It is quite incredible. In reality, there are no true fractals which appear in nature because though a mountain can be magnified to a hill, a boulder, and then a grain of sand, the size of the atoms are finite and eventually magnification will no longer yield a similar object with the same detail. It is also true that nature has no objects which can be described under plane geometry. Though a planet appears round it is very jagged and even man made structures are not smooth when magnified. Fractals can, however, be used to describe real objects such as coastlines or treeshapes much like plane geometry can describe the elliptical orbits of planets or the shape of a building.
Fractals are a fairly new concept in math and geometry, and quite different than the norm.
Normally the result of math or geometry is a number describing something. With fractals you get incredibly interesting pictures. I agree totally. Before I wrote this, I thought geometry was mildly interesting, but mostly boring computations using angles etc. But now that I have had a chance to study fractals, my interest in geometry, math and physics has definitely deepened.