8. Examples and Constructions

This chapter is dedicated to concrete examples of constant width sets and procedures on how to construct them. The most notorious convex body of constant width is undoubtedly the Reuleaux triangle of width h which is the intersection of three disks of radius h and whose boundary consists of three congruent circular arcs of radius h. In Section 8.1, we will see that the Reuleaux triangle can be generalized to plane convex figures of constant width h whose boundary consists of a finite number of circular arcs of radius h. They are called Reuleaux polygons. The plan for the rest of the chapter is the following: In Section 8.2, we will study the 3-dimensional analogue of the Reuleaux triangle, and in Section 8.3, we will construct Meissner’s mysterious bodies from it. In fact, in this section, we will use the concepts of ball polytope and Reuleaux polytope to construct 3-dimensional bodies of constant width with the help of special embeddings of self-dual graphs. In Section 8.4, we will give a procedure of finitely many steps to construct 3-dimensional constant width bodies from Reuleaux polygons, and in Section 8.5, we will construct constant width bodies with analytic boundaries.