Ted Mobius Transformations

Hey all, it's been a little over a week since my last post, but unlike this functor, I haven't forgotten about you. This week I'm going to introduce the titular Mobius Transformation. This bad boy, also known as a Linear Fractional Transformation, twist, turn, and stretch the sphere based on where 0, 1, and ∞ are mapped. These transformations also allow us to construct the Klienian group fractals.

As you can see the Ted Mobius Transformation adds age and a tux.

I want this post to be manageable and concise, but I also want to discuss a bunch of topics, like fixed and periodic points, and whether they're attracting or repelling, as those let us construct the Julia and Fatou set which are really awesome for building fractals. I think I'll just discuss what a Mobius Transformation is here, and then later I'll post about those points and sets.

A Mobius Transformation is a transformation from the Riemannian Sphere to the Riemmanian Sphere T(z) = (az+ b)/(cz+ d), where a,b,c,d are complex numbers such that ad-bc !=0. That last condition may remind a mathematical reader of a condition of an invertible 2x2 matrix.

Let's look at some inputs for z to see how our sphere changes. Some of the following text may be hard to parse over blogger (I'll try to format it in a way that's not too bad), but if you spend a second, you can easily convince yourself it's true. Maybe one day I'll get Latex and rewrite some of these lines.

T(∞)= (a∞ +b)/(c∞ +d) = a/c. That makes sense, as if a and c are infinitely large, then adding or subtracting a finite number won't change either of them. This also shows why we need to be on the Riemannian Sphere, because on the complex plane (without infinity) then there's no number we can plug into T(z) to get a/c.

T(0) = (a0 +b)/(c0 +d)= b/d. 

T(1) = (a1 +b)/(c1 +d)= (a+b)/(c+d).

These are some of the intuitive numbers to plug in, to see what we'd get. But now let's try some less initially intuitive but geometrically significant numbers.

T(-d/c) = (-ad/c +b)/(-cd/c +d) = (-ad +bc)/(-cd +dc) = (-ad +bc)/0) = ∞. But a tricky reader might want to test this function is well defined. "How do you know you'll never get 0/0?" Well, tricky reader, the only way we'd get 0/0 would be if -ad +bc =0, but as you'll remember by our definition of a Mobius Transformation, -ad +bc =0 can't be the case.

T(-b/a) = (-ab/a +b)/(-cb/a +d) = (-b +b)/(ad - bc) = 0/(-ad +bc) = 0. Remember, ad-bc can't equal 0, so it remains well defined.

This next one will be a bit more complicated looking, so bear with me. I'm going to color code fractions:
T((d-b)/(a-c)) = (a ((d-b)/(a-c)) +b)/(c ((d-b)/(a-c)) +d) = ((ad-ba)/(a-c) + (ba-bc)/(a-c))/((cd-cd)/(a-c) + (da-da)/(a-c)) = ((ad-bc)/(a-c))/(ad-bc)/(a-c)) = 1.

You can convince yourself now that these transformations work under composition and can be inverted.

Mobius Transformations can be thought of as a series of composition of different transformations. First take T(z) = z and scale it by c and translate it by d. Now you have T(z) = cz +d. Second, invert it so that T(z) = 1/(cz + d). Third, scale it by -1/c (almost like conjugating with the first step), and get T(z) = -1/c(cz + d). Finally, translate that by a/c: a/c - 1/c(cz + d). I'll do the algebraic manipulation below, or you can trust me that after some finagling, it comes out to T(z) = (az +b)/(cz +d). The key to the algebraic manipulation, which took me frustratingly way, way too long to see is realizing that ad-bc =1.

a/c - 1/c(cz + d) = a/c - (bc -ad)/c(cz + d)= (a(cz +d) + bc - ad)/ (c(cz+d)) = (caz + da + bc -ad)/(c(cz+d) = (caz +bc)/(c(cz+d)) = (az+b)/(cz+d).

That's really cool for several reasons. First, it makes finding the inverse very easy, because instead of composing T₄∘T₃∘T₂∘T₁(z), you can just do T^-1₁∘T^-1₂∘T^-1₃∘T^-1₄(z). Also if we take all of our Mobius Transformations to create a group under composition, with easily seen generators.

Pretty much, Mobius Transformations send three points to any other three points, and by doing so, determine the entire transformation of the sphere. This means we can determine exactly how the sphere acts based on where 0, 1, and ∞ are sent. But we can also look at other points. You can see yourself that given points a,b,c, T(z)= (z-a)/(z-b) * (c-b)/(c-a) sends a to 0, b to ∞, and c to 1.

