Alexis Nogelo's additional comments

Professor Banchoff,

In your response to my Chapter 5 reaction, you describe a way to draw an angle that is one-third the size of another. I have been trying to follow your instructions to do so and I am not quite reaching the point where I can see how each of the things that are supposed to be congruent are actually congruent. I can prove that angleTOA = angleTBC and that angleBCO = angleBOC, but I cannot seem to see why angleTOA = 1/2angleBCO. I will try with an actual compass and straightedge later (rather than a book and pen) to see if I can get a better picture to help me, although I know I should be able to prove it regardless of my poor drawing. In any case, it is a fun exercise. It implies that with a marked straight-edge we can in fact construct a regular enneagon, which for some reason surprises me. The marked straight-edge is a considerably more powerful tool than the unmarked straight-edge, I guess.

You also posed a question about the fewest number of struts connecting midpoints of the faces of a cube that are necessary to make that cube rigid. I am trying to think about this problem without actually building a cube to test different numbers of struts, so I will just relate to you some of my thoughts. Because so many representations of cubes with which we come in contact are made of solid material, it is slightly difficult to imagine in which ways a mathematical cube would be able to move if it had no struts supporting it. It is a question of degrees of freedom--it seems like there are three possible answers: 1, 2 or 3 (since each dimension is a 'degree of freedom' and a cube is three-dimensional). I do not think that 3 are necessary, although 3 would certainly be sufficient, since in the 0-dimension there is no freedom of movement, and 3 struts would reduce the cube to having the same freedom of movement as a 0-dimensional object. Reducing the possible answers to 1 and 2 does not exactly solve it, since I am not sure if line segments and squares are 'rigid', which would be equivalent statements, respectively. The more I think about it, the more I think that even a square is rigid (since it cannot move in the third dimension, twisting and bending, properties of not being rigid would be impossible), so if we reduce the cube to the movement of a square, we will have made the cube rigid. This means that we would need only 1 strut to make the cube rigid, and analogously 2 struts to make the hypercube rigid. Please let me know if this makes sense.