  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 $\geq 3$.
