Math 302

Journal Assignment for Monday, Nov 27, 2000

In class, we have been studying S^2 and H^2, that is, the two dimensional sphere and the two dimensional hyperbolic plane. These have three dimensional analogues, S^3 and H^3, or the three dimesional sphere and three dimensional hyperbolic space. Unlike S^2 or H^2, neither of these fits inside our three dimensional Euclidian world, even locally. This makes it hard to picture them! (Yes, I know that I said that our space is not actually Euclidian, but its pretty close!) The pictures that I have attatched are pictures of what these spaces look like. They pictures are drawn in perspective, so they show what your world would look like it you actually lived in one of these spaces. We assume that light rays travel along straigth lines, so the geodesics in the space determine your line of sight.
1. Figure 1 shows a tiling of the 3-dimensional sphere S^3 by regular polyhedra. (That is, the space has been completely filled with regular polyhedra). The polyhedra have been shrunk (all the same amount) so that you can see them more easily. The unshrunk polyhedra are represented by the thin lines.

   Questions: First, just spend a little time looking at the picture. What do you see that is interesting?

   How many edges meet at each vertex of the polyhedra? Can you tell how many faces/edges/vertices each one has?

   Which polyhedra look larger - those which are farther away or those which are closer?; Can you think of any ideas why this might be?  Examining a similar situation on S^2 might be of help.
   
   
   

2. Figure 2 shows a tiling of 3-dimensional hyperbolic space H^3 by regular dodecahedra. The "beams" are infinite straight lines. They are the same thickness at every point -- where they look thicker it is because they are closer.   Figure 3 is the same as Figure 2, but without the nice shading.;

   Questions:; Again, start by just looking at the pictures. What do you notice? Is anything interesting going on?

   How many lines meet at each vertex?  What does the angle between intersecting lines appear to be?   What shape is each face of the polyhedron?  Do the lines appear to be straight?  Do they appear to be infinite?  Can you think of reasons to explain any of these phenomena?