Life in Two Dimensions

In this chapter, we focus on triangulations of point sets in the plane, with two goals in mind. On the one hand, dimension two is the first “non-trivial case” of study (triangulations of 1-dimensional point configurations were studied in the previous chapter; see Example 2.3.7). Looking at plane triangulations will prepare us for more complicated situations to come. But the reader must be cautious in thinking that all results about triangulations in dimension two remain true in higher dimensions; that this is not the case will be emphasized in Section 3.6, where we contrast two-dimensional results with three-dimensional analogues.