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. $2$, $3$, $\ge 4$) curves. Each $C\in \Omega$ gives exactly $2$ coverings by sets of oriented curves which satisfy the local isomorphism property. They have opposite orientations.

arXiv commentMy new paper "Coverings of the plane by self-avoiding curves which satisfy the local isomorphism property" contains an improved version of the results and proofs of the present paper, as well as other results