Riemann sphere Puns

Hi I searched for the riemann sphere and it's relation to dividing by zero but I found nothing so if anyone have like a video or something it'll be appreciated

I really do not like the standard definition of the Riemann Sphere, given to students in a first class on complex analysis. I was wondering if people agree with me.

By "Riemann Sphere," I mean the same thing as "the extended complex plane," i.e. "C union {infinity}." Since Ahlfors is one of the most popular books for a first course in complex analysis, I will make reference to that in order to explain the problems I have with the definition.

One of the main reasons I do not like this definition is the way that one thinks about whether a function is analytic at infinity. From Ahlfors (given a function f:C->C) "Since f(infinity) is not defined, we treat infinity as an isolated singularity, and by convention it has the same character of removable singularity, pole, or essential singularity as the singularity of g(z) = f(l/z) at z = 0. " Here Ahlfors is trying to hide the fact that he is basically working on a manifold, and calls something that could seem relatively arbitrary a "convention." And the definition for a function being analytic at infinity looks very different from it being analytic at other places, despite the fact that the Riemann Sphere is very symmetric .

Another thing I do not like is the way that meromorphic functions are defined. Ahlfors defines a function to be meromorphic if it is analytic in an open set, except for poles. And he defines there to be a pole at a if the limit as the function approaches a is infinity. But the function approaching infinity near a is a priori a different infnity from the infinity of the Riemann Sphere. But then things are defined to make these the same. It seems to me like the wrong way of doing things.

The last major thing I do not like is the way mobious/linear fractional transformations are done. The first thing with this is that it feels very weird to just set f(-d/c)=infinity and f(infinity)=a/c when f(z)=(az+b)/(cz+d). Earlier, Ahlfors defines b+infinity=infinity and b*infinity=infinity (for nonzero b), but this just seems weird, extending arithmetic to infinity (which happens to make + and * into partial functions on the Riemann Sphere). The rules do not seem un-intuitive relative to ones prior intuition on infinity, but the fact that these extensions of the function f keep f a continuous map is not obvious. And the other thing with these functions is that they form a group action from GL_2. This seems completely out of nowhere, when the action is very fundamental.

The way I think that the

Hi everyone,

I just started out with manim, and one of the things I want to try is to make some animations of isometries of the hyperbolic plane and the Riemann sphere. I'm having some trouble getting started with this, and can't seem to find documentation/previous threads to help me.

With the hyperbolic plane, what I tried so far is using ApplyComplexFunction on NumberPlane(), but for some reason this doesn't work out (as I said, I only just started out, so there's probably an obvious reason for this). See, for instance, an example of applying z^2 which seems to not work properly (for instance - it moves the origin even though it's a fixed point. I added the code at the end of the post).

https://reddit.com/link/hp921q/video/er8rl4dmz7a51/player

Does anyone have an idea on what to try next? What can I use to model the upper half plane? About the Riemann sphere I'm even more clueless - I understand I can probably apply a function to a sphere, but it wouldn't really behave under the transformation as the Riemann sphere would.

I looked in the old 3b1b files and found a file name holomorphic.py. I tried running it but it didn't work, but by looking at the code I see that there's something called ComplexTranformationScene which could possibly be good for me (at least for the upper half plane), but I couldn't find documentation of it so I could read more about it (and I couldn't understand too much by myself from the code). Has anyone been able to use the holomorphic.py file/knows of this type of scene and can point me to where I can read more about it?

Thanks a lot in advance!

PS - the code I used for the animation:

class myscene(Scene):
	def construct(self):
		plane=NumberPlane()
		self.play(ShowCreation(plane))
		self.wait(1)
		self.play(ApplyComplexFunction(lambda z: z**2, plane), run_time=3)
		self.wait(1)

Someone mentioned this to me recently, and while I’ve been able wrap my head around some pretty complex ideas, I don’t understand at all how this can work.

(This question is also asked on Math Stack Exchange.)

My ultimate goal is to see how the point of infinity and an arbitrary transform in Riemann sphere can lead to what consequences in dynamical systems, and it seems that harmonic analysis plays a crucial role in between since it connects Fourier transform and spherical harmonics, Hilbert space and functional analysis, topology, group, graph and representation theory in one place. In the term of complex systems theory, this could be the super high-degree node.

