Finally someone explaining the 1-z part! I was pulling my hair out because I was looking at the wrong similar triangles and it's hard to look at something differently after your brain already decided how to see it. Your video was the only one after about a dozen that explained it! And every source online said things like "easily shown by similar triangles" or didn't explain it at all! Thank you!
Btw, while it doesn't bother me at all, isn't using the complex plane unnecessary and potentially adding a point of confusion for some viewers? I think stereographic projections can be done without reference to the complex plane, though complex functions etc are of course an amazing application of stereographic projection!
These videos have more fundamental topology without drowning the learner. A unique pedagogical presentation and one that encourages understanding the underlying topology. I feel that is a worthy approach! Thank you 😊
this video is amazing! I am studying complex anlaysis and every video on this topic is either too detailed or just animation without the actual calculations. Your video showed how the animation related to the calculation to make it really sink in
Why are 1's in these formulas? i.e. I'm not clear why the formulas all involve either +1 or -1 in some fashion. Why 1 and not some other number? I'm picturing light shining through a transparent sphere, and I wouldn't have thought something as clearcut as a simple factor of 1 would be at play, I would have thought it'd be some complicated mess involving pi or something. Thanks I appreciate it
Sure, the 1s come from the radius of the unit sphere. Around 1:06, the vertical axis is has a length of 1 unit since it reaches from the center of the sphere (the origin) to the North Pole N (top of the sphere). Next, P’s coordinate with respect to this axis is z, so the length of the vertical blue segment is the difference 1-z. In other words, the total colored part of the vertical axis has length 1, the orange part has length z, so the blue part has length 1-z.
Maybe…I learned the basics of homology for simplicity complexes first, then the basics of sheaf cohomology for schemes. But neither is something I ended up using beyond those classes, so (1) I don’t remember much, and (2) an algebraic topologist would have a better answer :)
@@ayzikdig1983 broadly commutative algebra, specifically whether certain properties of a ring ascend to certain extensions of that ring, and vice versa if certain properties an extension has would descend to the base ring. Do you have an area that you enjoy most yet?
Finally someone explaining the 1-z part! I was pulling my hair out because I was looking at the wrong similar triangles and it's hard to look at something differently after your brain already decided how to see it. Your video was the only one after about a dozen that explained it! And every source online said things like "easily shown by similar triangles" or didn't explain it at all! Thank you!
Btw, while it doesn't bother me at all, isn't using the complex plane unnecessary and potentially adding a point of confusion for some viewers? I think stereographic projections can be done without reference to the complex plane, though complex functions etc are of course an amazing application of stereographic projection!
Glad it was helpful!
Yeah you could do it without complex, it was just something from the complex variables course I teach, so I had that audience in mind.
Finally someone who can explain it simply !! thank you good sir !!
Glad it was helpful!
Excellent explanation. Clear, quick, simply clever.
Thank you, glad it was helpful!
These videos have more fundamental topology without drowning the learner. A unique pedagogical presentation and one that encourages understanding the underlying topology. I feel that is a worthy approach! Thank you 😊
Excellent excellent excellent. My complex analysis class threw these formulas at us without explaining them at all. Tysm
Glad it was helpful!
I love it, This channel is real gem.
Thank you for your kindness!
this video is amazing! I am studying complex anlaysis and every video on this topic is either too detailed or just animation without the actual calculations. Your video showed how the animation related to the calculation to make it really sink in
Thank you for the great feedback!
Amazing video, the animations and explanation made the derivations feel really intuitive. Well done!
Thank you!
thank you so much for this explanation, im doing my IB extended essay in mathematics and this was a giant help :)
Glad it was helpful!
Why are 1's in these formulas? i.e. I'm not clear why the formulas all involve either +1 or -1 in some fashion. Why 1 and not some other number? I'm picturing light shining through a transparent sphere, and I wouldn't have thought something as clearcut as a simple factor of 1 would be at play, I would have thought it'd be some complicated mess involving pi or something. Thanks I appreciate it
Sure, the 1s come from the radius of the unit sphere. Around 1:06, the vertical axis is has a length of 1 unit since it reaches from the center of the sphere (the origin) to the North Pole N (top of the sphere). Next, P’s coordinate with respect to this axis is z, so the length of the vertical blue segment is the difference 1-z. In other words, the total colored part of the vertical axis has length 1, the orange part has length z, so the blue part has length 1-z.
Best explanation by handsome man 🥰💯
can one study cohomology before homology?
Maybe…I learned the basics of homology for simplicity complexes first, then the basics of sheaf cohomology for schemes. But neither is something I ended up using beyond those classes, so (1) I don’t remember much, and (2) an algebraic topologist would have a better answer :)
@@DrMcCrady what is your expertiy?
@@ayzikdig1983 broadly commutative algebra, specifically whether certain properties of a ring ascend to certain extensions of that ring, and vice versa if certain properties an extension has would descend to the base ring. Do you have an area that you enjoy most yet?
@@DrMcCrady was sure you topology kinda guy
for now i am into algberaic topology