Complex numbers and branched covers: If you don't want to think about the math, just watch the video at the end. It's a spherical video, so that means you can click and drag the viewpoint around. Please do so, and don't be afraid to turn all the way around several times. [math] Consider the complex numbers, a + ib where a and b are real numbers and i squares to -1. One of the fun statements about the complex numbers that is not true about the real numbers is that for any polynomial P(z) = cnzn + cn-1zn-1 + ... + c1z + c0 where n is not equal to 0 and an is not 0, i.e. for any polynomial that isn't a constant function, there is some complex number a + bi such that P(a+bi) = 0. This is often called the fundamental theorem of algebra. Indeed, for most polynomials there are n distinct complex numbers a+bi such that P(a+bi) = 0. This is very different from the real case, where, for instance, x^2 + 1 = 0 has no solutions in the real numbers; for any real number x, x^2 is at least 0, so x^2 + 1 is at least 1 and hence can't equal 0. Now consider the function fc(z) = z2 - c. If fc(z) = 0, that tells us that z2 = c. Unlike the real case, for the complex numbers there is always at least one solution. Furthermore, if c is not equal to 0, we get two distinct solutions, since if z2 = c, then (-z)2 = z2 = c so solutions come in pairs if z is not equal to -z, i.e. if z is not 0. Let's look at the complex plane again. It stretches off into infinity in all directions without reaching it, and that makes it kind of unruly, so we will make a point at "infinity" just by declaring one; I'll denote it by ∞. So now every direction heads off to ∞. Perhaps a more concrete realization of this: consider, say, a circular rubber sheet, maybe a meter or so across. If you take the edge of the sheet and shrink and glue that edge together into a single point, you get a balloon, i.e. a sphere. Now every straight line on the original rubber sheet eventually hits the edge of the sheet, since the sheet is only a meter across, and since we've shrunk the edge to a single point, every straight line on the original sheet now hits that single point. That single point is ∞. Of course, the complex plane is a lot bigger than a meter across, but if we never actually measure distances then our complex plane + ∞ is just a sphere. This is called the "one-point compactification" of the plane, or the "Riemann sphere". But what can we say about polynomials now? Well, we should expect that ∞k is going to be ∞ for any positive integer k, so P(∞) = ∞ if P is not a constant function. And conversely, f∞(z) = z2 - ∞, so f∞(z) = 0 only if z = ∞. So let's examine our sphere. We have the map z -> z2, which maps points on the sphere to points on the sphere, in almost a 2-to-1 fashion. So it's almost like we're actually mapping two copies of the sphere to one copy. But it's not like we have one copy that is z and one copy that is -z. Consider z = 1. That gets mapped to 1. Moving counterclockwise around 0 gets us to z = i, which gets sent to -1, and then to z = -1, which gets sent to 1, and then to z = -i which gets sent to -1, and then back to 1. So our two copies of the sphere are actually joined together. One way to think about this is to take two copies of the Riemann sphere, slice them both along the positive-real axis from 0 to ∞, and then join the two copies so that the first quadrant of one sphere gets connected to the fourth quadrant of the other, i.e. positive-imaginary part of one sphere to the negative-imaginary part of the other. For the most part this just gets us a somewhat-more convoluted surface, but a short-sighted ant standing on it wouldn't be able to tell that anything funny is going on because you're just gluing the edge of one sphere to that of an identical sphere; except at 0 and ∞ where all of the quadrants come together. This surface is called a 2-fold branched cover of the Riemann sphere, because it's two copies of the Riemann sphere glued together so that most of the time it just looks like Riemann-sphere stuff, and then there are two points where things go bad. We call those bad points "branch points". [/math] Here's the promised video of a branched cover that Henry Segerman made of his apartment, with the branch points at the middle of the ceiling and the floor.