Weierstrass

After waiting patiently for more than two months, the Spring issue of Nautilus arrived. It’s fabulous in print, and I just finished reading Math’s Beautiful Monsters by Adam Kucharski, a fascinating insight into shaky foundations of Calculus, and how one pathological function by Karl Weierstrass brought its theory to knees by refusing to be differentiable, while being continuous. He says:

Many of the old guard wanted to leave Weierstrass’ monster in the wilderness of mathematics. It didn’t help that nobody could visualize the shape of the animal they were dealing with — only with the advent of computers did it become possible to plot it.

I was intrigued with the idea of plotting, and so I pulled Grapher and punched in the initial part of the infinite equation to see the irregularity for myself. Here’s a sample equation:

$$ f(x) = \sum_{n=1}^{\infty} \frac{cos(3^n x \pi)}{2^n} $$

As I zoomed in to peaks and troughs of the cosine curve in Grapher, I could see in awe how the curve kept morphing into a new shape, and how it would resist yielding derivatives.

Weierstrass function plot
Weierstrass pathological function plot.

It is fascinating to see how this equation jumps right out of conformity from the smoothness that Calculus expects.