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A self-avoiding plane-filling curve cannot be periodic, but we show that itcan satisfy the local isomorphism property. We investigate three families ofcoverings of the plane by finite sets of nonoverlapping self-avoiding curveswhich satisfy that property in a strong form. These curves are respectivelyinductive 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, thecoverings consist of a small number of curves (at most $6$), and in manyexamples only $1$ curve. We do not know presently if similar examples exist inspaces of dimension $\geq 3$.

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