First I have a straightforward mathematical question. On page 89, the book suggests that the cosine of 40 degrees can be found by solving the equation 8x^3 - 6x + 1 = 0.. apparently because of an identity for cos(x/3) in terms of cos(x). What is this identity, and how does it lead to the cubic polynomial? Also -- someone once told me that cos(20)*cos(40)*cos(80) = 1/8, which I verified to be true. Can this be somehow shown using the same polynomial techniques or trig. identities?
The discussion of "duality" made a lot of sense to me. I found myself wondering why duality always occured in sets of one or two. (The icosahedron is dual to the dodecahedron, and vice-versa, as is the case with the cube and the octagon.) Then it occured to me that one could obtain an infinite number of regular n-sided figures if this kind of reciprocality was not present. (By connecting the centers of the faces of a regular polyhedron, we are certain to obtain another regular polyhedron, I would think, which could go on forever if it did not observe the cycles which it does.)
I also want to go into a little more detail about the rotating die question which I considered a week ago. I thought about it some more, and came up a few answers, and a few more questions as well.. As a cube spins on its vertex, it definitely has a certain shape as seen from the side. This shape represents the shadow that the cube would project on a wall if one were to shine a light upon it from the side. However, parts of this shadow would be more solid than others, I realized. There would be an "inner core" which always blocked the light, but there would be some kind of distribution of matter as one extended from this inner core, finally tapering off to nothing along the boundaries of the outer shadow. It turns out that all of this can be really easily explained by looking at the slices of the cube perpendicular to the longest diagonal. The lengths of outer and inner shadows can be explained in terms of circles which are circumscribed and inscribed about these slices, respectively. (This makes a lot of sense if you think about it for a minute.. the radius of the circumscribed circle tells you how far the matter CAN reach, while the inscribed circle tells you where the matter ALWAYS reaches.)
So, I guess my real question is still how this can be applied to the fourth dimension, rotating a hypercube. The slices of a hypercube are triangular pyramids, truncated pyramids, and octahedrons, all of which have inscribed and circumscribed SPHERES. Could the radii of these spheres be used to describe what the rotation might look like? In rotating the cube, we can come up with a two-dimensional picture of the distribution of matter. (The rotation has radial symmetry, so all of the information can be condensed into a 2-d picture.) Would it not be possible (by analogy) to come up with a 3 dimensional map of some kind which would represent the distribution of matter during a hypercube rotation? If so, what would this map look like?? Before answering this, I think I need to understand more about rotations in four dimensions. In three dimensions, we rotate around a line.. while in four dimensions I think we rotate around a plane. Then we would have (four choose 2) = 6 rotations, since we are choosing two coordinates from {x,y,z,w}, such that these two coordinates define a plane of rotation. So.. where do I go from here?