What road map should I take? The suggested related topics are (1) Fourier transform of distributions, (2) spectral theorem (spectrum of the Laplacian), (3) harmonic analysis and (4) representations of locally compact groups. I have finished complex analysis (Needham), dynamical systems theory (Strogatz), and the first 6 chapters of Kreyszig's on functional analysis. I have some choices starting from here:

• Continue reading functional analysis:

• Keep reading Kreyszig. The rest of the book dedicates to spectral theory and I'm happy to keep reading, but since there are many other things to learn is this the optimized path?

• Switch to Rudin's book. It dedicates two parts specifically for (1) and (2), but it seems to mostly use Banach spaces?

• Switch to harmonic analysis:

• Dym & McKean, Fourier Series and Integrals. But it's too focused on Fourier analysis? It's also pretty old (about 50 years).

• Folland, A Course in Abstract Harmonic Analysis. Seem ideal, but is it too soon to read it now? And it doesn't talk about distributions.

• Related questions on harmonic analysis: Undergrad level: Math SE; grad level: Math Overflow, Reddit. Also What's a good primer from linear algebra to spherical harmonics?

• My another question on Physics Stack Exchange: How would behaviors of Riemann spheres represent characteristics of physical dynamical systems?

That really stretches my intuition. I am a recreational mathematician and I was wondering if there were better explainations than the one afforded by the wiki page.

The Riemann sphere warps distance on a plane, but leaves direction alone. Then it can represent the entire complex plane and infinity.

Can I assume that if yaw is untouched when representing a plane, then all rotations are untouched when representing a volume? Does that also mean that angles at intersections are preserved the way they are on the regular Riemann sphere?

As I always state, the question I ask is out of self study and not a formal class. Thus, technically, I'm not cheating.

I have reached this question:

> Prove that every circle in the extended complex plane is the image of some circle in S^2 under the stereographic projection φ.

I know that I can write a circle as

[; ( x - x_0 )^2 + ( y - y_0 )^2 = r^2 ;]

I figure it will be easier to start with a circle around the origin and then add the translation in the plane.

[; p^2 + q^2 = r^2 ;]
[; (2p/(1+r^2 ), 2q/(1+r^2 ), (r^2 - 1)/(r^2 + 1)) ;]

I can see how, setting r = 1, I get a circle, but why would this be a circle in the general case?

Comment: I can manage to activate LaTeX plugin under chromium.

In the fuction 1/x, as x approaches zero from the postive direction, y approaches infinity, and as x approaches 0 from the negative direction y approaches negative infinity

Imagine a triangle, it has 3 sides, imagine a perpendicular bisector for each side, the bisectors meet at one point, the distance between that point and any of the vertices of the triangle is the same, now, let's say of the angles of the triangle is obtuse, the point of intersection gets farther as the angle approaches 180°, when it hits 180°, the triangle become a line, and the bisectors never meet, but if it increases after 180°, the point of intersection comes back, from the other direction, it goes to infinity and comes back from negative infinity

https://youtu.be/zEGsq7H5egE

This a video about negative mass, go to 9:35

As the mass of the object aproaches infinity, postive and negative masses start to act the same

So, what, is infinity the same as negative infinity? I know this isn't a proof of anything, but i don't understand this phenomenon

Is it a number circle instead of a number line?

I am so confused

For example, I have recently learned about the existence of some group called the Grothendieck-Teichmuller group, where it is an open question whether or not it is isomorphic to the absolute Galois group of Q.

Does anyone know of any other examples of this?

So I am in a Topology class, and we will have to do a project for part of our grade later in the semester. We have a variety of topics to choose from, but we are allowed to potentially choose our own as well, with the instructor's permission. I am interested in learning a bit of Algebraic Geometry, and I am aware of the Zariski topology (though I haven't studied it at all) but I do not feel equipped to determine whether or not it would be a rich enough topic to do a major project on. Can anyone tell me if there are any applications of topology to AG (Zariski or otherwise) that might be accessible to an undergrad and provide a non-trivial amount of content to cover?

To mark the 414th episode of Omnibus (because I missed the round number of 400 and won't wait until 500), here's every speculation by Ken and John on the podcast about what the Futurelings will look like, smell like, be like, etc. This is from a notes file on my phone, so maybe I've missed some. Episodes that aren't listed contain no speculations.

(I shared a previous versions of this, but the posts are now archived so can't update it or comment)

Intro episode

Cockroaches, presumably.

European Starling

Starlings are the only bird left in North America, replacing the Eagle as the symbol of the country.

Live in a community of pacifism and pure energy... who have better taste than current people and revere the Long Winters, and use his track as a National Anthem.

Maybe half goat half person that like peeing on themselves.

Defenestration

Futurelings live in a world called Battlefield Earth.

