Given some of the comments to my last post and a meeting I had today with my ASU adviser Dr Paupert, I thought I should take this time to explain some further nuances and updates to my project.
My project will cover and focus on classifying Mobius transformations by symmetry. What this means is looking at what properties are kept constant under the group action of Mobius transformations. For example, to demonstrate symmetry around the Unit Circle, one must find a transformation that maps everything in the interval [0,1] to another element in [0,1], so though all the other points on the sphere may be dislocated, this circle is preserved.
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| This is a picture I took from a Rutgers lecture, if the shaded area were to be preserved, then when the mobius transformation acts on the Riemannian Sphere, it stays put |
However, though I intend on classifying these Mobius transformations, I also intend on using other resources such as Indra's Pearls: The Vision of Felix Klein to examine other objects, such as fractals (which are based in symmetry, free groups, Mobius maps, and Kleinian groups). Hopefully, I become adept at using math programming resources like Maple to visually compute examples (fortunately there are resources out there to aid me).
This is great for you! It means you get to see pretty pictures, and lots of them. To demonstrate, please take a look at some of these beautiful pictures taken from Indra's Pearls (found by googling "Indra's Pearls" in google images):
The University of Arkansas describes further these fractals, and cites three more beautiful pictures, which I think would make a good picture to have at the top of my blog.
"Kleinian Group fractals are fractals based on 2 pairs of Mobius transformations and allow you to produce Quasifuchsian, Single Cusp, and Double Cusp, Two-Generator Group fractals described in the book Indra's Pearls - The Vision of Felix Klein by David Mumford, Caroline Series, and David Wright"
Please comment if you have any questions, and I'd be happy to go further in detail about this as much as I can.
Best,
Connor
This sounds great! I love the pictures and can't wait to find out what they represent.
ReplyDeleteStill not sure how this all works but the pictures definitely make this blog worth visiting more often. Eager to learn more Connor. (throw in some sonnets every once in a while)
ReplyDeleteSo, what part of the analysis determines the color of the fractal?
ReplyDeleteI'm not entirely sure, but I know it tells us some important information. I believe it's something like how many iterations of your algorithm (rotation, dilation, and translation I believe) it takes for a point to escape.
DeleteThat's a really good question though for looking into fractals, because the number of iterations determines the color of the point, and usually points close together have similar colors, but when they don't those are the more interesting fractals
DeleteWhoah, is that actually a thing? I didn't think it mattered.
DeleteYeah I might use a blog post in a bit to go further in detail with it.
DeleteConnor,
ReplyDeleteMy name is Braden Bloom and I am writing to you from Lutheran High School in CO. Your project sounds interesting, but I am unsure if I really understand what you are doing exactly and what the purpose of it is. Could you elaborate a little more, please? Geometry is a subject that I have always liked and seemed to be good at, but what you are doing seems way beyond my head. Also, I'm not sure if this has anything to do with your project, or if you already know this bit of information, but I have read a few books in the past about the Golden Ratio - 1 to 1.618 - and I feel like it may relate to your project. I actually just watched a video on Moebius Transformations in hopes that I could better understand your project, but the entirety of your project just seems completely out of my field of knowledge and frame of reference. If you could just explain it in a simple manner so that I can understand, that'd be great because your project seems incredibly interesting!