Response from Prof. B.

Too bad about Prof. Gould not mentioning dimensionality explicitly. Some of our common students have found evidences of the notion in several of his projects, for example the underwater archaeology that tries to deduce the shapes of sunken vessels from the patterns of partially rotted timbers cut off at the water line, a good example of slicing.

The negative sign in the formula that looks like the Pythagorean theorem otherwise is motivated either by physics, where it arises from the hypothesis that the speed of light is constant, or from this rather artificial "stage lighting" construction that is meant to encourage students to explore the geometry of light cones.

I would really like to see you write down some equations for the rotating juggling clubs. The model is the curve traced out by a reflector stuck in the spokes of a bicycle moving along a straight road, and once you have equations for this motion, you can more or less superimpose them on the parabolic path traced out by the motion of the center of gravity of the thrown object. Look up the equation for the cycloid, and see what it looks like when you add the gravity complication?

It is true that for any fixed pendulum lengths, the set of possible positions correspond to the points on a circle of circles, except in the special case where one of the lengths is zero, when the space of configurations is given by a circle, the positions of the pendulum with non-zero length. Instead of pendulum motion, we can consider the positions of a reflector on a stationary wheel, and given two wheels rotating at different constant speeds, we can ask if the sequence of configurations ultimately repeats itself (which will happen if the ratio of the speeds is a rational number). In general, the path on the torus will not close up and repeat itself. The repeating cases are quite nice, leading to "torus knots".

If we didn't mention specifically that we get wave fronts for surfaces as well as for curves, we should do so now. These moving points of light are insubstantial so they pass through one another without difficulty, as in the case that came up in class, on the field trip at the Providence Art Club.

Your link doesn't work for me so I can't see what the involutes and parallel curves are that you are questioning. Let me know when the link works so I can answer.\