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 Ω of coverings of the plane by sets of disjoint self-avoiding nonoriented curves, generalizing the Peano-Gosper curves, such that: 1) each C ∈ Ω satisfies the local isomorphism property; any set of curves locally isomorphic to C belongs to Ω; 2) Ω is the union of 2 ω equivalence classes for the relation "C locally isomorphic to D"; each of them contains 2 ω (resp. 2 ω , 4, 0) isomorphism classes of coverings by 1 (resp. 2, 3, ≥ 4) curves. Each C ∈ Ω gives exactly 2 coverings by sets of oriented curves which satisfy the local isomorphism property. They have opposite orientations.