A couple of people have been commenting on the striking resemblance of the orbits to field lines. There's some nice remarks to make about this, which I'll put in a comment thread below.
Transverse sets of lines: In the context of orbits, the rotation orbits are transverse to the boost orbits. In the context of EM, you can consider equipotentials (contours of constant potential). These are transverse to the field lines, which are tangent to the gradient of the potential. 1/n
Then what is the right potential to take? We can consider the potential to be the difference of central potentials centred at different points. Given a central potential V_c(r) for charge q = 1 and radius r, then if the positive charge is located at C and the negative at D, our potential is V(X) = V_c(|C - X|) - V_c(|D - X|). 2/n
There's two reasonable choices of central potential. The most obvious (at least to me) is V_c(r) = 1/r (constants chosen to make this expression as simple as possible). The other is V_c(r) = log(r). The first corresponds to a point charge in 3D space. The second corresponds to a line charge. 3/n
The second choice is the correct one, and the equipotentials are known as the Apollonian circles. So what's left to show is that the x-orbits get projected onto Apollonian circles by stereographic projection. 4/n
To me, this isn't obvious without a calculation. There is at least one thing we can check, which is that the orbits remain circles after stereographically projecting. In fact, generally circles on the sphere get mapped to circles under the projection. One way to show this is using the result that Möbius maps preserve circles, and then using Möbius maps as transformations on the sphere to rotate your circle so that it obviously projects to a circle. 5/n, n = 5 for now
Beautiful video! I've always thought this is a (relatively) easily accessible topic that isn't explained clearly anywhere -- I was even considering making something on it so I'm very glad that you did
thx so much, videos like these really help me get a tiny hunch on topics related to stuff I find fascinating... They serve as beacons in the dark sea of expertise which i have no way of catching up to alone... thank you
Lovely, you have earned a new subscriber :) I have started studying twistor theory and spinors recently and these videos are great and I hope to see more on related topics!
wait a sec... Im in highschool so bare with my ignorance but... 7:58 look like like em. force lines on a magnet?? Dudes I never knew this connection, I had no idea!! Thanks for the vid!! Please keep popularizing science!
It is not exactly the same lines for point charges in 3d, those are not circles, but it does work in 2d with opposing pairs of infinite wires of opposite charge and/or current.
@@jeanf6295 it got me thinking feild lines would be a 3d projection of a higher dimension sphere.. But I don't have the math skills too even go near that idea. I don't even really understand this video, try as I might.
Electro is dual to magnetic -- photons or pure energy. Space is dual to time -- Einstein. Symmetry is dual to conservation -- the duality of Noether's theorem. Duality is a symmetry and it is being conserved according to Noether's theorem. AdS is dual to CFT. The conservation of duality (energy) will be known as the 5th law of thermodynamics -- Generalized Duality. Antipodal points identify for the rotation group SO(3) -- Stereographic projection. "Always two there are" -- Yoda.
When you talk about transformations of the celestial sphere, would it be correct to identify that with vision itself? In other words, is the sequence of successive of light cones (celestial spheres) coming into our eyes EXCLUSIVELY Möbius transformations, of a spherical family of light rays/photons? Penrose starts his twistor lectures by giving the example of imagining oneself in deep space, and states the way the stars move is conformal as you move around. What I'm not clear on is whether this conformal mapping of light is true only at high/relativistic speeds without gravity/mass around, or whether it's a general truth about vision no matter where you are/what you're seeing.
This is an absolutely beautiful presentation of diff geo. I think animations like these are really useful in this branch of math/phys but are sorely underused so this is well needed! Any chance you take do anything with Clifford algebras in general? Can't wait to see more so please don't stop when SoME ends!
Thanks! I don't have anything in mind for Clifford algebras in general so far unfortunately. The main reason is that I don't have a good 'visual' understanding of how Clifford algebras act. I do have more planned though! Including at least one on conformal symmetry :)
Hey, thanks for the kind comment! I make my animations using the community edition of Manim, a free use python library based on a library developed by 3b1b.
