Science fiction can transport you to other worlds, and so can mathematics. But few people ever see the mathematical worlds, because the natives speak a different language and the terrain is rough, full of abstractions and infinities. That's a shame, because there are some wonderful sights to see, and some fascinating ideas to try on. I'm a native, a professional mathematician, and if you're game, I'll take you on a tour.
I'm going to show you some topology, in part because it's the weirdest, wildest area of math, and in part because it's my home town, the math I spend my time studying. Topology is the study of spatial relationships that don't depend on measurement. "Connected," "between," and "inside" are topological ideas, "far" and "straight" are not. The best thing about topology is that topological properties are intrinsic to objects and don't require an external view, a "view from above." This is a subtle distinction, so be patient. We'll start slow and come back to it often.
The first real topology problem was discussed and solved by the great Swiss mathematician Leonhard Euler in 1736. The town of Konigsberg (now Kaliningrad) is cut by a branching river into four sections, and has seven bridges.
Euler wondered whether it was possible to take a stroll in Konigsberg and cross each bridge exactly once. What makes this a question of topology is the fact that the distances between the bridges, the size of the land areas, and the exact arrangement of the bridges make no difference to the problem; the only pertinent information is which bridges connect which sections of town. So, let's throw away the irrelevant details and reduce the problem to its essentials. Replace each section of town with a dot (called a vertex), and each bridge with a line (called an edge). Konigsberg becomes an abstract figure called a graph.
With the city reduced to a diagram, we can trace along the edges with a pencil instead of having to walk the streets. The problem is to run the pencil along every edge of the graph exactly once without lifting the pencil. Can it be done?
Konigsberg is just one graph. The more interesting problem is to figure out what makes it possible to trace some graphs and impossible to trace others. Below are two graphs (a fish and a house) that can be traced. Try them, and focus on where your path needs to start and end.
The key to tracing graphs is to notice that each time you come into a vertex by one edge you need to leave by another, so that each visit to a vertex uses two of its edges. As you fill in the graph, any vertex with an odd number of edges is going to cause problems -- there will be one edge left over. Let's call these vertices "odd." The only way to complete an odd vertex is to start or end your path at it. So, if you hope to trace the whole graph, you can only handle two odd vertices.
Two Traceable Graphs
We've just proved that tracing Konigsberg is impossible, since more than two (in fact, all four) vertices have odd numbers of edges. The other example graphs are all possible, provided you start at an odd vertex. But to thoroughly answer the graph tracing question, we'd like to say that any graph with two or fewer odd vertices is traceable. It's true, and not difficult to prove, but a bit lengthy. If you're game, here's a proof with some detail.
Tracing graphs is an old and maybe even innate human idea, as it shows up all over the world. Besides Western examples, there are traditions of graph tracing in some African societies, and in the Republic of Vanuatu in the South Pacific. These are a couple of sand tracings from the Tshokwe culture of southern Africa:
Moyombo Tree and the Path of a Chased Chicken
Here is another classic problem about graphs:
Can you connect each house to each of the three utilities so that none of your lines cross? (Lines can't run "under" houses or utility ovals, either.)
This is a topology problem, because it doesn't matter how long or meandering the utility lines become. And in fact, we don't care where the houses or the utility hookups are located (move them if you think it will make a difference!). So again, this is a question about a graph. There are three "house" vertices, and three "utility" vertices, and we have nine (count 'em) edges connecting houses to utilities. The question: Can we draw the graph without any crossing edges?
There is a fundamental difference between the Konigsberg bridge problem and the utilities problem. The Konigsberg problem is internal to the graph. It is a question about how the graph is connected, and the two-dimensional drawing of the graph is unimportant. The utilities problem is an external one, as it is precisely a question about how the graph is drawn. This contrast is crucial, so let's illustrate it with rodents.
Imagine a hamster in a Habitrail set up like the Konigsberg graph. The hamster's worldview is entirely one-dimensional (if it helps, imagine the tubes are opaque so it can't see the outside world). The hamster can move forwards or backwards through its tunnels, with the occasional choice of left or right at a fork. Although it can't see the graph from the outside, the graph tracing question makes perfect sense: the hamster wants a route to survey every tunnel of its domain exactly once.
