 '12 at 2:21
@Peter: something something non-integer Hausdorff dimension something something? – Rahul Apr 19 '12 at 2:25
@RahulNarain Planar Cantor-type sets have Hausdorff dimension anywhere between 0 and 2 (see the animation here ), and there is no reason to exclude the 1-dimensional set from this family. In fact, it's the most interesting member of the family from some points of view. – user31373 May 19 '12 at 21:27

There exits other type of fractals which are generate using no only one function like in the clasic sense, we can generate fractal for a semigroup of rational functions, this was my undergraduate thesis, here some images. enter image description here enter image description here

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The following links give some general information about several types of fractals:

Here are examples of many different types of fractals:

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