Chana – Proofs in the Pudding

Infinite Dust: the Counterintuitive Cantor Set

May 25, 2023Chana 1 Comment

Chana Lyubich

Often, we study smooth, differentiable functions with the assumption that they are an adequate mathematical tool to describe our world. However, they are not always sufficient. A well-known example of this is the coastline paradox, which observes that the length of a coastline is not well-defined: rather, it depends on how the measurement was made. This is because a coastline has fractal properties, and does not resemble a differentiable curve. Indeed, fractals can be used to describe many concrete phenomena in nature, despite often having strange, counterintuitive properties!

One of the first examples of a fractal is the Cantor Set, which was discovered by Georg Cantor in 1883. (Incidentally, Georg Cantor is also the founder of set theory.) While the construction of the Cantor Set is deceptively simple, it results in some seemingly incompatible properties.

We can construct the Cantor Set $K$ as follows:

Let us take the segment $[0, 1]$ and remove the open middle third (that is, we remove the interval $(\frac{1}{3}, \frac{2}{3})$ ). We will label this first gap as $G^0$ and the two remaining segments as $I_{0}^{1}$ and $I_{1}^{1}$ . In this labeling method, the superscript indicates the “level” of the segment and the subscript indicates whether it is the left (0) or right (1) segment.… Read the rest