Squaring the Circle: A Timeless Classic Still Alive and Thriving

The classical problem of squaring the circle - namely, the long-pursued task of drawing a square with the same area as a given circle, using only compass and straightedge throughout a finite number of steps - was definitively proven impossible during the 19th century. Approximate constructions remain feasible; for example, rational approximations to any desired level of accuracy can in theory be translated into geometrical constructions. A naive approach, though, would yield impractical procedures requiring finite but overwhelming quantities of steps. Mathematicians therefore have been, and still are, fascinated by the quest for constructions that achieve exceptional precision with respect to the length of the process. In this paper we explore the topic: recalling the underlying approximation of the involved transcendental constants; describing the implemented ranking system, including the adoption of the quantitative approach devised by Lemoine to objectively assess the complexity of geometrical constructions; reviewing some historical milestones and discussing more recent contributions; finally, presenting some original findings, that is, novel approximation formulas - representing noteworthy disclosures in their own right - along with new constructions that substantially expand the boundaries of the conquered territories within this framework.