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Taran Knutson
เข้าร่วมเมื่อ 30 พ.ค. 2015
The two points that lie on every circle (???) #SoME3
Chapters
0:00 - Preamble
0:51 - Chapter 1. Geometry of points and lines in RP^2
10:28 - Chapter 2. Algebra of points and lines in RP^2
15:01 - Chapter 3. Equations of conics
21:08 - Chapter 4. Conics in RP^2
24:42 - Chapter 5. Projective duality of conics
25:59 - Chapter 6. The behavior of circles at infinity
29:16 - Chapter 7. Circles as special conics
Enormous thank you to Desmos for allowing us to make the animations in this video. Desmos is an incredibly powerful online graphing calculator available at desmos.com/calculator. Desmos usually only works in 2D, though, so to make 3D projects like those in this video, you'll have to implement it yourself.
Desmos links:
Chapter 0-1: www.desmos.com/calculator/znudurapg5
Chapter 3: www.desmos.com/calculator/3w44mnsav6
Chapter 4: www.desmos.com/calculator/x7retiwufp
Chapter 5: www.desmos.com/calculator/4sltqjp0oc
0:00 - Preamble
0:51 - Chapter 1. Geometry of points and lines in RP^2
10:28 - Chapter 2. Algebra of points and lines in RP^2
15:01 - Chapter 3. Equations of conics
21:08 - Chapter 4. Conics in RP^2
24:42 - Chapter 5. Projective duality of conics
25:59 - Chapter 6. The behavior of circles at infinity
29:16 - Chapter 7. Circles as special conics
Enormous thank you to Desmos for allowing us to make the animations in this video. Desmos is an incredibly powerful online graphing calculator available at desmos.com/calculator. Desmos usually only works in 2D, though, so to make 3D projects like those in this video, you'll have to implement it yourself.
Desmos links:
Chapter 0-1: www.desmos.com/calculator/znudurapg5
Chapter 3: www.desmos.com/calculator/3w44mnsav6
Chapter 4: www.desmos.com/calculator/x7retiwufp
Chapter 5: www.desmos.com/calculator/4sltqjp0oc
มุมมอง: 34 894
วีดีโอ
Schubert Calculus and Quiver Varieties - ICM 2022 Allen Knutson
มุมมอง 1.7K2 ปีที่แล้ว
Schubert Calculus and Quiver Varieties - ICM 2022 Allen Knutson
Connecting CRT to Partial Fractions and Jordan Canonical Form #SoME2
มุมมอง 4.6K2 ปีที่แล้ว
This video was made for the Summer of Math Exposition 2. Check out 3Blue1Brown's video about this event: th-cam.com/video/hZuYICAEN9Y/w-d-xo.html I'll update this description when the playlist of submitted videos are out.
@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 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.
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
honest to god this video blew my mind im binging all your videos for sure
oh sht nvm
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.
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.
we need huggbees voiceover
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
Excellent. More, please!
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.
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
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
This is, without a doubt, my favourite video on this website, period. Amd I have been here for a long time
DANG
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 :)
4:01
5:11 +/- ☆
6:26
6:50
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!
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.
gold
THIS WAS BRILLIANT!!! 🎉 Had me at the edge of my seat at every chapter!!! Definitely subscribing, liking and sharing!
I'm glad The Algorithm showed me this vid. It would be a great complimentary video to introductory chapters of 'Elliptic Tales'.
I understood just enough to get a sense of awe and wonder.
26 years of my life in 26 minutes of yours :)
That circle isn't so great. I've seen greater.
The conversation format reminded me of "Outside In"
this video is really underrated, it's cool and simple. great job
superb video, it make me think about Plato's dialogue with Socrates and Théétète
I feel like this was inspired to some degree by outside in
This dialogue is superb!
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.
Love the conversational format!
Although I think you should explain (or at least) mention quotion constructions when talking about "identifying" or "pasting" points etc.
Desmos
I took "Algebraic Geometry" in my Masters program and struggled to understand it. THIS makes it start to come together... Thank you!
Obviously, the vector is cyclic since applying the transformation on it over and over cycles it through a basis of the space....
nifty
I love the so me2 series
The framing of this as a conversation was really good for following along! Loved it!
wait how... ok i get it. month later - wait how?? )))
Es increíble este video, me sorprende la calidad que tienes. Es sorprendente, ojalá llegues a ser un gran divulgador.
¡Gracias!
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.
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 th-cam.com/video/wO61D9x6lNY/w-d-xo.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.
Amazing presentation style, I saw a lot of the thoughts in my head being echoed by the two speakers, that's good writing!
Amazing presentation style, I saw a lot of the thoughts in my head being echoed by the two speakers, that's good writing!
Cool video. What did you use to make it?
Desmos (see links in description) and DaVinci Resolve
Really really superb!
I love how passive-aggressive they sound but then they're like "yeah that's actually cool"
Loved it
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
so cool!
I’m getting “Sphere inside out” vibes
mindblowingly well made!!! Keep up the AMAZING work, thank you! <3
Good video ! XD reminds me of the conversations in the video "Turning a sphere inside out"