Mathematics has always been a beautiful discipline, but its appeal is of an austere kind that takes some effort to fully appreciate. But computers and simulations now enable people to create visualizations of the beauty that they could formerly only create in their minds. The beautiful patterns generated by the Mandelbrot set is one example.
Here is another nice example, how something known as Mobius transformations can be made more easily understood by using simulations to look at the same transformations in a new way.
I see how having the sphere above the plain makes the transformations easier to understand.
Holy crap that is so elegant.
I never learned the Mobious transformations, but I can imagine how seeing this video before getting into the math would really help keep everything in context.
Wow.
Makes me want to go back to school and finish a degree in maths.
Wow, this is amazing. I recently reviewed Möbius transformations in my topology class, and I had never realized the correspondence of Möbius transformations and movements of the sphere.
I needed this.
I’m just starting to get into real math (i.e. the stuff they don’t bother to teach to physics undergrads, i.e. the fun stuff), and it’s still making my brain hurt (especially since I’m basically wandering around without access to a professor for the next year), so sometimes a visual aid like this is a huge help.
(I want to make a model like that sphere in the animation, now…the position of the light would have to change with respect to the position of the colored bits, but that could be done. Maybe two nested spheres with a fluid separating them…)
OK, since geometry is apparently more important than sleep…
I’m trying to get my head around inversions, particularly using the rotating-sphere version. It seems at first glance like this involves a one-to-one mapping between points on the sphere and points on the plane. If this is the case, it involves a mapping of a surface of finite area (the sphere) onto a surface of infinite area (the plane); intuitively it seems like the set of points on the sphere should be a smaller infinity than the set of points on the plane…
…and in any case, it seems like when using the relevant math to model actual phenomena, there would probably be a very strong “preference” for the spherical or planar expressions of the model?
There is indeed a trick involved in the one-to-one mapping, but it’s the other way around: The sphere has one point (the north pole) that does not correspond to a point in the plane, but has to be added to it as the “point at infinity”. Mapping an infinite area to a finite area or vice versa is rather easy; for a one-dimensional example, the arctan function maps all numbers to those between -π/2 and +π/2.
Both the plane and the sphere have uncountably infinitely many points; that’s the same size of infinity for both. Compare this post by our esteemed host for an explanation of different sizes of infinity.
I’m no expert on Moebius transformations, but I would indeed expect that the choice of model depends on what’s easier to work with for each possible application -- with the added bonus that shifting models at will may simplify matters.
I think Mano’s next lengthy series should be a 230 part series on all of the three dimensional space groups.
I can’t wait for P n m m!
Yeah, I had the feeling that my not-very-mathematically-trained intuition was failing me in terms of exactly the sorts of things I remembered from that post. It’s encouraging that I was at least correct in my suspicions about the incorrectness of my initial instinct. More to learn!