The first thing the image at 8:00 reminded me of was a dipole field. I wonder if there’s a way to map between dipoles and stereo graphic projections of the Riemann sphere
Man, those orbits for the Mobius maps reek of electromagnetism - they look just like dipolar field lines! I wonder if a similar object, like S^3, might have orbits that resemble other physical phenomena
Electro is dual to magnetic -- photons. Space is dual to time -- Einstein. Symmetry is dual to conservation -- the duality of Noether's theorem. Duality is a symmetry and it is being conserved according to Noether's theorem. AdS is dual to CFT. The conservation of duality (energy) will be known as the 5th law of thermodynamics -- Generalized Duality. Antipodal points identify for the rotation group SO(3) -- Stereographic projection. "Always two there are" -- Yoda.
@@hyperduality2838 I'm not really sure what you're trying to say with all that, but there are some issues I'd like to bring your attention to. Noether's theorem is about conservation laws that arise from *continuous* symmetries, not discrete ones. Duality is a discrete relationship, and it isn't a physical quantity that can be conserved - the concept just doesn't apply here because it's a relationship between different physical quantities. Also we don't live in an anti-deSitter universe, and the 1st law of thermodynamics already says energy is conserved.
@@nzuckman Continuous (classical) is dual to discrete (quantum). Classical reality is dual to quantum reality synthesizes true reality -- Roger Penrose using the Hegelian dialectic. Duality is a continuum between two opposite or opposame absolutes. You make some valid points but waves can become particles in physics and vice versa. Waves are dual to particles -- quantum duality. Enantiodromia is the unconscious opposite or opposame (duality) -- Carl Jung. Concepts are dual to percepts -- the mind duality of Immanuel Kant. Mathematicians create new concepts or ideas all the time from their perceptions, observations or measurements (intuitions) -- a syntropic process, teleological. Converting perceptions into ideas or conceptions is therefore a continuous process -- thinking. I would argue that thinking is an ongoing or continuous process. Energy is duality, duality is energy. Potential energy is dual to kinetic energy -- gravitational energy is dual. Converting potential energy into kinetic energy is a continuous process and not discrete. Apples fall to the ground because they are conserving duality (energy). The conservation of duality (energy) will be known as the 5th law of thermodynamics -- Generalized Duality. Action is dual to reaction -- Sir Isaac Newton or forces are dual. Attraction (sympathy) is dual to repulsion (anti-pathy), stretch is dual to squeeze, push is dual to pull -- all forces are dual. If forces are dual then energy must be dual:- Energy = force * distance -- simple physics. Energy is duality in physics -- the 5th law of thermodynamics! "May the force (duality) be with you" -- Jedi teaching. "The force (duality) is strong in this one" -- Jedi teaching. There are new laws of physics which you may not be aware of, but you are asking the right type of questions! Questions are dual to answers. Space is dual to time -- Einstein. Energy is dual to mass -- Einstein. Space translation is dual to time translation -- Noether' theorem. Moving through space and time are continuous processes are they not.
Can't believe you have done such a magnificent job. I am deeply impressed by your work, sir. Especially I really want implement this video, torus twist and stars in the sky you posted on your websites by myself. May I take a look at the code please?
5:00 Actually, the North and South poles would be the same point, no? Mapping a point on the sphere to the infinite plane, and the plane cutting through the equator of the sphere, ends up mapping the North and South poles to the same "points" at infinity, making the North pole and the South pole the same point.
Not quite. The North Pole is at (0,0,1), and the South at (0,0,-1). The ray that joins these two hits z = 0 at (0,0,0). Or verify using the explicit formula. It's a good exercise to try and show the formula in order to get a handle of the geometry.
Space is dual to time -- Einstein. Symmetry is dual to conservation -- the duality of Noether's theorem. Duality is a symmetry and it is being conserved according to Noether's theorem. AdS is dual to CFT. The conservation of duality (energy) will be known as the 5th law of thermodynamics -- Generalized Duality. Antipodal points identify for the rotation group SO(3) -- Stereographic projection. "Always two there are" -- Yoda.
@@execute6200 i remember that time in the movie where Morbius transformed and said "it's mapping time" and mapped everyone truly the most map of all time
Space is dual to time -- Einstein. Space/time symmetries are dual to mobius maps. AdS is dual to CFT. Time dilation is dual to length contraction -- Einstein, special relativity. "Always two there are" -- Yoda.
