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given in order to uniquely identify a point in that space. This definition is sufficient for most familiar spaces (such as all subsets of Euclidean spaces), but breaks down in the case of some more interesting figures[1]. One of the cases in which this definition becomes fuzzy is the case of the Pythagoras Tree described above: because of the way the figure is structured, it behaves in some formal ways as a two-dimensional figure, and in other ways as a not two-dimensional figure.

The notion of topological dimensionality refines the intuitive concept of dimensionality. A full discussion of topological dimension is beyond the scope of this chapter, but the basics of the idea are easy enough to grasp. Topological dimensionality is also sometimes called “covering dimensionality,” since it is (among other things) a fact about how difficult it is to cover the figure in question with other overlapping figures, and how that covering can be done most efficiently. Consider the case of the following curve[2]:

Fig. 2.2

  1. Additionally, it’s difficult to make this definition of dimensionality more precise than the very vague phrasing we’ve given it here. Consider a curve embedded in a two-dimensional Euclidean plane—something like a squiggly line drawn on a chalkboard. What’s the dimensionality of that figure? Our intuitions come into conflict here: for each point on the curve, we have to specify two numbers (the Cartesian coordinates) in order to uniquely pick it out. On the other hand, this seems to just be a consequence of the fact that the curve is embedded in a two-dimensional space, not a fact about the curve itself—since it’s just a line, it seems like it ought to just be one-dimensional. The intuitive account of dimensionality has no way to resolve this conflict of reasoning.
  2. This figure is adapted from one in Kraft (1995)