Now let the hamster out of its Konigsberg Habitrail and build it one which models the utilities problem. Give it three "house" rooms and three "utilities" rooms, and run one tube from each house to each utility. Now you want to ask the hamster if any of the tubes cross, when seen from above. But from the hamster's point of view, this is a ridiculous question! It knows very well which tunnel to take from a particular house to a particular utility, but can't "see from above" to decide if the various tubes are crossing. We'll see that the external nature of the utilities problem gives rise to some surprising variations, but first we need a tour of two-dimensional topology.
A photographer goes bear hunting. From camp, he walks one mile due South, then one mile due East, then shoots a bear, and, after walking one more mile, is back at camp. What color was the bear?
A plane is two-dimensional, meaning it takes two numbers to describe one of its points. As an example, 1200 East 57th Street describes a location in Chicago and requires two numbers. Stand on the Sears Tower, turn 16 degrees left of due south, and look 7.3 miles away. You'll see the same spot in Chicago, and it still took two numbers to describe.
Your shower is a nice example of a two-dimensional system. You can describe the water flow with two numbers, the amount of hot and the amount of cold, and most showers come with knobs to do just that. Some showers have a pull/turn lever that specifies temperature and water volume, which gives a different description of the same water flow but is still clearly two-dimensional.
Freedom of motion is a good point of view for understanding dimension. The hamster in a tube has only one free motion, forwards or backwards, which is one dimension of freedom (the points where his tunnels branch are not free motions, but restricted choices, and don't add a dimension). On a flat field, the hamster has two full dimensions of freedom: forwards/backwards and side-to-side. A hamster with wings (a bat?) adds up/down as a third dimension of freedom.
In two dimensions the most familiar object is the infinite flat plane, or Euclidean plane, from high school geometry. In fact, a mathematician defines a two-dimensional "manifold" as an object with the property that a small piece of the object looks like a small piece of the plane. For example, a sphere is a two-dimensional manifold. The surface of the Earth is a sphere, but you see only a small piece of it, small enough that from your perspective the Earth looks flat.
In fact, how do you know the Earth is round? You've seen pictures from space, but that's an external view, and not one you're likely to experience. But even the ancient Greeks knew we live on a sphere. Their evidence was the changing night sky at different latitudes, and the fact that during lunar eclipses the Earth's shadow on the Moon was round. In the third century B.C., the librarian Eratosthenes even measured the circumference of the Earth by calculating the angle made by the Sun's rays at different latitudes. Yet all of this evidence is rooted in external phenomena, using the fact that the sun and stars have a "view from above" and observing the consequences. To get an internal proof that the world is round, you need to observe a large part of it. Making a map of a big chunk of the planet would demonstrate its curvature, but only with sufficiently accurate tools. Most convincingly, you could walk in a straight line until you return to where you started. That would be solid evidence that the world is not a plane, and would require no external view, nor even a measurement.
The bear problem, if you haven't solved it yet, depends on the curved Earth. Really, it's a problem with our North-South-East-West coordinate system for the Earth, which seems to give a nice grid but actually fails miserably at the poles. Starting at the north pole, the photographer could march one mile in any direction (always South!), walk East for a mile (which requires steadily turning to keep the pole at his left), and then return to his polar camp, still one mile away. The only bears he's likely to see would be white ones.
Moving on from the sphere, another good example of a two-dimensional manifold is the torus, most familiar to us as the surface of a donut. As in the case of the Earth, it is the surface that is of interest. The Earth is really three-dimensional -- you can dig into it, birds and airplanes can leave it, but if you want to stay on the surface, you're stuck with two dimensions of movement. The same is true of a donut. The two-dimensional torus is not the doughy insides, it's the glaze.
