An Investigation of the Four Vertex Theorem and its Converse

In the study of curves there are many interesting theorems. One such theorem is the four vertex theorem and its converse. The four vertex theorem says that any simple closed curve, other than a circle, must have four vertices. This means that the curvature of the curve must have at least four local maxima/minima. In my project I explore different proofs of the four vertex theorem and its history. I also look at a modified converse of the four vertex theorem which says that any continuous real-valued function on the circle that has at least two local maxima and two local minima is the curvature function of a simple closed curve in the plane. The modified converse has a rather long history and was only resolved in 1997.