A simple problem that has a solution that is far deeper than expected!

Transcription

1 The Water, Gas, Electricity Problem A simple problem that has a solution that is far deeper than expected! Consider the diagram below of three houses and three utilities: water, gas, and electricity. Each house needs to be connected to each utility. So we draw lines or curves to represent the connections. The problem is that in this world, lines are not allowed to cross each other. Can we connect each house to each utility? Play with this problem for a while. You will probably come to the conclusion that it can t be done. There are nine connections that need to be made, and just as soon as you make the eighth, it always seems to block off the route that you need to make the ninth. But can you prove that it is impossible? And what about generalizations? Could you solve it on a sphere? On a Mobius strip? On a torus? To begin to solve these problems, we are going to have to systematically disassemble the plane into small pieces that are easy to deal with. We will break a plane into a union of cells each of which is a very simple piece. That way the plane will become a cell complex that is put together by a few simple rules, but which will allow us to answer questions like this one. To start with, a 0-cell is just a single point. A 1-cell is a line segment or curve segment, with two endpoints. The two endpoints can be glued together, which would yield a loop. Notice that the boundary of a 1-cell is a collection of 0-cells. It is usually two 0-cells, but these can be glued together into a single cell. Now, how nice does our 1-cell have to be? For instance, we definitely want it to be continuous, not wild like sin(1/x) or something. We will allow only curves that are differen- 1

2 tiable. We could make an angle by joining two curves in the middle there s nothing saying our functions need to be differentiable when we stick them together just that each piece is differentiable by itself. Now if we move up, a 2-cell would be a square, or basically some nice piece of a surface. Notice that its boundary is a curve (perhaps not differentiable) so that it is a combination of 1-cells and 0-cells glued together. The important thing is that it not have any holes in it! For instance, an annulus is not a 2-cell. We could keep going up like this to as many dimensions as we wish, but we re working with two dimensions, so we ll stop at two for the time being at least. Let s build some familiar surfaces with cells. A cube is easy. It consists of six 2-cells. Each is a square with four sides and four corners, which are 1-cells and 0-cells respectively. Now two squares meet along their sides. So the 1-cells tie two squares together. So there are 4 6 sides, and each side is part of two squares, so there are 24/2 = 12 1-cells. Similarly, each 1-cell has two ends, which are 0-cells, but three 1-cells meet at each corner of the cube, so altogether there are 12 2/3 = 8 0-cells. We knew all this the cube has six faces, twelve edges, and eight vertices. Could we make a sphere? Sure! Take two hemispherical surfaces. These will be our 2-cells. Sew them together along their boundaries. Now the boundard could be just a single 1-cell looped to join itself end-to-end. That is, we could have one 1-cell and one 0-cell where the ends are attached together. Alternately, we could make the boundary circles of our hemispheres be split into as many pieces as we would like. The diagram shows the equator split into three 0-cells and three 1-cells. Can we make a Mobius strip? Sure! take two squares and glue them together along their edges, making sure to twist one of them. The edges in the diagram below will be glued together so that the arrow match up. Note that there are two 2-cells, six 1-cells, and four 0-cells in this shape. 2

3 What about the plane itself? Well, since it goes on forever, it doesn t really fit into this construction process. We could just say that since everything we are doing takes place in a finite area of the plane, we could just take a single large square that encompasses everything, and this we would have a single 2-cell together with its boundary, which could be any number of 1-cells and 0-cells (as long as it s the same number!) joined together in a loop. In this way, the plane is like the sphere, with one of the 2-cells taken away. A different approach would be to tile the plane with squares, having infinitely many of each kind of cell in a repeating pattern. We don t want to do this. Infinity is too complicated. A third option is to allow some of our 2-cells to go out to infinity. This shouldn t bother us much, seeing as all the really means is that we re keeping an ideal point around. And since the things we re considring, like the WGE problem, don t go off to infinity anyway, we can always draw out pictures so that nothing actually happens within the infinite 2-cell anyway, so we don t need to worry about its properties. This is the way we ll handle it. Here s a weird thought experiment for you. Start with a single 0-cell. Now attach to it a 1-cell so that the ends are connected together, forming a loop. Now the boundary of a 2-cell is a loop, so we can attach the 2-cell to what we have already, and get a proper plane (with boundary). But no one said we had to attach it one-for-one! Stretch the 2-cell out so that the boundary is twice as long as the loop. Now attach half of the boundary to the loop as usual, and then keep going, attaching the other half of the boundary to the loop again! That way, our loop will bound our 2-cell on both sides! What have we created? Hint: the gluing pattern is as shown below: 3

