SPACE-FILLING CURVES AND HAUSDORFF DIMENSIONS

Abstract. We construct a continuous curve from the interval [0, 1] into the n-dimensional cube [0, 1]n in Rn under which the entire cube is filled by the image of a subset of [0, 1] of Hausdorff dimension r, for any positive integer n ≥ 2 and any real number r between 0 and 1. A space-filling curve is a continuous function from the one-dimensional unit interval [0, 1] onto the two dimensional unit square [0, 1]2 = [0, 1] × [0, 1]. In 1879, Netto showed that [0, 1]2 cannot be a homeomorphic image of [0, 1]. Hence the discovery of a space-filling curve (and consequently the fact that [0, 1]2 is a continuous image of [0, 1]) is surprising. After Peano’