The third paragraph on page 32 has the topic sentence, "This object [the Sierpinski Gasket] has the remarkable property that doubling its size produces a figure composed of three copies of the original figure." This seems an interesting fact, but I don't see exactly how this means that the figure has a non-integral dimensionality. It lies in a plane, fer gosh's sake. You could draw a circle around it. I see that the central triangle of the new gasket, created by doubling the edge length would be empty, but the figure would still cover four times as much surface area as the original gasket, just like a square with doubled edge lengths produces a figure with four times as much area.
The thing that throws me most about this example is that I am not accustomed to saying that a figure has a dimensionality. I am more accustomed to referring to the dimensionality of spaces. That is to say, if we're talking about the dimensionality of something, it seems to me odd unless that something is in some sense a possible "set of all points" for some geometry, a little world which other figures might inhabit. The fact that this figure is just a set of points, in fact a set of points which lies entirely in a plane, makes me feel funny to even be questioning its dimensionality. Every point of it can be uniquely identified with an ordered pair of coordinates. How is that not 2-dimensional?
What does it mean to say that a space (or figure, or whatever) has a non-integral dimensionality? What properties would such a space have? Is it possible for mortal humans such as myself to imagine such a space? What's the 2.3-volume of a 2.3-cube? Is any real number a valid dimensionality, or only certain ones?
The issues brought up here make me realize that something perhaps very helpful would be a rock-solid, mathematically verifiable and useful definition of dimension. The fact that the term "dimensionality" can be meaningfully applied to a planar figure surprised and confused me. So what sort of property is dimensionality? To what mathematical objects does it apply? What tests can we perform to determine the dimensionalities of such objects? We've been talking an awful lot about dimensionality in this course for a few weeks now; so just what is it that we've been talking about?
There seems to be a massive bird's nest of issues brought up by this bit of the chapter, and there is apparently no satisfying resolution to them to be found in Chapter 2. The implications of non-integral dimensionality are overwhelming to me.