# Beyond 3-D, Chapter 2

## David Akers

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