@AllenKnutson Don't give me that "I'm not sure how to do a 6×6 determinant" jazz. I know for a fact that you can do 8×8. (Granted, the matrix was at least half zeroes, if memory serves, and evaluated to 8 factorial, but still impressive.)
@@tomkerruish2982 That wasn't the lie for the video that I had the most trouble mouthing. It was "I thought the complex plane was the same as the real plane" 26:32 . Yuck. Taran carried off the subsequent sigh very well I think.
@@AllenKnutson I listened to it again. That was a very good, heartfelt sigh. Another good piece of acting was when you feigned incredulity about being able to just change the rules in math. Heck, I'm pretty much a Platonist, and even I say that's how you go on a journey of exploration and discovery.
Honestly one of the greatest math videos I've ever seen, everything is explained really clearly in a novel and very effective format. And its real math not over simplified pop math. Please make more videos! This was amazing!
@@AllenKnutson Honestly one of my only problems with this video are that the subtitles are confusing in some parts. I had to re-watch parts of the video more than 5 times to figure out who exactly is saying what. Just using "A" and "T" to distinguish who's saying what doesn't really work very well. You can actually color RUclips subtitles, (I don't know how, though. I've only ever seen it on Tom Scott's videos, so I'd ask him how exactly he does it) so I'd suggest using that rather than your current way of, distinguishing who's speaking.
There's actually a lot of tricks available in this format. E.g. Taran can make some advanced/esoteric point, which in a one-voice video might run the danger of derailing the main thread, and I can go "WhatEVER" and bring the focus back to the central story.
Computer vision algorithms tend to work in RP3. It was fun seeing how the some of the concepts used there can be visualized. I really enjoyed the animations of the antipodal spots and great circles on the sphere.
This video was fucking illuminating for me, i studied projective space in geometry and i didn't have the right imagine of how to think the projective space or projective conics, thank you very much
I love the dialogue format that is taken in this video! It’s very intuitive and answers many questions which a viewer might have while also being extremely engaging
This gave me flashbacks to the distant past, when I used to work on elliptic curves. I think I spent a week or so making graphs of projective elliptic and hyperelliptic curves. It was definitely a week well spent.
The conversation style of teaching math is so useful. This should go up with the other great works like turning a sphere inside out and Donald Knuth's book Surreal Numbers.
Like everyone else here, I'm rating this 10/10. This is the most accessible video I've come across on algebraic and projective geometry. Sadly I'm only an engineer and lack so much mathematical foundation, but this refreshing and intuitive explanation will certainly help me I'm my research :)
Unbelievable video, thanks. As someone with a maths background who is a bit embarrassed I never learned any projective geometry this was really clear and interesting.
Very underrated project. It's a really amazing way to teach students about these topic that are generally difficult for them to grasp at first. Well done
Well hells bells, lads. I understand things now that I didn't understand before by watching this video. Super neat. I'm absolutely going to watch this again with the hopes that it'll happen again.
This video is amazing! I have yet to learn anything about projective/algebraic geometry, and this video got me hooked immediately and completely blew my mind. This is just beautiful mathematics. I also really like the format of the video as a dialogue, it is very relaxing in a sense. Can't wait for this video to blow up
Wow this is really incredible!! and I totally buy and approve all the comparisons with Outside In (of which I was one of the creators..) great explanations of deep math. love it.
This was wayyy up there in terms of mathematics video quality. I just had my second semester mathematics exams. Already looking forward to geometry in the 4th semester. I am seeing a lot of similarities between the snippets from what my friend told me about that course (largely focused on hyperbolic, spherical, / just non Euclidian geometry) and this video.
this really feels like one grad student just explaining a thing to another. I feel like I've been in that discussion a few times lol. also this video feels like reading a textbook chapter and I kinda wish there some exercises
this is a gem, though I do get lost at a few points: 1. 6:37 why there's "got to be" a point passing through itself three times 2. 16:48 the space of answers for exactly what? Curves passing through serveral points, lines tangent to several curves and etc?
