Search
Random Quotation
Thus metaphysics and mathematics are, among all the sciences that belong to reason, those in which imagination has the greatest role. I beg pardon of those delicate spirits who are detractors of mathematics for saying this .... The imagination in a mathematician who creates makes no less difference than in a poet who invents.... Of all the great men of antiquity, Archimedes may be the one who most deserves to be placed beside Homer.
Discours Preliminaire de L'Encyclopedie, Tome 1, 1967. pp 47 - 48.
Loci: Convergence
show printer friendly
send to a friend
Limit Points and Connected Sets in the Plane
Introduction
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.