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

How Complex Could Geometry Be?

Okay I spent way too long trying to think of an interesting title. I initially wanted to make a pun based on psychological complexes--"Argand had a real zero complex" (see, I've got absolutely nothing... if you've got something, feel free to use the comments...)--but then I said the word "complex" too many times that I completely forgot what it meant, and I had to google "complex plane" to make sure that was a thing.

See, math, it's dangerous (maybe more than I thought). Oh well, this title should be sufficient. If anything, it thematically connects this post to my last post's brief mention of how complex analysis is awesome for geometry, symmetry, and pretty much everything.

Last post I gave a brief introduction to symmetry. I figured I could take the time and make another post to detail the basics of complex analysis. Again, this will be long and will simply give an introduction to the use of complex numbers, so feel free to skip through it if you know it already.. Soon, I'll make another post for group theory. 

Complex numbers are awesome. They are very similar to the real numbers, but by the way we construct them, allow for greater geometric significance. First let's begin with i.

i is the square root of -1, defined by the equation i² = -1. This allows us to define complex numbers, where we can more fully account for algebra. For example, when we are restricted to just the reals, the equation x⁴ - 1 = 0 has just two solutions--1 and -1--but in the complex numbers it has 4 (the same amount as the degree of the polynomial)--1, -1, i, and -i.

Now let's define complex numbers

Take R2

No, not you

Excuse me, the real plane, R²

It has an x-axis and a y-axis, and every point in R² can be represented by a coordinate pair, (a, b).

The complex plane is very similar, instead of an x- and a y-axis, it has a "real" and an "imaginary" axis:

Here, still every point can be represented (a, b), if we understand (a, b) --> a + ib. This is the form of a complex number--a is the real part, and ib is the imaginary part.

Every complex number, as I alluded to in my last post, has an argument and a modulus. The argument is the angle, and the modulus is the distance between (a, b) and the origin, (0, 0). Thus, the modulus of z = a + ib is |z|² = a² + b².

Complex numbers can also be represented by z = r cis(θ)= r(cos(θ)+ isin(θ)) = r e^iθ, where r is a real number and represents the modulus, and 0≤ θ≤ 2π where θ represents the argument.

The usual algebraic operations work with complex numbers, and it is easiest to view i as a variable with the special case that i² = -1.


Let w = a + ib and v = c + id be complex numbers. Addition and subtraction are clear: w+ v = (a+b) + i(b+ d). Multiplication, less so, but still intuitive: wv = (a + ib)(c + id) = ac + iad+ ibc - bd = (ac - bd) + i(ad + bc).

Now that we see multiplication, let me introduce you to our friend the complex conjugate! Each complex number z = a + ib has a complex conjugate = a - ib. Notice how this is the reflection of z across the real axis!

From wikipedia

Why this is so special is that if we ever want to get a real number from a complex one , all we have to do is multiply z by is conjugate (hence the line in the sonnet in my first post "Conjugating in C, you made me realize)! (a + ib)(a - ib) = a² + b².

Now, this gives us motivation behind division. v != 0, w/v = (a+ ib)/(c+ id), and multiplying by the conjugate of v over the conjugate of v, gives us ((ac+ db)+ i(bc - ad)) / c² + d².

In polar coordinates, these operations becomes much smoother. Multiplication of w= r₁ e^iθ₁ and v = r₂ e^iθ₂, wv= (r₁r₂) e^{i(θ₁+ θ₂)}. Notice that the multiplication of the conjugate would result in some z = (r r ) e^{i(θ- θ} = r² e⁰ = r², which is exactly what we'd think it'd be.

Inversion is easy to see. We are looking for a z^-1 such that zz^-1 = 1. If z = r e^iθ then by what we just saw before, we are looking for some z^-1= p e^iΦ such that rp = 1 and θ+ Φ= 0. Thus, p = 1/r and Φ= -θ.

This post got pretty long, so I'll give you a break and will post about mapping in the complex plane and the Riemmanian sphere in a separate post later.

Connor

Symmetry | yrtemmyS

Part of my week was spend abroad at MIT! It is insanely beautiful there--despite being very cold. Fortunately, the blizzard didn't phase me much. Also shout-out to the dining hall chefs of Next House for making some great stir fry! 

The following picture is proof that I did at least some work (that is, while I wasn't partying it up with the gracious residents at Next House over study breaks with some tasty, tasty cheese). 

Picture taken in the middle of finding more out about Stereographic Projection

Anyway, while I was taking an Uber with my awesome cousin Jonathan to a entertaining Improv show and the world renown Mike's Pastry, I realized just how much symmetry surrounds us, and I thought this would be a good opportunity to blog about the basics of symmetry.

Take MIT's mascot, Tim the Beaver.

Get it? Beaver, like nature's engineers?
Whatever, engineering is too applied anyway.

His name TIM is MIT spelled backwards, which is an example of bilateral symmetry, that is, symmetry under reflection. Imagine Tim the Beaver now with a vertical plane separating him in half. If we had some sort of magical machine that would copy every molecule of his left half and move it horizontally and by the same distance to the right side, then that would reflect the left half of him to his right half and give us bilateral symmetry. 

