Möbius Transformations Revealed [HD]

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Strong rec Sam Nead · 2014-08 · Mathematics Stack Exchange · on “Is it possible to build a metal wire sphere whose shadow projects the 2d conformal maps of the Riemann Sphere on a flat wall?”

Yes. This is illustrated at minute 2:00 in the following video. [this video] -- the entire video is really excellent, by the way.

Strong rec Mark McClure · 2015-01 · Mathematics Stack Exchange · on “What is unique about the Möbius transform?”

Möbius transformations correspond to rigid motions of the sphere in a very natural way. Möbius Transformations Revealed [LINK] (also on YouTube [this video] ) is the definitive illustration of this. Here's simpler illustration: In this animation, the graticule of lines that we see on the plane and on the sphere form the image of a square in the complex plane. If we project the square to the sphere via stereographic projection [LINK] , we get the slightly deformed graticule that you see on the sphere. If we then rotate or translate the sphere, the graticule is moved in a very simple way. If we project this back to the sphere, we get the graticule that you see on the plane. This happens to be the i

Recommends edo arad · 2013-08 · Mathematics Stack Exchange · on “geometric view of a complex function.”

Another way to see this is using Möbius transformations [LINK] . Let $\phi(z)=\frac{z-1}{z-2}$ be a Möbius transformation. we know that these transformations map circles to circles and that composition of Möbius transformations is done like matrix multiplication, hence the inverse of $\phi$ is easy to find: $\phi^{-1}(z)=\frac{2z-1}{z-1}$. Then all you need to do is compute the image of $|z|=3$ under $\phi^{-1}$. (actually, I don't think this method is better for computation, but it makes the fact that it turns out to be a circle quite trivial) To see why Möbius transformations maps circles to circles, I really recommend viewing this [this video] .

Recommends Jesse Madnick · 2013-09 · Mathematics Stack Exchange · on “Interesting math-facts that are visually attractive”

Here is a beautiful video about sphere eversion [LINK] . Here is a beautiful video about Möbius transformations [this video] . Here is a gallery of surfaces in differential/algebraic geometry [LINK] . There are dozens of beautiful images here -- and there is so much to say about all of them. Examples: Here are pictures of the Weierstrass function [LINK] and of $f(z) = \text{exp}(1/z)$. The Weierstrass function is continuous but nowhere differentiable. The second function provides a visual example of Picard's Theorem [LINK] in action. Both of these are pretty mind-blowing, I think. And lastly, here is a picture of the phenomenon of holonomy [LINK] , which is a topic I'm considering researching. No

Recommends exchange · 2018-03 · Mathematics Stack Exchange · on “Is every conformal transformation a stereographic projection of an isometry of the inverse projection?”

Here [this video] is a really nice video that shows the following: Suppose we have a subset of $\mathbb{C}\cup \{\infty\}$. Then we can project it onto the 2-sphere via the inverse stereographic projection. If we rotate or translate the sphere in the surrounding $\mathbb{R}^3$ and apply the stereographic projection again, then the result amounts to a Möbius transformation of the initial subset. Here is a screenshot to illustrate this: $\hskip2in$ [LINK] Furthermore, on this wiki-page on conformal maps [LINK] , the 4th paragraph of the subsection "Complex analysis" says that "A map of the extended complex plane (which is conformally equivalent to a sphere) onto itself is conformal if and only if i

Recommends HelloGoodbye · 2024-07 · Mathematics Stack Exchange · on “Stereographic projection combined with rotation is a Möbius transformation”

Not that this helps you show mathematically that the rotation of the sphere under the stereographic projection is a Möbius transform, but this video [this video] illustrates it quite nicely.

Cites Mark McClure · 2013-05 · Mathematics Stack Exchange · on “Plotting in the Complex Plane”

First, the graph of a complex function is by definition $\{(z,w)\in{\mathbb C}^2: w=f(z)\}$, which lives in 4D. Thus, there's not a "simple" way to visualize this. Nonetheless, there are a number of standard tricks to help us visualize the function. Perhaps the most standard is to illustrate the effect that the function has on some standard domain or set of domains. If you type "plot 1/z" into WolframAlpha, for example, you'll see a number of images related to the function, one of which is labeled "Complex map" and looks like so: This illustrates how a grid of lines throughout the square $[-1,1]\times[-1,1]$ is affected by the map $z\rightarrow 1/z$. This is a bit like trying to understand t

Cites Sam Nead · 2014-08 · Mathematics Stack Exchange · on “Is it possible to build a metal wire sphere whose shadow projects the 2d conformal maps of the Riemann Sphere on a flat wall?”

This can be done. For example, see [LINK] The technical name for this way of "casting shadows" is called stereographic projection. See [this video] [LINK] [LINK] [LINK] [LINK] and so on.

Cites anon · 2016-09 · Mathematics Stack Exchange · on “Of rotations, $S^2$, the Riemann sphere, $\text{SU}(2)$, $\text{SO}(3)$, and the precise relationship between two maps.”

The fact that using matrices for fractional linear transformations creates a group action (turns matrix multiplication into function composition) may seem like divine inspiration if projective spaces are not mentioned to motivate the situation. Given a vector space $V$, there is an associated projective space $\mathbb{P}^1(V)$ which collects together all of the "lines," i.e. one-dimensional subspaces. When $V=\mathbb{R}^n$ or $\mathbb{C}^n$ we write $\mathbb{RP}^{n-1}$ or $\mathbb{CP}^{n-1}$. In the case that $n=2$, we have a complex projective line $\mathbb{CP}^1$. We may put an equivalence relation on $\mathbb{C}^2\setminus0$, saying $v\sim w$ whenever $v$ and $w$ generate the same complex