  We consider $n$-folding triangular curves, or $n$-folding t-curves, obtained
by folding $n$ times a strip of paper in $3$, each time possibly left then
right or right then left, and unfolding it with $\pi /3$ angles. An example is
the well known terdragon curve. They are self-avoiding like $n$-folding curves
obtained by folding $n$ times a strip of paper in two, each time possibly left
or right, and unfolding it with $\pi /2$ angles.
  We also consider complete folding t-curves, which are the curves without
endpoint obtained as inductive limits of $n$-folding t-curves. We show that
each of them can be extended into a unique covering of the plane by disjoint
such curves, and this covering satisfies the local isomorphism property
introduced to investigate aperiodic tiling systems. Two coverings are locally
isomorphic if and only if they are associated to the same sequence of foldings.
Each class of locally isomorphic coverings contains exactly $ 2^{\omega }$
(resp. $2^{\omega }$, $2$ or $5$, $0$) isomorphism classes of coverings by $1$
(resp. $2$, $3$, $\geq 4$) curves. These properties are partly similar to those
of complete folding curves.
