Let's dive into the beauty of toric birational transformations. The definition of a toric algebraic variety might be complicated, but in the projective case, there is a nice translation of the toric structure into a polytope. A polytope is simply a generalization of a polygon (or a polyhedron) to higher dimensions. It can be manipulated quite easily, and the birational transformations of the projective variety it encodes induce an intelligible modification of the polytope itself. Through an example, let's apply the method of Cox, Little and Schenck to construct the associated coordinate ring.