Insectile claw into record grooves to listen to Omnibus.

Marathon of 1904

Giant. Megafauna.

Blocks of cheese, or maybe Ents?

The Pig War

All living in utopian Cascadia.

No Facebook... and no faces either.

Smell-o-vision

Giant moles, smelling the podcast, not hearing it.

The Rachel

May be wearing a haircut called the Ken Jennings.

Tesseract

Live on potato starch... worship the potato.

Water Wars

3.5 feet tall horned toads that lick moisture off rocks.

Listening thousands of feet underground at the water table.

Gadsby

Live in hives.

Eat massive 12,000 pound ducks that last the entire futureling’s life.

Click mandibles to write letters.

Secret Order of the Double Sunrise

Listening from Singapore which has a billion people, part of a global hive.

Have gossamer dragonfly wings due to evolvolution.

Houston and New York are part of the same megalopolis.

They all have an AI John Roderick in their treehouses.

Darien Gap

Sentient fish who live underwater and look at submerged trees.

Heil Honey I’m Home!

No such thing as race in futurelings.

All rooms only have three walls due to the main architectural influence being family sitcoms.

Monrovia

Old futurelings watch Jeopardy hosted by Alex Trebec’s head in a jar of nutrients.

Might all be part of the Canadian Hegemony.

Mandibles clicking in excitement.

Live in 55,000 small nation states.

Kohoutek

Giant Cockroaches again

Twitter became SkyNet and Omnibus is recorded for it.

Sentinelese

All left handed, or left mandible

In the show Gravity Falls, they reveal a die with an infinite number of sides.

How would this work when you roll it? Since there are an infinite number of sides, would there ever be an outcome? Can you calculate the probability of said roll? Other thoughts?

Thanks!

https://gravityfalls.fandom.com/wiki/Infinity_sided_dice

Do your worst!

It really does, I swear!

For context I'm a Refuse Driver (Garbage man) & today I was on food waste. After I'd tipped I was checking the wagon for any defects when I spotted a lone pea balanced on the lifts.

I said "hey look, an escaPEA"

No one near me but it didn't half make me laugh for a good hour or so!

Edit: I can't believe how much this has blown up. Thank you everyone I've had a blast reading through the replies 😂

Hey Kurt,

the world is filled with cowards,

tis a bitter truth, they say,

but what is cowardice?

i would like to know,

and what is prudence?

where do the two merge,

where do they diverge,

that subtle question affronts us;

but alas, we are mostly unaware,

as layer upon layer of emotion,

leads us here and there,

alas, we know not where,

because the larger picture,

even if one is there,

is hidden under these layers,

as abstract as prayers,

and as precise as well;

sometimes it coerces,

somewhere it disrupts,

but all these are concepts,

that we know not off,

and so we believe,

whatever catches our fancy,

out of whatever is prevailant in society,

"and so it goes......",

as your time traveler once said,

the world moves on,

and we with it;

but the gift of prophecy!

is that not the gift of life?

to predict the chaos,

and to hold back entropy,

that is what the mind is it seems,

a pinnacle of balanced chaos,

a true anti dark-side force,

real life of-course,

that some call a life force,

others a universe;

a weird place where infinities collide,

and merge from within the abstract,

to form mayhaps a mandelbrot set,

within a Riemann sphere of possibilities;

whatever the fuck this is,

it does feel good to feel,

and for a dork like Dada,

that is enough;

A. Z. Dada •°•°•°•°•°•°•°•°•°•°•°•°•°•°•°

https://www.reddit.com/r/OCPoetry/comments/prrl5e/thoughts_from_a_professional_poet/hgbqw8b?utm_medium=android_app&utm_source=share&context=3

https://www.reddit.com/r/OCPoetry/comments/q5zxka/i_married_the_same_man_twice/hgbt4vy?utm_medium=android_app&utm_source=share&context=3

They’re on standbi

Buenosdillas

Pilot on me!!

Dad jokes are supposed to be jokes you can tell a kid and they will understand it and find it funny.

This sub is mostly just NSFW puns now.

If it needs a NSFW tag it's not a dad joke. There should just be a NSFW puns subreddit for that.

Edit* I'm not replying any longer and turning off notifications but to all those that say "no one cares", there sure are a lot of you arguing about it. Maybe I'm wrong but you people don't need to be rude about it. If you really don't care, don't comment.

When I got home, they were still there.

What did 0 say to 8 ?

" Nice Belt "

So What did 3 say to 8 ?

" Hey, you two stop making out "

I won't be doing that today!

You take away their little brooms