Geodesics Teacher Support: Part 2

In the diagrams that follow, the square on the left is the gluing diagram of a torus. On it is the geodesic from A to B. The fact that the geodesic is broken into two pieces indicates that A and B are on opposite sides of the cut line of the torus. The slope of the geodesic is 1/2. It would seem very logical for a student to ask what happens if an attempt was made to move from A to B along a path with slope of 2. This path too will go over the cut line at the top of the square and wrap back over the bottom. However since the slope is so steep it will make one complete revolution of the torus before it connects with B. This is illustrated on the center diagram below. The bottom square represents the gluing diagram of the torus. The broken segment from A to its top represents the path to the cut line of the torus. The middle square represents a complete revolution around the torus on its gluing diagram which still doesn't take us to B. The top square shows the completion of the path to B. The square at the right is the composition of the three squares in the middle to show the path of slope 2 on the gluing diagram of the torus. Care should be taken to point out to students that while this shows another path from A to B, it is a geodesic, but not the shortest path. A comparison of the paths on the left square and the right square should make this obvious.

This example shows that there is more than one geodesic. Question: Can we find a geodesic from A to B using any slope? Consider the following example.

The first square below shows the geodesic from A to B along a path of slope 2. Again, the fact that it is broken into two pieces indicates that it crosses the cut line of the torus. The group of three squares shows an attempt to move from A to B along a path of slope -1. The bottom square of the three represents the original geodesic and shows the start of the path of slope -1. The two squares above it show the path looping around the torus and finally coming back to A without ever passing through B. If we were to continue this process from point A again, it would merely cycle us through the same path. We will never get to B following a path from A with slope of -1. Students should easily conclude that not all paths work. Many paths will leave us spinning throughout our gluing diagram without ever reaching our destination.

Let us return to the first example where it was possible to move from A to B along paths of either slope 1/2 or slope 2. Consider the path with slope 1/2 which gave us the geodesic. By gliding the fundamental domain of the torus several times both horizontally and vertically it is easy to completely fill the plane as shown in the following diagram. We refer to this as tiling the plane. The diagram below shows such a tiling. The shaded region in the middle represents the original fundamental domain of the torus with the geodesic marked on it. It is finite in nature. The fundamental domain serves as the pre-image for infinitely many glides which fill the plane. Each square is an exact duplicate of the original mapped back onto itself. As in The Shape of Space video which showed infinitely many duplicates of the spaceship and two stars, this space contains only two points and infinitely many images of them.