1. This is definitely not supposed to be obvious. It can be proven but it's not important for this video 2. The number of conics through five points in particular, but also the space of lines that go through two distinct given points _and_ the space of points that lie on two distinct lines _and_ the space of lines tangent to two circles are all zero-dimensional. Thanks!
This is a very well-done video! Two notes: - You introduced the p_m notation near the start, but never brought it up again. Is this the same as [1,m,0]? In this case, the two points on every circle are p_i and p_-i. - I second the comment someone else made that you should see if you can color-code the subtitles.
The circle thing makes sense given that every scaling and translation of the hyperbola x^2-y^2=1 contains the points p_1 and p_-1. Circles are scalings and translations of the equation x^2-(iy)^2=1.
It's just a name. We're not going to do algebra with it, e.g. try to "multiply" two slopes together. You're worried that you got a hold of the slope as a/b, and in other contexts it's safer to say "that ratio is undefined" than to say "that ratio is infinity". In _this_ context, the reason that people like "infinity" as the name for the vertical slope is that it suggests the right "topology on the space of slopes". Concretely, you should think that just as if we consider lines with slopes 5.1, 5.01, 5.001, ... we'll sneak up on a line with slope 5, if we consider lines with slopes 10, 100, 1000, ... we'll sneak up on a line with slope infinity.
While I definitely saw Outside In nigh 30 years ago ruclips.net/video/wO61D9x6lNY/видео.html I had completely forgotten that it is done as a dialogue. We were more inspired by the flow of actual conversations (between the two of us, and with other people), as I'm sure the Outside In people were.
@22:30, since there's TWO antipodal points at infinity (at the Northeast and Southwest), it's not a Riemann sphere correct? But could perhaps be thought of as two Riemann spheres superimposed over each other, with diametrically opposite points at infinity?
(1) Correct RP^2 is not a sphere (let's skip the "Riemann" adjective). (2) I don't think it's that good an idea to think of it as two spheres S & T superimposed; that would be like saying there's a map from S union T -> RP^2. (Which I suppose there is, but) it makes more sense to say there's a two-to-one map from S -> RP^2, where each spot on S gives the spot pair on RP^2. Bringing in a second sphere T is a weird distraction, at that point.
@@AllenKnutson Thank you. I guess I'm confused because I've been watching some Penrose videos and he often implies that the celestial sphere, which to my mind is just vision itself(?), is a Riemann sphere, but that the Riemann sphere is "not really a metric sphere" (his words). But at the same time, I thought projective space (RP^2) was supposed to be what we actually see around us day to day. Do you have any thoughts on this? It's not clear to me if points or lines at infinity are abstractions or what's really out there in front of us right now... certainly things appear to get smaller as they get further away, and artists need projective geometry to make art appear realistic. Is there a clear delineation between what geometry is abstraction and what is our actual experience?
@@erawanpencil I don't think I know what "RP^2 is supposed to be what we actually see around us" is supposed to mean. For a one-eyed person (for simplicity), who therefore can't judge distance, what they see around them might as well be scaled to all be at distance 1, which puts it on a sphere, not onto an RP^2. The RP^2 is relevant for one-eyed people whose skulls are transparent, and so when they see an object in some direction, they not only can't tell if it's near or far, they also can't tell if it's in front or in back of them.
The traditional quadratic formula applies to quadratic equations in one variable. If we were looking at a general quadratic equation in two variables ax^2 + bxy + cy^2 + dx + ey + f = 0, there'd be no way to factor it. But in the case at 24:20, the equation is the homogenization of a quadratic equation in one variable, so the traditional quadratic formula can be applied.
Hmm, I'm not sure how to explain this in another way. The green line has some slope m, and by definition this means that it has the point p_m on it. The purple line is parallel, so it has the same slope, and therefore also has p_m on it. They share the point p_m in the same way that non-parallel lines share their intersection point.
![Eminem - Temporary (feat. Skylar Grey) [Official Music Video]](http://i.ytimg.com/vi/ZaK9Wi5ho0o/mqdefault.jpg)
s,ubcl,./scraibe
Smash that, like, button
@AllenKnutson Don't give me that "I'm not sure how to do a 6×6 determinant" jazz. I know for a fact that you can do 8×8. (Granted, the matrix was at least half zeroes, if memory serves, and evaluated to 8 factorial, but still impressive.)
