For many patterns that display self-similarity, the Hausdorff dimension can be found. Sometimes the dimension is calculated and approximate - as is the case with the Feigenbaum attractor - but often its closed form in terms of known mathematical constants be obtained - for instance, the Hausdorff dimension of the Cantor set is $ \log_{3}(2)$.
I am interested in the inverse problem: suppose we consider a mathematical constant like $\pi^{-1}$ or $\gamma e$. Can we always find and describe a fractal whose Hausdorff dimension is equal to this preset number?
We can construct a fractal similar to the Koch curve with arbitrary dimension $1<D<2$.
Consider starting with a line segment of length 1. We divide it into 2 line segments at a slight angle. Then replace each of those segments again with the opposite angle. And so on.
If we pick the relative length of the line segments such that after 2 iterations we get a equilateral triangle, we have the "normal" Koch curve with dimension $D=\log_3 4$. Otherwise we get a curve with an arbitrary dimension $1 <D <2$.
If we pick $f$ as the length fraction of the original line segment that we divide into 2 segments, then the length of the curve with respect to dimension $D$ after $n$ iterations is: $$L_n = 2^n (f^n)^D = (2f^D)^n$$ It means that the dimension $D$ of the curve is such that $$2f^D=1 \implies f=e^{-\ln 2/D}$$ Fill in $D=\gamma e$ to get the corresponding curve.
Similarly we can construct a Cantor set with $D=\pi^{-1}$ as @Claude pointed out.