While I found the fourth dimensional slicing techniques discussed in Chapter 3 to be the most interesting, I was also impressed by some of the techniques used to describe 3-dimensional slicing -- especially those which involved symmetry. The slice which divided a 3-dimensional cube "vertex first," creating a hexagon, reminded me of a problem from an old math test that I took. The problem was as follows (and perhaps this could be used as an advanced exercise..):
Imagine a large cube which has been partitioned into 27 equal sub-cubes. Now slice the large cube with a plane perpendicular to its diagonal such that the cube is divided exactly in half. How many of the 27 sub-cubes are hit by this slice? (One can use symmetry to determine the answer)
A random thought which also occured to me . . consider the equation which is used to generate the Mandelbrot set: f(z) = z^2 + a, where z and a are complex variables. There are really FOUR variables involved here, since both z and a contain real and imaginary components. Is the mandelbrot set (obtained by varying a, and iterating z) a two-dimensional slice of a four-dimensional object? Are Julia sets (obtained by fixing a and then iterating) other 2-d slices of this same object? I am a little confused.