Tim's name is even bilaterally clever: TIM | MIT. Each letter, M, I, and T, is equidistant from the line in the middle, and when reflected MIT gives TIM, and vice versa.

This is the type of symmetry you probably think of when you imagine something symmetric, as we as people love bilateral symmetry. Vehicles like planes, boats, and cars are usually bilaterally symmetric, and people almost universally think other with bilaterally symmetric faces are, ceterus peribus, prettier.

But there are two main other kinds of symmetry, too!

The second kind is translational symmetry. This is the kind of symmetry you get when you move picture

s over some distance. It's essentially just repeating some abstract motion over some object. An object is said to have translational symmetry if when physically translated somewhere, the the view of an object as a whole remains constant even though the parts are translated.

For the simplest analogy, imagine being in the middle of the desert and there's only a straight road which winds from horizon to horizon, until it bends out of sight. If you were to take each point which makes up the road and translate it a foot downwards, then though each part of the road is moving, the overall object remains the same.

If that wasn't clear, here is a picture from Mumford's Indra's Pearls of Vision, the wonderful book which this project is based out of, and which you can find a link to in the side bar (caption from the book as well).

Two rows of flowers moved along by the same translations.
The flowers are quite different but the symmetry of the two rows is the same

Translational symmetry may sound fairly basic, but it has profound implications. Aside from applications which you see every day like architecture, it is key to algebra and analysis.

Another interesting aspect of translational symmetry is wallpaper groups or crystallographic restriction. These are the different patterns which can tessellate the plan without overlap. I don't know too much about this, but perhaps it will be of some study later. For more information, check out wikipedia (really use wikipedia as a first hand resource for anything you're unsure of in this blog--that's why I included a wikipedia widget in the side bar).

Finally, the third kind is rotational symmetry. This is the symmetry you get if you have an object that can be rotated around some point but still look the same.

Take a square and rotate it about its middle 90, 180, 270, or 360 degrees, you'll get the same image back. If you have a pentagram, you'll have 5 rotational symmetries. An octagon, 8. If you keep going, you'll see that a circle has an infinite amount of rotational symmetry.

Rotational symmetry is why the complex plane is so useful for geometric interpretations of groups, and is why I will be using it extensively. A complex number is determined by an argument (angle) and a modulus (length)--together z = r * e ^i*θ--and multiplying complex numbers not only increases their modulus, but it also increases their argument.

Take z =  e ^i*θ. Its square, z², is e ^-2θ. Obviously the argument of z² is twice that of z, so though it's been rotated, but it is still overall the same. This is similar to other complex properties like roots of unities which I won't go into here.

All together, these help build some symmetry or dihedral groups, as the rotations correspond to different ρ's (pronounced "rho") and the reflections to τ's (pronounced "tao"). But I will post on symmetry groups later.

Why symmetry is important is that it allows us to better understand transformations (functions or maps both in the same geometry) on the plane. Symmetries can be thought of as pattern-preserving transformations.

Let T be a transformation on the plane R² (so that T sends every point in R² to a point in R²).

If T is a reflection of the form T(x,y) = (-x, y), then every point in R² has is x coordinate flipped over the y axis. Obviously this is not every kind of reflection on the plane, as the transformation depends over what line it is being reflected, but I'll just provide reflections over the y-axis here, and leave the rest as an exercise for the reader.

If T is a translation of the form T(x,y) = (x + a, y + b), then every point in  R² has its x coordinate mapped to the number a ahead of it and its y coordinate mapped to the number b ahead of it. If for example, our particular translation is T(x,y) = (x +1, y), pretty much the graph is shifted over 1 unit to the right.

As a quick aside for the mathy people, the set of translations in R, T = {a | a in S_R, x -y = xa - ya}, x,y in S_R, forms a subgroup of the symmetry group of R.

Proof: i) identity is pretty obvious.
ii) inverse: let a be in T, so (xa^-1) - (ya^-1) = x(a^-1a) - y(a^-1a) = x - y,
iii) closure: let a, b be in T, so xab -yab = xa - ya = x -y.

If T is a rotation of the form T(x,y) = (x cosθ - y sinθ, x sinθ + y cosθ), then every point is being rotated anticlockwise around the origin by the angle θ. Though it looks more complicated, if we plug in some numbers we intuitively know, we see this makes sense: T(0,1) by 90 degrees should map to (-1,0), and if we plug it in with θ=90, we see T(0,1) = (-sin90, cos90) = (-1,0). T(1,0) by 90 degrees should map to (0,0), and if you plug it in, it checks out. Same with T(0,-1) mapping to (1,0). Thus we can conjecture (correctly I might add, but you can prove it yourself) that when using θ = 90, then T(x,y) = (-y,x).

Okay this went on longer than I thought it would, and though I still have material to blog about (composition, groups, conjugation, etc.), I can just make a new post in a few days. I should be able to post in detail about fractals within the month.

See you soon, 

Connor