Topology: An Inquiry-Based Approach

One fun application of the Hausdorff metric is in fractal geometry. You may be familiar with objects like the Sierpinski triangle or the Koch curve. These objects are limits of sequences of sets in $\mathcal{H}({\mathbb{R}}^2)\text{.}$ We illustrate with the Sierpinski triangle. Start with three points $v_1\text{,}$ $v_2\text{,}$ and $v_3$ that form the vertices of an equilateral triangle $S_0\text{.}$ For $i$=1,2, or 3, let $v_i=\left[\begin{array}{c}{a}_i\\ b_i\end{array}\right]\text{.}$ For $i$=1,2, or 3, we define ${\omega }_i:{\mathbb{R}}^2\to {\mathbb{R}}^2$ by

Then ${\omega }_i\text{,}$ when applied to $S_0\text{,}$ contracts $S_0$ by a factor of 2 and then translates the image of $S_0$ so that the $i^{\text{ th }}$ vertex of $S_0$ and the $i$th vertex of the image of $S_0$ coincide. Such a map is called a contraction mapping with contraction factor equal to $\frac{1}{2}\text{.}$ Define $S_{1,i}$ to be ${\omega }_i(S_0)\text{.}$ Then $S_{1,i}$ is the set of all points half way between any point in $S_0$ and $v_i\text{,}$ or $S_{1,i}$ is a triangle half the size of the original translated to the $i^{\text{ th }}$ vertex of the original. Let $S_1=\bigcup _{i=1}^3{S}_{1,i}\text{.}$ Both $S_0$ and $S_1$ are shown in Figure 17.20. We can continue this procedure replacing $S_0$ with $S_1\text{.}$ In other words, for $i$ = 1, 2, and 3, let $S_{2,i}={\omega }_i(S_1)\text{.}$ Then let $S_2=\bigcup _{i=1}^3{S}_{2,i}\text{.}$ A picture of $S_2$ is shown in Figure 17.20. We can continue this procedure, each time replacing $S_{j-1}$ with $S_j\text{.}$ A picture of $S_8$ is shown in Figure 17.20.