@@tomkerruish2982 That wasn't the lie for the video that I had the most trouble mouthing. It was "I thought the complex plane was the same as the real plane" 26:32 . Yuck. Taran carried off the subsequent sigh very well I think.
@@AllenKnutson I listened to it again. That was a very good, heartfelt sigh.
Another good piece of acting was when you feigned incredulity about being able to just change the rules in math. Heck, I'm pretty much a Platonist, and even I say that's how you go on a journey of exploration and discovery.
yoink
Honestly one of the greatest math videos I've ever seen, everything is explained really clearly in a novel and very effective format. And its real math not over simplified pop math. Please make more videos! This was amazing!
Dang with feedback like that we won't be able to help ourselves! Thanks a lot!
@@AllenKnutson Honestly one of my only problems with this video are that the subtitles are confusing in some parts. I had to re-watch parts of the video more than 5 times to figure out who exactly is saying what. Just using "A" and "T" to distinguish who's saying what doesn't really work very well. You can actually color RUclips subtitles, (I don't know how, though. I've only ever seen it on Tom Scott's videos, so I'd ask him how exactly he does it) so I'd suggest using that rather than your current way of, distinguishing who's speaking.
@@nikkiofthevalley In our last video we did it by audio channel. People _really_ didn't like that!
I love how passive-aggressive they sound but then they're like "yeah that's actually cool"
This is done very well. I like how you use a conversation to help motivate every step.
There's actually a lot of tricks available in this format. E.g. Taran can make some advanced/esoteric point, which in a one-voice video might run the danger of derailing the main thread, and I can go "WhatEVER" and bring the focus back to the central story.
Computer vision algorithms tend to work in RP3. It was fun seeing how the some of the concepts used there can be visualized. I really enjoyed the animations of the antipodal spots and great circles on the sphere.
This video was fucking illuminating for me, i studied projective space in geometry and i didn't have the right imagine of how to think the projective space or projective conics, thank you very much
Yay!
This is, without a doubt, my favourite video on this website, period.
Amd I have been here for a long time
DANG
I love the dialogue format that is taken in this video! It’s very intuitive and answers many questions which a viewer might have while also being extremely engaging
This gave me flashbacks to the distant past, when I used to work on elliptic curves. I think I spent a week or so making graphs of projective elliptic and hyperelliptic curves. It was definitely a week well spent.
The conversation style of teaching math is so useful. This should go up with the other great works like turning a sphere inside out and Donald Knuth's book Surreal Numbers.
Like everyone else here, I'm rating this 10/10. This is the most accessible video I've come across on algebraic and projective geometry. Sadly I'm only an engineer and lack so much mathematical foundation, but this refreshing and intuitive explanation will certainly help me I'm my research :)
Good video !
XD reminds me of the conversations in the video "Turning a sphere inside out"
Unbelievable video, thanks. As someone with a maths background who is a bit embarrassed I never learned any projective geometry this was really clear and interesting.
this is so cool and so well explained!
the dialogue makes it so easy to follow.
Amazing presentation style, I saw a lot of the thoughts in my head being echoed by the two speakers, that's good writing!
Very underrated project. It's a really amazing way to teach students about these topic that are generally difficult for them to grasp at first. Well done
The framing of this as a conversation was really good for following along! Loved it!
I'm glad The Algorithm showed me this vid.
It would be a great complimentary video to introductory chapters of 'Elliptic Tales'.
Well hells bells, lads. I understand things now that I didn't understand before by watching this video. Super neat. I'm absolutely going to watch this again with the hopes that it'll happen again.
This video is amazing! I have yet to learn anything about projective/algebraic geometry, and this video got me hooked immediately and completely blew my mind. This is just beautiful mathematics. I also really like the format of the video as a dialogue, it is very relaxing in a sense.
Can't wait for this video to blow up
I love this conversational style of math videos
superb video, it make me think about Plato's dialogue with Socrates and Théétète
I took "Algebraic Geometry" in my Masters program and struggled to understand it. THIS makes it start to come together...
