Exploring Lill's method: beyond graphical solution - ISISS M. Casagrande (Pieve di Soligo)

The aim of this article is an in-depth study of Lill’s method, an ingenious graphical method of finding the roots of polynomials of any degree developed by Austrian engineer Eduard Lill and
published on the Nouvelles annales de mathematiques in 1867 where the proof is left to the reader. Initially we analyze the original method to better understand how it works and we produce some proofs about its fundamental properties and a couple of results: we recognize a nice connection with the well known Ruffini’s method for factoring polynomials and we use its geometrical properties to represent particular algebraic numbers and to give an expression for the number π. Finally we generalize the method and by exploiting its properties we show how it allows to study the problem of inscribing regular polygons inside other regular polygons.