Wish You Were Sphere

First of all, I just found this link, and thought it'd really nicely explain some of the purpose questions behind this whole project for anyone out there skeptical of the point of math, especially when it's so abstract.

This is the continuation of the last post, How Complex Could Geometry Be? Here, I'll talk about mappings in the complex plane, the Riemannian sphere, and stereographic projection. I'll take this time now to apologize preemptively to my blogging group, for the quite frankly ridiculous amount of posts you have to read. All I ask is that you remember a lot of it is pictures, and I like to think this is pretty interesting stuff.

Of course, huge shout out to the wonderflorious Mrs. Bailey, who is largely responsible for my knowing what I do. To ease the non-math readers, this post will be more heuristic, but if anyone wants more mathematical rigor, make a comment and I will either answer there or make a new post if need be.

I don't want to take about complex mappings too much, but I want to point out some significant, overarching geometric implications.

Remember back to complex multiplication. When two complex vectors, w= a*e^ix and v= b*e^iy, are multiplied, the resulting product wv= ab*e^i(x+y). This would geometrically look as if it were spiraling out, as the modulus keeps getting larger and the argument does too, but loops back to 0 every time it reaches 2π.

Now, mapping may get kind of, heh, complex, sometimes--especially with lots of numbers to consider, but it becomes easier if we use colors to heuristically describe it (foreshadowing, this color mapping may be relevant later when it comes to determining fractals...). The following pictures of color mappings are from Levente Locsi's seminar Colorful visualization of complex functions.

Imagine that the complex plane is this picture:

We are going to see, visually, how that color-plane changes based on applying different transformations.

If T(z) = z, then obviously the result is:

That's just the identity function. If we look at the conjugate function (where every point becomes it's conjugate), the color-map looks like:

Remember, the conjugate to a+ ib is a- ib, which is just the reflection over the real axis, which is what happens in the picture. Pink swaps with red, and blue swaps with black. Now let's look at the square function, T(z) = z².

Notice now how essentially we've duplicated out map! If we were to cube it, we'd have three iterations of the pattern pink-blue-black-red, if we were to raise z to the fourth, we'd have four, and you get the picture. The reason this is super interesting is that it connects complex analysis further to geometry and also to group theory! As I briefly mentioned last post, if we have the equation x⁴- 1= 0, there are 4 solutions: 1, -1, i, and -i. If we were to plot these solutions on the complex plane, we'd get a square!

And if we were to have the equation, say, x⁵- 1= 0, then there'd be 5 solutions, which means instead of a square, we'd get a pentagon!

We could keep going, or we could generalize this to say for some real number a, xⁿ- a= 0, in the complex plane, it corresponds to an n-gon! How cool is that?!

Now this next picture is totally irrelevant to anything else in this post, and only will be included because it's an example of color mapping on the complex plane AND it's a really cool picture, and I promised you really cool pictures.

If we start out with the following as our color representation of the complex plane:

Then applying the transformation T(z) = 1/cos(z) gives us:

Again, I don't really have analysis to go along with this picture, I just think it looks cool. :)

Anyway! Though it's really hard to imagine mappings on the complex plane numerically, fortunately, it becomes much easier and solves the whole "infinity" problem with the Riemmanian sphere!

What we do is essentially wrap the complex plane around the unit sphere, and then glue it together at the north pole, and tada! we have the Riemannian Sphere, where everything is better. This is done by a process called stereographic projection.

Let's go through this again, slower. Imagine the unit sphere, such that the unit circle lies on the complex plane. 

Just focus on the picture of the sphere for right now, the rest will be relevant in a bit.

Now, imagine creating a line that goes through the north pole, (0,0,1) and towards some point (x,y, 0) on the complex plane. If we ignore the north pole, then the line from the north pole to the point (x,y, 0) on the complex plane only intersects one point on the sphere, (a, b, c). Stereographic projection essentially is the process is moving all the points on the complex plane to their corresponding point on the sphere.

Now a colorful picture:

This shows you which parts on the plane map to which parts on the sphere. The light blue ring seems to be
the unit circle--or everything with modulus 1. Notice how all the points with a modulus less than 1 are
mapped to the lower half of the sphere, and everything with modulus greater than one to the top half

But what about infinity? In order for us to account for infinity, then we have to pretend like whenever on our complex plane a line would go through infinity--in any direction--then we just map it to the north pole, (0,0,1). Think about it; it makes sense. If you look at the colorful picture above, you could imagine that the black line would "intersect" infinity when it's tangent to north pole. Also, this let's us think of 0 and ∞ as opposites, after all, they are the opposite poles on the Riemannian Sphere.

Remember how your math teacher would always yell at you for trying to divide by zero? Well, in the Riemannian Sphere, the following properties hold:

Let a be a finite, run-of-the-mill number, a+ ∞ = a- ∞ = a*∞ = ∞/a = ∞. (Obviously. If you're confused about a- ∞ = ∞, remember, as we just defined ∞, it's infinity in all directions on the plane, which includes what you might think of as -∞).

But now a/ ∞ = 0 and a/ 0 = ∞! This makes sense, as if you took, say, 1 and parceled it up infinite times, you'd have nothing left. As for a/ 0 = ∞, think about how 1/.5 = 2, because if, for two numbers a,b, b is less than 1, then a/b will be larger than a. Logically, if you keep making b smaller, a/b will get larger. So, it reasons, when you get to 0, and have an appropriately defined ∞ term, you'll get ∞.

But, alas, you still can't divide 0/0 or add, subtract or divide ∞ from itself. Sorry, but us math geeks haven't figured that one out just yet.

It's worth quickly noting how cool circles are on the Riemmanian sphere. Rather than me just tell you everything about it, try looking at the below picture I drew to think about when which circles and lines map to circles from the complex plane to the Riemannian sphere! If you want a tip, the answer, to discuss the proof, or anything related--head on down to the comments!

So now you know enough complex analysis for me to introduce the eponymous Mobius Transformations! Without stealing thunder from a post which will be able to devote more time and words to the subject, Mobius Transformations are just transformations which act on the Riemannian Sphere to itself in the form T(z) = (az +b)/(cz +d)!

These have many cool properties I look forward to sharing at a later time!

Connor