Peano-Gosper curves and the local isomorphism property

Abstract

We consider unbounded curves without endpoints. Isomorphism is equivalence up to translation. Self-avoiding plane-filling curves cannot be periodic, but they can satisfy the local isomorphism property: We obtain a set $\Omega $ of coverings of the plane by sets of disjoint self-avoiding nonoriented curves, generalizing the Peano-Gosper curves, such that: 1) each $C \in \Omega $ satisfies the local isomorphism property; any set of curves locally isomorphic to $C$ belongs to $\Omega $; 2) $\Omega $ is the union of $2^{\omega }$ equivalence classes for the relation "$C$ locally isomorphic to $D$"; each of them contains $2^{\omega }$ (resp. $2^{\omega }$, $4$, $0$) isomorphism classes of coverings by $1$ (resp. $2$, $3$, $\geq 4$) curves. Each $C \in \Omega $ gives exactly $2$ coverings by sets of oriented curves which satisfy the local isomorphism property. They have opposite orientations.