So here is the graph with some geodesics. As you can see it's very symmetrical. This graph can be decomposed to only show the geodesics starting at (0,0) and ending at (1,1) and also to show the geodesics starting at (1,0) and ending at (0,1), but here is the superimposed version:
My first proposal at a solution is: Glue (0,0) to (1,1) and similarly glue (0,1) to (1,0) and look at that shape, but I'm not sure this is helpful. I have the equations for each of the geodesics in the graph, but the main issue is finding the shape that results from defining an equivalence relation between the points (0,0), (1,1) and (0,1), (1,0). Another idea I had was to mold the whole graph into a torus and see what that looked like. Molding the graph into a torus would treat ALL four points as equivalent because they'd end up at the same point, right?
Here is a much simpler and enlarged version of the graph: As you can see each curve that originates from (0,0) goes to (1,1) and each curve that originates from (0,1) goes to (1,0) but the starting and ending points don't really matter. 