There's no immediate conserved quantity as inversion is a discrete, rather than continuous symmetry, so Noether's theorem doesn't apply here. But inversion (as in z -> 1/z) can still be interpreted as a very interesting transformation from the point of view of string theory. The way to do this is to consider the trajectory of a closed string (a loop) through spacetime. It sweeps out a cylinder, the worldsheet. One can always pick a 'good set' of coordinates where the worldsheet can be viewed as the complex cylinder C* = C\{0}, where increasing time corresponds with increasing radius, while the periodic spacial coordinate is the argument of the complex number. Then inversion becomes the time reversal operator, so conserved quantities are those invariant under time reversal!
Space is dual to time -- Einstein. Symmetry is dual to conservation -- the duality of Noether's theorem. Duality is a symmetry and it is being conserved according to Noether's theorem. AdS is dual to CFT. The conservation of duality (energy) will be known as the 5th law of thermodynamics -- Generalized Duality. Antipodal points identify for the rotation group SO(3) -- Stereographic projection. "Always two there are" -- Yoda.
A couple of people have been commenting on the striking resemblance of the orbits to field lines. There's some nice remarks to make about this, which I'll put in a comment thread below.
Transverse sets of lines: In the context of orbits, the rotation orbits are transverse to the boost orbits. In the context of EM, you can consider equipotentials (contours of constant potential). These are transverse to the field lines, which are tangent to the gradient of the potential. 1/n
Then what is the right potential to take? We can consider the potential to be the difference of central potentials centred at different points.
Given a central potential V_c(r) for charge q = 1 and radius r, then if the positive charge is located at C and the negative at D, our potential is
V(X) = V_c(|C - X|) - V_c(|D - X|). 2/n
There's two reasonable choices of central potential. The most obvious (at least to me) is V_c(r) = 1/r (constants chosen to make this expression as simple as possible). The other is V_c(r) = log(r). The first corresponds to a point charge in 3D space. The second corresponds to a line charge. 3/n
The second choice is the correct one, and the equipotentials are known as the Apollonian circles. So what's left to show is that the x-orbits get projected onto Apollonian circles by stereographic projection. 4/n
To me, this isn't obvious without a calculation. There is at least one thing we can check, which is that the orbits remain circles after stereographically projecting. In fact, generally circles on the sphere get mapped to circles under the projection. One way to show this is using the result that Möbius maps preserve circles, and then using Möbius maps as transformations on the sphere to rotate your circle so that it obviously projects to a circle. 5/n, n = 5 for now
Thank you so much for your submission! (final video announcement with winners / runner ups out now, by the way)
my favorite part about the Möbius Map is when Möbius möbs into Möbius and said "It's Möbin' Time!"
Underrated comment
thinked that might be a non-zero probability of a comment like under this video.
And in fact it is
From what I've seen, it's valid to write ö as oe, and let me tell you, I've seen a lot of Moebin' Time this past month.
This was exactly, what I was looking for. Thx
super presentation man, please keep posting more...
Awesome video! Excited to see more, especially on CFT!
Beautiful video! I've always thought this is a (relatively) easily accessible topic that isn't explained clearly anywhere -- I was even considering making something on it so I'm very glad that you did
Glad to see Möbius in the context of transformations instead of the strip.
thx so much, videos like these really help me get a tiny hunch on topics related to stuff I find fascinating... They serve as beacons in the dark sea of expertise which i have no way of catching up to alone... thank you
My new favorite channel!
Oh snap! Can't wait to learn about these mobius maps!
Lovely, you have earned a new subscriber :) I have started studying twistor theory and spinors recently and these videos are great and I hope to see more on related topics!
wait a sec... Im in highschool so bare with my ignorance but... 7:58 look like like em. force lines on a magnet?? Dudes I never knew this connection, I had no idea!! Thanks for the vid!! Please keep popularizing science!
I hope there will be a next part!
Also funny how 7:36 and 7:57 look like pairs of electric and magnetic charges and the corresponding field lines.
I noticed this too! I'll have to do some investigating to see how close this relation is
It is not exactly the same lines for point charges in 3d, those are not circles, but it does work in 2d with opposing pairs of infinite wires of opposite charge and/or current.
@@jeanf6295 it got me thinking feild lines would be a 3d projection of a higher dimension sphere..
