<A HREF="https://pdfs.semanticscholar.org/4928/071439ea10140b21dd0a0ccd124812c84f31.pdf">Energy Form on a Closed Fractal Curve</A> (2004). See Proposition 2.2, attributed to Ref. 15, which I did not find online:

*The Koch snow flake is the union of three Koch curves of Hausdorff dimension $D=\ln 4/\ln 3$ and Hölder exponent $\beta=\ln 2/\ln 3$.*