Hausdorff Dimension, Its Properties, and Its Surprises

1. INTRODUCTION. The concept of dimension has many aspects and meanings within mathematics, and there are a number of very different definitions of what the dimension of a set should be. The simplest case is that of R d : in order to distinguish points in R d , we need d different (real) coordinates, so R d has dimension d as a (real) vector space. Similarly, a d-dimensional manifold is a space that locally looks like a piece of R d. Another interesting concept is the topological dimension of a topological space: every discrete set has topological dimension 0 (e.g., any finite sets of points in R d), an injective curve has topological dimension 1, a disk has dimension 2 and so on. The idea is that a set of dimension d can be disconnected in a neighborhood of every point by a set of dimension d − 1: curves and circles can be disconnected by removing isolated points, disks can be disconnected by removing curves and circles, etc. A formal definition is recursive, starting conveniently with the empty set: the empty set has topo-logical dimension −1, and a set has topological dimension at most d if each point has a basis of open neighborhoods whose boundaries have topological dimension at most d − 1. All these dimensions, if finite, are integers (we will ignore infinite-dimensional spaces). An interesting discussion of various concepts of dimension, different in spirit from ours, can be found in the recent article of Manin [17]. We will be concerned with a different aspect of dimension, having to do with self-similarity of " fractal " sets such as those shown in Figure 1. As Mandelbrot points out [16, p. 1], " clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line, " so many objects occurring in nature are not manifolds. For instance, the fern in Figure 1 is constructed by a simple affine self-similarity process, and people have tried to describe the hairy systems of roots of trees or plants in terms of " fractals " , rather than as smooth manifolds. Similar remarks apply to the human lung or to the borders of most states and countries. The concept of Hausdorff dimension is almost a century old, but it has received particularly prominent attention since the advent of …