Chapter 2 presented many clever geometric ways to figure out the volumes of various figures. Especially interesting were the derivations which worked for any n-dimensional object.
Dan Margalit suggested in his paper that one might want to consider calculating the hypervolume of a hypersphere, if possible. We can do this by using the same concepts of "slicing" discussed in Flatland, with the aid of some calculus. Finding the volume of a sphere provides a simple example of the use of "slicing" to sum up lower-dimensional pieces to find volumes of higher-dimensional ones. For this case, we can find the volume by summing up all of the infinitely thin spherical shells which together form the volume of a sphere. We simply take the integral of the surface area (4pi*r^2) with respect to r, giving us (4/3)pi*r^3, the volume of the sphere. One might be tempted to say that, by analogy, the "volume" of a hypersphere could be found by summing up solid spheres. We would then have the integral of 4/3*pi*r^3 = pi*r^4/3. This formula displays the correct scaling with respect to r, but it is not correct. In fact the derivation is much more complicated, and results in the actual formula, hyper-volume = (1/2)pi^2*r^4. The correct derivation can also be extended to find the hyper-volume of an n-dimensional ball. (I don't really understand how one gets this, but I believe it is possible.)
I also want to react to comments made by Keith Adams, who was upset over the application of the term "dimension" to figures within a space, rather than spaces. While I share his uneasyness about this, I have actually seen this use of "dimension" myself while working in the Physics Department at Emory University. My boss used to term to describe certain features of fractal-like atomic surface morphologies which he had been studying. To see his definition of "fractal dimension," click here.
David Akers