Peano-Gosper curves and the local isomorphism property

TL;DR

This paper explores plane-filling, self-avoiding curves that satisfy the local isomorphism property, generalizing Peano-Gosper curves, and classifies their coverings and isomorphism classes.

Contribution

It introduces a set of plane coverings by self-avoiding curves that generalize Peano-Gosper curves, with a detailed classification of their local isomorphism classes.

Findings

Each curve set satisfies the local isomorphism property.

The set of coverings has a rich structure with continuum many classes.

Each curve set admits exactly two oriented coverings with opposite orientations.

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.

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