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 $1<s<2$. We simply generate a self-similar set based on the iterative scheme implied by the following picture.
Note that $r$ is parameter satisfying $1/4<r<1/2$. The resulting fractal has dimension $$\log(4)/\log(1/r).$$ When $r=1/4$, the result is an interval with dimension $1$, when $r=1/2$, the result is a solid triangle with dimension $2$, and any dimension between these is obtained as $r$ ranges from $1/4$ to $1/2$. We can illustrate this evolution as follows:
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.
