FRACTALS AND DIFFERENTIATION

If you are familiar with calculus, you might have noticed that one aspect of fractals makes it impossible to analyze them using the most basic calculus technique. Consider differentiating a regular function. When you do that, you assume that as you take smaller and smaller pieces of a curve it gets closer and closer to being a line. Now take a fractal. As you magnify it, the direct opposite happens. Instead of becoming less curved, it stays exactly the same!

NONDIFFERENTIABILITY

Fractal curves are said to be nondifferentiatable. That is, you cannot draw a line which is tangent to a fractal with fractal dimension between 1 and 2. Similarly, you cannot find a plane which is tangent to a fractal with dimension from 2 to 3. What is surprising about that is that continuous curves, to which fractal curves belong are usually differentiatable. Nondifferentiatable functions were first introduced by Weierstrass in 1872. They were called Weierstrass functions after him. The simplest one is:

Don’t get scared by the infinity sign on top of the summation. In practice, to get accurate values it is enough to run n from 0 to the largest number that will not cause an overflow. Below is the graph of this function. Notice that just like fractals, Weierstrass functions do not get simpler with magnification.