 From a classic brain teaser as the following, three houses A, B, and C must be connected to three utilities, W, G, and E. Safety regulations require that the lines from the house to the utilities don't cross. Is it possible to connect all three houses to all three utilities? Now, a normal person might try out several different connections, and give up after a while, but mathematicians are not normal people. And in fact, we have graph theory. There are three houses and three utilities, so there are six vertices, and since every house must be connected to every utility, there must be nine edges. Also, since houses don't connect to each other, and utilities don't connect to each other, this is actually a bipartite graph, so all cycles are even, and if the graph is planar, in other words, if we can make the connections without crossing, then every face must include at least four edges. So we know that if a planar graph faces have at least four edges, we must have the edges less than or equal to twice the vertices minus four. Contra-positively, if E is greater than 2V minus four, the graph cannot be planar. And so we check. We know the number of edges and vertices. And so we find, consequently, the graph cannot be planar. We must cross edges, and solving the utilities problem is impossible on a plane. While it might not be possible to solve the problem on a plane, the houses and utilities are, in the real world, on a sphere. So some things that might not be possible on the plane might be possible on the sphere. For example, in 1494 Pope Alexander VI split the world between Spain and Portugal. The rest of the world wasn't consulted. But every educated person knew that the world was spherical, so if you go far enough on the Spanish side of the line, you'd get to Portuguese territory. Unfortunately, spheres and planes have very similar topological properties. In fact, we can map a plane surface to a sphere using a polar rejection, and we'll do that as follows. So we can map a sphere onto the plane, then join a line NP from the north pole of the sphere to a point on the plane, and the line passes through a point on the sphere, which is the projection of the point on the plane to a point on the sphere. So a planar graph can be mapped onto the sphere, and vice versa. Now there is one problem at the north pole. In projective geometry we posit a point at infinity, which corresponds to the north pole, but that takes us to a different area of mathematics, which we'll talk about later. This problem sometimes shows up in video games. In an open world game, a player can explore a world. But how can the game-maker keep the player from going off the map? One possibility is to build a wall or limit the player's ability to travel in any particular direction. And another is to identify edges. Literally this means that saying two edges are the same edge. So if you cross the edge, you just appear on the other side. This was a common feature of video games from the 1980s. If we identify the left and right edges of a rectangle, it's as if we've rolled it up so the left and right edges coincided. So a plane with two opposite edges identified is the same as a cylinder. Is it possible for a graph that can't be drawn on the plane or the sphere to be drawn on the cylinder? Unfortunately, no. There's a cylindrical projection that maps every point on a cylinder to a point on the sphere and vice versa if we include a point at infinity. We'll leave that as an exercise for the viewer to figure out. What if we also identify the top and bottom edges of a rectangle? Identifying the top and bottom edges corresponds to rolling it into a cylinder. Then identifying the left and right edges corresponds to joining the ends of the cylinder. This produces a donut, although mathematicians use the term torus. And here's the important thing. A torus and a cylinder have very different topological properties. Imagine a horizontal line across a rectangle whose left and right edges have been identified. Even though you can leave from the right and enter on the left, you still can't cross the line. We'll keep things simple for now. It's a boundary between the two regions. But if we also identify the top and bottom edges, you can pass from one side to the other by exiting from the bottom. You then re-enter from the top on the other side of the line. So the line is no longer a boundary between two regions. In fact, suppose we draw a vertical line. You can leave from the left and re-enter on the right, then leave from the top and re-enter on the bottom, then leave from the right and re-enter on the left. So even two lines aren't enough to separate a region from any one of these places you can get to any other. So now let's consider the utilities problem, but this time on a torus. So we can join A to gas, water, and electric. Since we can leave from one edge and re-enter on the other, we can join B to electric, then from B to the top edge to water, and from B off the top edge to gas. Then we can join C to electric, then C off the top edge to water, and then C off the right edge to gas. And since we can leave from one edge and re-enter on the other, we can solve the utilities problem. Now you might wonder, while we could solve the utilities problem if we lived on a torus, we don't live on a torus. So why does it matter? We'll take a look at that next.