Or going a step further, you can see that given poitns a,b,c,d,e,f, if T(z) = w is a Mobius transformation, and (z-a)/(z-b) * (c-b)/(c-a) = (w-d)/(w-e) * (f-e)/(f-d), then the transformation maps a to d, b to e, and c to f.

Take a look at this picture I drew representing a Mobius Transformation abstractly:

I'll post more later about fixing points, specific Mobius Transformations (like ones that preserve symmetries, ie, which one maps the unit circle to itself, etc), and their relation to fractals. This post should just introduce Mobius Transformations, there's still much more to go!

Abelian Grapes

Hey all! Exciting news, I've been attending a weekly differential geometry seminar at ASU, and I finally understood a lecture (about conjugacy limits in Cartan subgroups of SL(n, R), looking specifically at n=3, and then applying use of the hyperreal numbers). I don't know much about Cartan subgroups or Lie groups in general, but I could follow along the lecture and parse, understand, and reflect on examples and terms and such. Cool stuff! I guess I am learning math, who knew?

My favorite part about math is the terminology--for example, take the hyperreal numbers, *R, (equivalence classes of sequences of real numbers constructed by taking all real series and denoting their convergent point a hyperreal number--and apparently if it doesn't converge like (0,1,0,1,0...) then you just assign it to either 0 or 1? I don't know all the ins and outs of them). The hyperreal numbers *R are cool because they're non-Archimedian. Numbers like the reals are Archimedian because if there's a number a < 1/n, where n is any natural number, then a as to be 0. But the hyperreals have infinitesimals like ϵ= (1, 1/2, 1/4, ...), which is smaller than everything, but bigger than 0. This also means the hyperreals have infinitely large numbers, like 1/ϵ--which should remind you of how the Riemannian sphere acts with infinity and 0. And then *R has appreciable numbers, which are just real numbers plus infinitesimals α = a+ ϵ, a in R. Pretty cool stuff. Turns out the hyperreals are a much easier way to conjugate by a series of matrices, which is needed to do to find conjugacy limits of things. But the reason I bring this up is because some of the terms are pretty cool. If you have an appreciable number, α = a+ ϵ, and you want to go back to the reals, you just "take the shadow of α," which, though it just means to take the real part, a, of α, makes it sound really cool, like it's from some children's TV show about children's card games. And another cool term, galaxy, is represented Galϵ(x) = {y in *R, | |x- y| <= kϵ}, k in R. During the lecture, when she introduced "galaxies," I turned to the guy to my right and said "I love math," and he nodded in agreement.

Anyway, this post should be fairly short and easy to grasp, especially compared to the other posts, and the posts about to come. This will just introduce the concept of a group.

A group is a set G with a binary operation, * : G x G -> G, such that the following three properties

1. For g,h,i in G, (g * h) * i = g * (h * i), ie, * is associative
2. There exists an identity element, e, in G such that, g in G, g * e = e * g = g
3. For each g in G, there exists an inverse g-1 in G such that g * g-1  = g-1 * g = e

But remember that because * is a binary operation, then for G = {a,b,c} to be a group, a*b has to also be in G as well. This property is called closure and is essential to bear in mind.

Furthermore, a group G is abelian if its binary operation is commutative, that is, gh=hg, g,h in G.        

Common groups include the integers under addition represented as (Z, +) or referred to Z mod n, consisting of the elements {0,1,2,...,n-1}. To form the multiplicative group (Z, •), one must remove the 0 element, as to satisfy the inverse requirement of a group--g in Z, 0•g = 0 =/= 1.

Symmetry involves and necessitates group theory. Given a finite set A, its permutations (bijections from A->A) result in interesting properties, with application to geometry. If A = {1,2,...,n}, the group of possible permutations of A is called Sn, the symmetry group of n.

If n=3, then we may model geometrically the elements of S3 with the vertices of a triangle. 

from Troy University

Here, various permutations of Sn yield different transpositions. For example, mapping 1->2, 2->3, and 3->1 results in a rotation by 120°, but the mapping 1->1, 2->3, 3->2 results in a reflection across the angle bisector of vertex 1.

Also, groups are key for complex analysis. Not only are there many groups involving C, but the big one, which this project is titled for, is called the Linear Fractional Group, ie all the Mobius Transformations together. Matrices are a great way to represent groups, and in the next post about Mobius Transformations, I will probably also introduce the analogous matrices.

EDIT== By the way, the title is a joke based on