But I don't have the math skills too even go near that idea. I don't even really understand this video, try as I might.
Electro is dual to magnetic -- photons or pure energy.
Space is dual to time -- Einstein.
Symmetry is dual to conservation -- the duality of Noether's theorem.
Duality is a symmetry and it is being conserved according to Noether's theorem.
AdS is dual to CFT.
The conservation of duality (energy) will be known as the 5th law of thermodynamics -- Generalized Duality.
Antipodal points identify for the rotation group SO(3) -- Stereographic projection.
"Always two there are" -- Yoda.
When you talk about transformations of the celestial sphere, would it be correct to identify that with vision itself? In other words, is the sequence of successive of light cones (celestial spheres) coming into our eyes EXCLUSIVELY Möbius transformations, of a spherical family of light rays/photons? Penrose starts his twistor lectures by giving the example of imagining oneself in deep space, and states the way the stars move is conformal as you move around. What I'm not clear on is whether this conformal mapping of light is true only at high/relativistic speeds without gravity/mass around, or whether it's a general truth about vision no matter where you are/what you're seeing.
just seen the CFT video, cool visual
Reaaly great video, hope you will continue like this ! ;) And am I the only one to see electric and magnetic field lines at 7:30 and 7:52 ??? oO
This is an absolutely beautiful presentation of diff geo. I think animations like these are really useful in this branch of math/phys but are sorely underused so this is well needed! Any chance you take do anything with Clifford algebras in general? Can't wait to see more so please don't stop when SoME ends!
Thanks! I don't have anything in mind for Clifford algebras in general so far unfortunately. The main reason is that I don't have a good 'visual' understanding of how Clifford algebras act.
I do have more planned though! Including at least one on conformal symmetry :)
Subbed, really enjoyed your vids, well presented and easy to follow
This is an absolutely awesome video! Thanks for sharing your knowledge in such a digestible way. May I ask, how do you do those sleek animations?
Hey, thanks for the kind comment! I make my animations using the community edition of Manim, a free use python library based on a library developed by 3b1b.
Is there a way I can personally message you? Great job.
The first thing the image at 8:00 reminded me of was a dipole field. I wonder if there’s a way to map between dipoles and stereo graphic projections of the Riemann sphere
Yeah, that's right! I've put some explanation of this in a comment thread which I've pinned.
@@OnePlusOneSpace Oh amazing, I'll have a read!
Man, those orbits for the Mobius maps reek of electromagnetism - they look just like dipolar field lines! I wonder if a similar object, like S^3, might have orbits that resemble other physical phenomena
Electro is dual to magnetic -- photons.
Space is dual to time -- Einstein.
Symmetry is dual to conservation -- the duality of Noether's theorem.
Duality is a symmetry and it is being conserved according to Noether's theorem.
AdS is dual to CFT.
The conservation of duality (energy) will be known as the 5th law of thermodynamics -- Generalized Duality.
Antipodal points identify for the rotation group SO(3) -- Stereographic projection.
"Always two there are" -- Yoda.
@@hyperduality2838 I'm not really sure what you're trying to say with all that, but there are some issues I'd like to bring your attention to. Noether's theorem is about conservation laws that arise from *continuous* symmetries, not discrete ones. Duality is a discrete relationship, and it isn't a physical quantity that can be conserved - the concept just doesn't apply here because it's a relationship between different physical quantities. Also we don't live in an anti-deSitter universe, and the 1st law of thermodynamics already says energy is conserved.
@@nzuckman Continuous (classical) is dual to discrete (quantum).
Classical reality is dual to quantum reality synthesizes true reality -- Roger Penrose using the Hegelian dialectic.
Duality is a continuum between two opposite or opposame absolutes.
You make some valid points but waves can become particles in physics and vice versa.
Waves are dual to particles -- quantum duality.
Enantiodromia is the unconscious opposite or opposame (duality) -- Carl Jung.
Concepts are dual to percepts -- the mind duality of Immanuel Kant.
Mathematicians create new concepts or ideas all the time from their perceptions, observations or measurements (intuitions) -- a syntropic process, teleological.
Converting perceptions into ideas or conceptions is therefore a continuous process -- thinking.
I would argue that thinking is an ongoing or continuous process.
Energy is duality, duality is energy.
