Beyond the Third Dimension--Chapter 8
1. Is an n-tuple just a set of n coordinates?
2. Having never taken multivariable calculus, i had no idea how simple it could be to carry certain concepts, such as the pythagorean theorem and the unit square, into higher dimensions. Maybe when I read the final project by the group that is upgrading geometry I will learn more about the actual mathematics of fourth dimension.
3. I understand better now how the "hidden hexagon" relates to the fourth dimension. When you move through the unit hypercube from the origin to the sixteenth vertex, you get a series of coordinates growing from 0 to 4. In the exact center, you find a regular octahedron. I wonder if 4-dimensional creatures are as surprised about the presence of this octahedron as we all were to find the regular hexagon in the center of our cube...
I think in general I find it easier to grasp things in the fourth dimension when they are spelled out in terms of coordinates and motion through a graph, since I cannot really picture a hypercube at all on my own.
4. Why is it called the golden ratio? I know that the five-pointed star has had religious significance in the past, what with it being 5 points all connected to one another in a continuous line. Is the term related at all to Christianity or any kind of mysticism?
5. H.S.M. Coxeter sounds like an interesting man. Did he contribute anything else to the world of visual representation or to mathematics? Was he primarily a mathematician, or was he an artist?
I did not understand how to read the graphs of the complex figures, although I thought they were very pretty.