Search Loci: Convergence:
Perhaps I could best describe my experience of doing mathematics in terms of entering a dark mansion. You go into the first room and it's dark, completely dark. You stumble around, bumping into furniture. Gradually, you learn where each piece of furniture is, and, finally, after six months or so, you find the light switch and turn it on. Suddenly, it's all illuminated and you know exactly where you were. Then you enter the next dark room....
PBS Nova program: The Proof
Limit Points and Connected Sets in the Plane
Intuitively a set S in the plane is connected if for every pair of distinct points p1 = (x1,y1) and p2 = (x2,y2) and in S there is a curve consisting of points in S with endpoints p1 and p2. This definition was sufficient until the end of the nineteenth century, when an implosion of imaginative examples of curves in the plane demanded a more rigorous definition. One purpose of this paper is to trace this aspect of the early history of connected sets based on the formal definition that replaced the intuitive one. To illustrate the need for such a rigorous definition, we introduce a curve called Mullikin’s nautilus that was first constructed in 1919. Since the formal definition relies on limit points, the nautilus not only illustrates its necessity but serves as an instructive vehicle for introducing limits in the plane to students who have studied limits on the real line. Therefore the second purpose of this paper is to present material directly applicable to the classroom.
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