Arcs with Positive Measure and a Space-Filling Curve

Abstract

In this note we construct a one-parameter family of arcs in R2 with positive two-dimensional Lebesgue measure, which converge uniformly to a space-filling curve. By an arc we mean a homeomorphism of I = [0, 1] into I x I. The existence of such maps is not news, but our procedure for constructing them is so transparently simple that it can be presented in undergraduate or graduate courses in advanced calculus, real analysis, or topology. Furthermore, each of these curves is the uniform limit of a recursively defined sequence of very simple functions. The examples of space filling curves of Peano or Hilbert, which are generally cited (e.g., [1], [21, or [3]), are also constructed recursively, but the recursions are more complicated. All of them pass from one stage to the next by splicing together reduced and rotated copies of an initial figure. In our construction, no rotation is necessary to get them to fit, so the recursion has a much simpler description. Our construction is similar to one found in Section 10.10 of [1]. This work was motivated by problem E 2975, posed by F. B. Jones in the December, 1982, issue of the American Mathematical Monthly. We wish to thank the referee for several helpful suggestions. Let us first describe a typical member of our family of arcs intuitively. Imagine a necklace composed of four congruent squares and three connecting line segments arranged as in FIGURE 1. In each square we put another necklace, but one in which the + shaped region is proportionately much thinner. By leaving the original