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It's hard to condense fractals down to 10 pages, I think you did mostly ok.  But I'd restructure it to talk about fractals in nature first, as motivation for defining (box-counting) dimension as a measure of "roughness", then move on to synthetic fractals and show how dimension can be calculated analytically for them.

I'm not 100% sure, but Benoit Mandelbrot could maybe do with a circumflex?  Whether he created or discovered the set is a matter for philosophical debate I guess.  A classic paper is his "how long is the coastline of Britain". His TED talk is pretty good too.

When talking about the Koch curve (or should that be von Koch curve?) it might be worth mentioning that it's continuous but (almost everywhere?) non-differentiable, so it's "pathological" in another way than having infinite length.

The IFS fern image could do with some rectangles showing the affine transformations.

Are your two quadratic Julia set images to the same scale?  BTW they're pictures of the filled-in Julia set, rather than the Julia set itself (which is the boundary between the Fatou components).

You could mention the escape time colouring method (seeing as you have pictures with rainbow colours) and its utility in exposing details that would be too small to see (the Mandelbrot set is connected, but it isn't obvious with binary in/out colouring).

No mention of chaos seems awkward, but I know it's over-length already...  a good example of chaos in fractals with practical applications in numerical analysis is the Newton fractal (extreme sensitivity to initial conditions at the boundary, where all the domains meet).

Finally, "surface area of 100 yards" is ill-dimensioned, which is pretty much the only flat-out error I spotted.

Hi Giorgio,

Good job!  In addition to Claude's comments, I offer these:

On your first page, you refer to Hausdorff's notion of dimension, but you don't define it until later.  I suggest adding some indication that topics of dimensionality will be discussed later, so that the reader isn't left to feel that he or she should already know the Hausdorff dimension.

You also state that fractals are too complex to be described with classical Euclidean (proper noun, should be capitalized) geometry.  I think I understand the spirit of your comment, that fractals are more than simple curves and polygons.  Let, your first examples (Cantor set, von Koch curve, Sierpinski triangle) all are described with notions of classical geometry.  Indeed, I think the only way most folks garner an understanding of fractals is by extending their understanding of classical geometric figures.  Also, non-Euclidean geometry is generally associated with curve space, whereas the vast majority of fractals are based on flat space, so I think saying that they can't be described with classical Euclidean geometry is too extreme.

Overall, very nicely done.

Kerry