An FASS-curve is a curve which is space filling, self-avoiding, self-similar and simple. Simple means here that the curve can be drawn in one stroke. FASS-curves in the euclidean plane have a fractal dimension of D=2. As a consequence they can be derived from aperiodic substitution tilings by applying an appropriate decoration. The opposite way is also possible in many cases, so that curve and tiling form a dual structure.