Can we construct a Koch curve with similarity dimension $s\in[1,2]$? We can make a Koch curve $K$ with similarity dimension $s\in \mathbb Q \cap [1,2]$ by writing $s=\frac{p}{q}$, and constructing such a generator that by scaling with the factor of $2^q$, we'd find $2^p$ of it inside the scaled one. Then $$\dim_\text{similarity}K=\log_{2^q}2^p=\frac{p}{q}=s.$$ Example: a generator of a Koch curve with $s=\frac{3}{2}$: ![]( Obviously, the procedure needs $s$ to be rational. Is there a way to construct a Koch curve for any real $s\in[1,2]$?

It's fairly easy to construct a "Koch-like curve" of any dimension $s$ in the open interval $1
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Note that $r$ is parameter satisfying $1/4
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Depending on your meaning of "Koch-like curve", you might say that we've already hit the values $1$ and $2$. There is also a notion of borderline fractal that might be more "Koch-like" but is not strictly self-similar. I might be able to elaborate on that, if there is interest.