In my experience, some of the most interesting parts of math have been trying to picture what will happen when a three dimensional object is cut by a plane. Often times the result of a slice is difficult to understand.
A very useful technique is starting at a slice you understand, and then rotating the slicing plane just a little bit away from the angle you were at, and then examine what hapens to your slice. Imagine you've sliced an elipsoid (a non-circular skipping stone, w/ different lengths for the three axes) through its center and have gotten an elipse. What happens if you rotate the angle of your slicing plane so that your next slice that goes through the center is a bit different than the first? Maybe your new angle shortened and extended the respective axes of your first elispe slice, bringing your new slice closer to a circle. In fact, for any elipsoid at least two angles of planer intersection will reward a circle when sliced through the center of the elipsoid. Prove to yourself that this must be true. (after that, solve for the critical angle given that the equation of an elipse is [ (x^2/a^2) + (y^2/b^2) + (z^2/c^2) = 1]
Following up on the slicing aspect of the chapter, the figure the book did not go over was what the slices of a 4 simplex (hyper-triangular pyramid), starting at a tetrahedron (which is same as a vertex), then a triangular face, then a trianguler edge. Should be interesting.
In terms of a contour map, find yourself a bagel and and slice it from a skewed angle to get a tactile sense of how the slices work. I think I need to do that as well.
Your exercise about the ellipsoid is a very good one, and one that is sometimes difficult for people to believe when they first encounter it. But it has to work by the intermediate value property--if we rotate the slicing plane about the middle length axis, then at one stage the other axis is smaller and at another it is larger, so somewhere in the middle it has to be just right. It is interesting to note that all the planes parallel to the plane through the origin with circular sections also have circular sections, and the two planes in such a parallel family that are tangent to the ellipsoid with three unequal axes will be tangent at special points called "umbilics". They play a special role in the analysis of surfaces in Math 106, Differential Geometry, and to a certain extent in Topology (Math 141) as well.
So, what _do_ we get from slicing the hypertriangular pyramid, or other cones for that matter? Is there something that can be said in general?
Has anyone sliced a bagel so as to get two intersecting circles in the slice? A worthy project.
Prof. B.