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Fractals

In mathematics, fractals are defined as objects that have a self-similar structure. This means if we zoom at a specific section of the object, it looks similar to or at least has a similar structure to the original object. 

Due to their self-similar structures, fractals are naturally very rough objects and cannot be approximated by smooth structures. This is perhaps the reason they have not been received that much attention in mathematics. One reason is that , historically, mathematics has been built around smooth objects. Roughly speaking an object is smooth if it approaches a line or plane or a flat object as we keep zoom in. This implies that the roughness of the objects disappears after enough zooming. 

Mathematically speaking such smooth objects are called Manifolds and are the well-behaved objects to work with. Almost all the familiar shapes such as cones, circles, lines, curves, ball, sphere, etc. are all examples of smooth objects. In such objects, the difference between a set and a flat objects approximating that set vanishes locally at each point. And this is exactly the definition of a differential of a function. 

For differentianble objects, one can define a well-defined dimension as the dimension of the approximating plane. For example, any curve can be approximated by a line so it should have a dimension of 1. Or a flat surface in 3D such as a cone or sphere can be approximated by a plane which is 2D object so it has a dimension of 2.

Fractals are not smooth, so they do not have any inherent dimension. However, one can define a packing dimension for fractals as the density of balls of a specific radius used to cover the set. To be more specific, suppose we need N(r) balls of radius r to cover a set. Then we can define the dimension of the set as d(r) = -logN(r)/log(r) as r becomes smaller and smaller.

For example, in a line, we need around N(r) = L/(2r) balls of radius r to cover it. So -log N(r) = -log(L/2) + log(r). Thus, d(r) = 1 - log(L/2)/log(r) which approaches 1 as r goes to zero. One can show that the dimension remains 1 for any smooth curve.

Now consider another object known as Serpinsky triangle illustrated in the following picture. One can show that the perimeter of this triangle scales like (4/3)^n and has a dimension of 4/3 rather than 1.