Potential energy is dual to kinetic energy -- gravitational energy is dual.
Converting potential energy into kinetic energy is a continuous process and not discrete.
Apples fall to the ground because they are conserving duality (energy).
The conservation of duality (energy) will be known as the 5th law of thermodynamics -- Generalized Duality.
Action is dual to reaction -- Sir Isaac Newton or forces are dual.
Attraction (sympathy) is dual to repulsion (anti-pathy), stretch is dual to squeeze, push is dual to pull -- all forces are dual.
If forces are dual then energy must be dual:-
Energy = force * distance -- simple physics.
Energy is duality in physics -- the 5th law of thermodynamics!
"May the force (duality) be with you" -- Jedi teaching.
"The force (duality) is strong in this one" -- Jedi teaching.
There are new laws of physics which you may not be aware of, but you are asking the right type of questions!
Questions are dual to answers.
Space is dual to time -- Einstein.
Energy is dual to mass -- Einstein.
Space translation is dual to time translation -- Noether' theorem.
Moving through space and time are continuous processes are they not.
Thank you!
Can't believe you have done such a magnificent job. I am deeply impressed by your work, sir. Especially I really want implement this video, torus twist and stars in the sky you posted on your websites by myself. May I take a look at the code please?
very cool!
Awesome visuals
What do you use for your graphing animations. It's really clean and looks awesome
The animations are all done in manim CE (community edition)
And then math said "It's morbin time"
5:00 Actually, the North and South poles would be the same point, no? Mapping a point on the sphere to the infinite plane, and the plane cutting through the equator of the sphere, ends up mapping the North and South poles to the same "points" at infinity, making the North pole and the South pole the same point.
Not quite. The North Pole is at (0,0,1), and the South at (0,0,-1). The ray that joins these two hits z = 0 at (0,0,0). Or verify using the explicit formula. It's a good exercise to try and show the formula in order to get a handle of the geometry.
Space is dual to time -- Einstein.
Symmetry is dual to conservation -- the duality of Noether's theorem.
Duality is a symmetry and it is being conserved according to Noether's theorem.
AdS is dual to CFT.
The conservation of duality (energy) will be known as the 5th law of thermodynamics -- Generalized Duality.
Antipodal points identify for the rotation group SO(3) -- Stereographic projection.
"Always two there are" -- Yoda.
Amazing!
Awesome video!
morbius map?????
It's mapping time
@@execute6200 i remember that time in the movie where Morbius transformed and said "it's mapping time" and mapped everyone
truly the most map of all time
OHHHH, THE space-time IS GOING TO MORB
Space is dual to time -- Einstein.
Space/time symmetries are dual to mobius maps.
AdS is dual to CFT.
Time dilation is dual to length contraction -- Einstein, special relativity.
"Always two there are" -- Yoda.
Is there a conserved quantity associated with the inverse transformation?
There's no immediate conserved quantity as inversion is a discrete, rather than continuous symmetry, so Noether's theorem doesn't apply here.
But inversion (as in z -> 1/z) can still be interpreted as a very interesting transformation from the point of view of string theory. The way to do this is to consider the trajectory of a closed string (a loop) through spacetime. It sweeps out a cylinder, the worldsheet. One can always pick a 'good set' of coordinates where the worldsheet can be viewed as the complex cylinder C* = C\{0}, where increasing time corresponds with increasing radius, while the periodic spacial coordinate is the argument of the complex number.
Then inversion becomes the time reversal operator, so conserved quantities are those invariant under time reversal!
Space is dual to time -- Einstein.
Symmetry is dual to conservation -- the duality of Noether's theorem.
Duality is a symmetry and it is being conserved according to Noether's theorem.
AdS is dual to CFT.
The conservation of duality (energy) will be known as the 5th law of thermodynamics -- Generalized Duality.
Antipodal points identify for the rotation group SO(3) -- Stereographic projection.
"Always two there are" -- Yoda.
@@OnePlusOneSpace Continuous (classical) is dual to discrete (quantum) -- physics.
You are conserving duality.
morbius maps??!?!?!?!??!!
2:21 the plit
ratio jumpscare 3:57
az + b + get boosted
1:14 ofthe corect pne
/pane?
4dlate
Morbuis