Coverings of the plane by self-avoiding curves which satisfy the local isomorphism property

Abstract

A self-avoiding plane-filling curve cannot be periodic, but we show that it can satisfy the local isomorphism property. We investigate three families of coverings of the plane by finite sets of nonoverlapping self-avoiding curves which satisfy that property in a strong form. These curves are respectively inductive limits of: 1) $n$-folding square curves such as the dragon curve, obtained by folding $n$ times a strip of paper in $2$ and unfolding it with $\pi /2$ angles; 2) $n$-folding triangular curves such as the terdragon curve, obtained by folding $n$ times a strip of paper in $3$ and unfolding it with $\pi /3$ angles; 3) generalizations of Peano-Gosper curves. In each family, the coverings consist of a small number of curves (at most $6$), and in many examples only $1$ curve. We do not know presently if similar examples exist in spaces of dimension $\geq 3$.