The fractal “Frog”

In [1–3] some analytical properties were investigated of the Von Koch curve $\Gamma_\theta$, $\theta\in(0,\frac\pi4)$. In particular, it was shown that $\Gamma_\theta$ is quasiconformal and not AC-removable. The natural question arises: Can one find a quasiconformal and not AC-removable curve essentially different from $\Gamma_\theta$ in the sense that it is not diffeomorphic to $\Gamma_\theta$? The present paper is an answer to the question. Namely, we construct a quasiconformal curve, calling the “Frog”, which is not AC-removable and not diffeomorphic to $\Gamma_\theta$ for any $\theta\in(0,\frac\pi4)$.