With respect to the Golden Ratio, check out Andrew Miller's week 11 page and my response thereto.
The pendulum story is a bit complicated, i n that we are concerned not just with the positions of each pendulum but also their velocities. If we have synchronized pendulum motion, then this determines one point on a circle of circles for each time, and this gives a curve on a torus in four-space. Even more interesting curves occur when one pendulum is rotating at twice the speed of the other, or if three times the speed of one is twice the speed of the other. Maybe we can set this up on the computer and demonstrate these configurations.
T he collection of planes in four-space turns out to be a four-dimensional configuration space of great interest to geometers and topologists. I have co-authored several papers that deal with this space in one way or another, but most of these articles are somewhat inaccessible I'm afraid. I should probably put one of them into HTML anyway, just as an illustration of what can be done with the material we have been studying in this class.