The Cardano-golden ratio and the associated curves

Abstract The aim of this paper is to introduce and study the cubic real polynomials P having as Cardano resolvent exactly the quadratic equation providing the well-known golden ratio Φ. One obtains that these polynomials form a 1-parameter family and the unique positive root of the depressed case is called Cardano-golden ratio . We generalize this cubic depressed polynomial to arbitrary grade n ≥ 3. Also for this depressed polynomial a cubic curve is naturally associated. A regular curve and a regular surface in ℝ 3 , both called golden , are defined and studied from the point of view of differential geometry.