The Sierpinski curve viewed by numerical computations with infinities and infinitesimals

The Sierpinski curve is one of the most known space-filling curves and one with the highest number of applications. We present a recently proposed computational methodology based on the infinite quantity called grossone to investigate the behavior of two different constructions of the Sierpinski curve. We emphasize that, adopting this point of view, we have infinitely many Sierpinski curves depending, contrarily to traditional analysis, on each specific starting configuration. Of particular interest are some power series expansions in the new infinitesimal quantities emerging from the study of the considered curves.