4 Answer: we have created a projective plane! Or another way to think about it, we have created a model of spherical geometry. This object can t really exist in regular 3-dimensional space without passing through itself. That s why it s so hard to imagine! OK, now that we ve played around a little, let s do some topology. Topology is the study of things that do not change when we bend or warp things. Certainly, breaking an object into cells is just drawing lines and points on the object, so we re doing something topological. Thus, something about the cell structures shouldn t change as we make different structures for the same object. So let s say that we have an object that we have divided into cells. Let s say we want to change that cell structure. What can we do? Well, anywhere along a 1-cell we could break it into two 1-cells glued together at a 0-cell. We could also undo this if there is a 0-cell with exactly two different 1-cells coming out of it we could fuse them into one single 1-cell (assuming we can smooth out any angle at which the two meet). They must be two different 1-cells, otherwise we re erasing all the 0-cells on that 1-cell, leaving it with no end points, which is not a 1-cell. And we can t do this merge if there are three or more 1-cells sticking out of the 0-cell either what would happen to the ones that don t get merged? So this operation creates or destroys one 0-cell and one 1-cell. Another thing we could do is that if we have two different 0-cells connected by a 1-cell, we can contract them together. That is, we eliminate the connecting 1-cell, and merge the two 0-cells. We can also do the reverse of this. Given any 0-cell, we can split it by replacing it with two 0-cells connected together by a 1-cell, and separating the 1-cells that originally connected to it into two groups, with each group (which could be empty!) connecting to one of the two new 0-cells. Again, this either creates or destroys one 1-cell and one 0-cell. A final thing we can do is to pick two 0-cells (which may both be the same one!) on the boundary of any 2-cell. Now connect them with a 1-cell. This has the effect of cutting the 2-cell into two pieces, each of which is a 2-cell. So we can add a 1-cell and a 2-cell at the same time. We can undo this as well, merging two 2-cells, provided we can make the angles smooth. So we can subtract one 1-cell and one 2-cell. The upshot is that anything that we do to legally change the cell structure of a shape changes the number of 1-cells, but it also either changes the number of 2-cells or of 0-cells by the exact same amount. This tells us that if F is the number of 2-cells, E the number of 1-cells, and V the number of 0-cells, then F E + V must remain constant. We use these letters to honor the names of the parts of the cube: faces, edges, and vertices. For instance, we have seen that for a cube, F = 6, E = 12, V = 8. So F E + V = 2. We will call this the Euler characteristic of the cube. For a sphere, we have done it two ways: F E + V = = = 2. As far as topology is concerned, the sphere and the cube are essentially the same thing! And so is the plane (assuming we include the the ideal point; if we don t, then planes and spheres are different, because planes have to have infinite pieces, and spheres can t). I claim this let s us prove that WGE can t be solved on a plane. If we could, we might as well work on a sphere. Now let s say there was a solution. Each house and utility could be thought of as a vertex, and each line an edge. So we would have a cell structure for the sphere with six vertices and nine edges. This requires five faces. Now each face has at least four sides. Why? Start at one vertex of a face (say a utility) and walk to a neighboring vertex (which will be a house). Now keep walking around the 4

5 boundary and you will walk to a different utility becuase there is no need to have two lines from the same utility to the same house! Now you will walk to a different house, and then to a utility (which might now finally be the original one). So each face has at least five sides, so there are at least 20 total sides. When counting sides, each edge is counted twice, because it is part of the boundary for exactly two different faces. Still, this means that there must be at least 20/2 = 10 edges, and we know there are only nine! Reality check: if we do one of the diagrams where all of the houses are connected to all of the utilities except we are missing one connection, is this a legal cell structure? It is easy to check, just by drawing and counting, that it is. We get six vertices, eight edges, and four faces, each with four sides, leading to 4 4/2 = 8 edges, and everything checks. 5