Thank you!
THIS WAS BRILLIANT!!! 🎉 Had me at the edge of my seat at every chapter!!! Definitely subscribing, liking and sharing!
This dialogue is superb!
Excellent. More, please!
Wow this is really incredible!! and I totally buy and approve all the comparisons with Outside In (of which I was one of the creators..) great explanations of deep math. love it.
Yesss, I’ve been wanting to learn about circular points for so long but I couldn’t understand, THANKSS
powerful outside in energy
This was wayyy up there in terms of mathematics video quality. I just had my second semester mathematics exams. Already looking forward to geometry in the 4th semester. I am seeing a lot of similarities between the snippets from what my friend told me about that course (largely focused on hyperbolic, spherical, / just non Euclidian geometry) and this video.
honestly incredible 🫢
mindblowingly well made!!! Keep up the AMAZING work, thank you!
Wow.... Please make more math videos... I'm blown away....❤
i have the same talk in my mind, two person with that mood, amazing know you both or just you.Thanks for be brave and upload
The conversation format reminded me of "Outside In"
Love the conversational format!
Although I think you should explain (or at least) mention quotion constructions when talking about "identifying" or "pasting" points etc.
Es increíble este video, me sorprende la calidad que tienes. Es sorprendente, ojalá llegues a ser un gran divulgador.
¡Gracias!
this really feels like one grad student just explaining a thing to another. I feel like I've been in that discussion a few times lol. also this video feels like reading a textbook chapter and I kinda wish there some exercises
Well, there's the one at 17:19
This is so cool, youtube hasnt made justice yet
Edit: no this is not cool, thats one of the best math videos ive seen so far!
Good video! I got a little lost on chapter 3 so I will have to rewatch it latter.
I feel like this was inspired to some degree by outside in
Really really superb!
this is a gem, though I do get lost at a few points:
1. 6:37 why there's "got to be" a point passing through itself three times
2. 16:48 the space of answers for exactly what? Curves passing through serveral points, lines tangent to several curves and etc?
1. This is definitely not supposed to be obvious. It can be proven but it's not important for this video
2. The number of conics through five points in particular, but also the space of lines that go through two distinct given points _and_ the space of points that lie on two distinct lines _and_ the space of lines tangent to two circles are all zero-dimensional.
Thanks!
You can use backticks (" ` ", next to the "1" key) on both sides of a part of text in the label to make it write in LaTeX!
Omgggg thank you
Loved it
I love the so me2 series
so cool!
gold
It's excellent introduction to projective geometry. The takeaway is that both of you aren't punctual at all which adds difficulty to understanding.
Hi and thanks. What do you mean by punctual here?
it looks like you're using desmos for graphics, you can add (some) latex in point labels using ` ` e.g `p_2` (just fyi in case you didn't know ❤)
We did _not_ know and that is good to know, thanks! Yes it's Desmos and if you want to play with it, see links in the description.
This is a very well-done video! Two notes:
- You introduced the p_m notation near the start, but never brought it up again. Is this the same as [1,m,0]? In this case, the two points on every circle are p_i and p_-i.
- I second the comment someone else made that you should see if you can color-code the subtitles.
The circle thing makes sense given that every scaling and translation of the hyperbola x^2-y^2=1 contains the points p_1 and p_-1. Circles are scalings and translations of the equation x^2-(iy)^2=1.
Yeah p_m = [1,m,0] now that you mention it. That would indeed have been good to make explicit. Sigh.
Great video
we need huggbees voiceover
nifty
I’m getting “Sphere inside out” vibes
Cool video. What did you use to make it?
Desmos (see links in description) and DaVinci Resolve
1:23 I too prefer thinking about vertical lines as having slope with infinity but isn't that technically not correct? It should be undefined
It's just a name. We're not going to do algebra with it, e.g. try to "multiply" two slopes together. You're worried that you got a hold of the slope as a/b, and in other contexts it's safer to say "that ratio is undefined" than to say "that ratio is infinity".