To get the topologist's picture of a torus, you need to live on one. You can imagine walking around on a donut-shaped planet, but that image is burdened with a very external view of the situation. Instead, let's try and understand what life would be like inside a two-dimensional torus universe. This is hard, so start with the easier question: what would life be like in a one-dimensional circle universe? Don't take the external view of a circle drawn on the floor. Take the hamster's view, where living inside a circle means the Habitrail is a loop, and the hamster can run around it forever. If the loop is big enough, the hamster might even believe he lives in an infinite straight tube.
Now let's try the torus. One way you may have inhabited a torus is through video games. Consider Pac-Man, which is played on a flat screen. In Pac-Man, when the characters move off of one edge, they come back onto the opposite side. The magic teleportation from one side to the other has the same effect as if the screen were rolled up and glued into a cylinder.
In the even older game "Asteroids," the edge gluing is even more apparent, since the objects aren't confined to a maze. And with Asteroids, the bottom of the screen is connected to the top as well. This has the same effect as if the screen were rolled into a cylinder, and then bent again to glue the two circular ends together, forming the familiar donut shape.
Life aboard the Asteroids ship would be strange. If you've played Asteroids, you know that bullets shot out the front of the ship will wrap around the screen and come back toward your ship from behind. This would be true of light rays, as well, which means that looking out the front window you would see the back of your ship in the distance. And not just once, either, but infinitely many views arranged in a grid. The effect would be quite similar to standing in a square room with mirrors on all four walls.
If the torus was large and visibility less than perfect, those images would be too far away to be seen, and there would be no evidence that you lived on a torus. You would believe your world was a flat plane, infinite or possibly with an edge at some unfathomable distance. In fact, your world view would be quite similar to that of most Christians during the dark ages, when the Earth was widely believed to be flat, reaching out to its "four corners" and ringed with ocean.
Trying to picture life inside of a space different from our own is a difficult exercise, not easily grasped in one sitting. If your interest is piqued, read the science fiction classic Flatland, by E. A. Abbot, which chronicles the adventures of A. Square, a resident of a two-dimensional world. Written in 1884, it's now in the public domain. A print version is also available.
Let's return to our utilities problem and its topological nature. Hopefully you've played with it enough to convince yourself it cannot be done. But what if these houses were built on the Earth? Now, at least theoretically, you could run a utility line around the back side of the planet. Does this help solve the problem? Alas, no. Suppose you could draw your solution on a balloon, representing the earth. Pick a spot away from the lines, carefully pop the balloon at that spot, and then flatten it out. As long as the balloon is stretchy enough, the damage from popping will stay near the pinprick and not ruin any of our lines, giving us a nice flat solution to the utilities problem. In short, if you can't solve the utilities problem on a plane, you can't solve it on a sphere either.
Now consider the utilities problem on a torus. You could try and draw a solution on a donut, but it would be messy and a waste of a good snack. Instead, unroll the torus and use the video-game model, where the surface is a square. Remember, the left and right edges connect, as do the top and bottom edges. Now a perfectly good utility line would look like:
On the torus, the utilities problem does have a solution! (Find it yourself, or peek at the solution.) The existence of a solution has two consequences. First, the problem really does involve two-dimensional topology, because the existence of a solution depends on what sort of surface the lines are drawn on. Second, the torus and the sphere are fundamentally different. This may seem obvious looking at a donut and a Ding-Dong, but an inhabitant of the donut planet would have a tough time telling his world from a sphere.
Finally, let's go one step further, and consider three dimensions. To imagine living in a three-dimensional space is easy, because that's where you live, sitting on the Earth as it moves through space. Our view of the universe is limited enough that space appears largely flat, but the universe may have topology we have yet to observe. Possibly the whole thing is a big sphere which closes up. Possibly there are places where traveling far enough in one direction brings you back to where you started, like the ship in Asteroids.
If we could step outside the universe and gaze back, any loops or connections would be easy to spot. But like hamsters in tubes, our view is limited. And based on the view so far, the universe is flat, goes on forever, and any three-dimensional topology is purely the domain of the mind.
Bryan Clair is a professor of mathematics at Saint Louis University.
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