Hausdorff measures: the ultimate collection of measuring cups

Hausdorff measure $\mathcal{H}^\alpha (E)$ of a set $E$ is a nonnegative number that measures its ``$\alpha$-dimensional'' size. If $E$ is a curve, $\mathcal{H}^1 (E)$ is the \textbf{length} of the curve, as we know it from Calc 2. But $\mathcal{H}^2 (E) = 0$, meaning our 2-dimensional goggles will not notice this set. Hausdorff measure applies to any arbitrary set $E$ and more surprisingly, $\alpha$ does not have to be an integer. Thus you may measure $\sqrt{2}$-, $\frac{2}{3}$-, or even $\pi$-dimensional size of a set. I will show you a $\frac{\log 8}{\log 3} = 1.8928$-dimensional carpet from your Calculus 2 book!