In _this_ context, the reason that people like "infinity" as the name for the vertical slope is that it suggests the right "topology on the space of slopes". Concretely, you should think that just as if we consider lines with slopes 5.1, 5.01, 5.001, ... we'll sneak up on a line with slope 5, if we consider lines with slopes 10, 100, 1000, ... we'll sneak up on a line with slope infinity.
@@AllenKnutson I like this explanation, thanks. Totally with you on this, I like saying infinite slope as opposed to undefined.
Great video! Were you inspired by the classic video about turning the sphere inside out?
Actually no
While I definitely saw Outside In nigh 30 years ago ruclips.net/video/wO61D9x6lNY/видео.html I had completely forgotten that it is done as a dialogue. We were more inspired by the flow of actual conversations (between the two of us, and with other people), as I'm sure the Outside In people were.
@22:30, since there's TWO antipodal points at infinity (at the Northeast and Southwest), it's not a Riemann sphere correct? But could perhaps be thought of as two Riemann spheres superimposed over each other, with diametrically opposite points at infinity?
(1) Correct RP^2 is not a sphere (let's skip the "Riemann" adjective).
(2) I don't think it's that good an idea to think of it as two spheres S & T superimposed; that would be like saying there's a map from S union T -> RP^2. (Which I suppose there is, but) it makes more sense to say there's a two-to-one map from S -> RP^2, where each spot on S gives the spot pair on RP^2. Bringing in a second sphere T is a weird distraction, at that point.
@@AllenKnutson Thank you. I guess I'm confused because I've been watching some Penrose videos and he often implies that the celestial sphere, which to my mind is just vision itself(?), is a Riemann sphere, but that the Riemann sphere is "not really a metric sphere" (his words). But at the same time, I thought projective space (RP^2) was supposed to be what we actually see around us day to day. Do you have any thoughts on this? It's not clear to me if points or lines at infinity are abstractions or what's really out there in front of us right now... certainly things appear to get smaller as they get further away, and artists need projective geometry to make art appear realistic. Is there a clear delineation between what geometry is abstraction and what is our actual experience?
@@erawanpencil I don't think I know what "RP^2 is supposed to be what we actually see around us" is supposed to mean. For a one-eyed person (for simplicity), who therefore can't judge distance, what they see around them might as well be scaled to all be at distance 1, which puts it on a sphere, not onto an RP^2. The RP^2 is relevant for one-eyed people whose skulls are transparent, and so when they see an object in some direction, they not only can't tell if it's near or far, they also can't tell if it's in front or in back of them.
wait how... ok i get it. month later - wait how?? )))
Desmos
24:20 sorry i cant understand why the equation being in two variables can be a problem can you explain it to me ?
The traditional quadratic formula applies to quadratic equations in one variable.
If we were looking at a general quadratic equation in two variables ax^2 + bxy + cy^2 + dx + ey + f = 0, there'd be no way to factor it.
But in the case at 24:20, the equation is the homogenization of a quadratic equation in one variable, so the traditional quadratic formula can be applied.
By any chance, are you related to world-renowned juggler Allan Ivar Knutson?
How did you guess? Allen by the way
@taranknutson175 I went to Caltech and lived in Dabney Hovse. Let him know that Benzene says "Hi." Also that I flamed out of grad school. Twice.
@@taranknutson175 Also, tell him major congratulations on writing a paper with Tao.
@@taranknutson175One more thing: I'm still watching your video. I like what I've seen so far!
Hey Tom. I'm "A:" in the subtitles. @@tomkerruish2982
That circle isn't so great. I've seen greater.
4:01
5:11 +/- ☆
6:26
6:50
3b1b viewer here. you lost me at 1:50. maybe think about explaining more or differently what you're doing or a different visualization. All the best.
Hmm, I'm not sure how to explain this in another way. The green line has some slope m, and by definition this means that it has the point p_m on it. The purple line is parallel, so it has the same slope, and therefore also has p_m on it. They share the point p_m in the same way that non-parallel lines share their intersection point.
Amazing presentation style, I saw a lot of the thoughts in my head being echoed by the two speakers, that's good writing!