The two points that lie on every circle (???)

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/znu...
Chapter 3: www.desmos.com/calculator/3w4...
Chapter 4: www.desmos.com/calculator/x7r...
Chapter 5: www.desmos.com/calculator/4sl...

[@taranknutson175]

s,ubcl,./scraibe

[@AllenKnutson]

Smash that, like, button

[@tomkerruish2982]

​@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.)

[@AllenKnutson]

@@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.

[@tomkerruish2982]

​@@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.

[@user-gu2fh4nr7h]

yoink

[@officiallyaninja]

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]

Dang with feedback like that we won't be able to help ourselves! Thanks a lot!

[@nikkiofthevalley]

​@@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 KZbin 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.

[@AllenKnutson]

@@nikkiofthevalley In our last video we did it by audio channel. People _really_ didn't like that!

[@huhneat1076]

I love how passive-aggressive they sound but then they're like "yeah that's actually cool"

[@cashkurtz5780]

This is done very well. I like how you use a conversation to help motivate every step.

[@AllenKnutson]

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.

[@RobertFerro3]

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

[@logo2462]

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.

[@MouhibBayounes]

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

[@DiegoMathemagician]

Good video ! XD reminds me of the conversations in the video "Turning a sphere inside out"

[@d.l.7416]

this is so cool and so well explained! the dialogue makes it so easy to follow.

[@casperspook4415]

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 :)

[@aviralsood8141]

Amazing presentation style, I saw a lot of the thoughts in my head being echoed by the two speakers, that's good writing!

[@chokza0238]

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

[@AllenKnutson]

Yay!

[@thatapollo7773]

This is, without a doubt, my favourite video on this website, period. Amd I have been here for a long time

[@AllenKnutson]

DANG

[@mindvr]

I'm glad The Algorithm showed me this vid. It would be a great complimentary video to introductory chapters of 'Elliptic Tales'.

[@joda7697]

The framing of this as a conversation was really good for following along! Loved it!

[@CognitiveOffense]

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.

[@daniellewilson8527]

I love this conversational style of math videos

[@cyberscriptor]

superb video, it make me think about Plato's dialogue with Socrates and Théétète

[@shahenohanjanyan3545]

This dialogue is superb!

[@lwmarti]

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.

[@WithinEpsilon]

THIS WAS BRILLIANT!!! 🎉 Had me at the edge of my seat at every chapter!!! Definitely subscribing, liking and sharing!

[@harelkariv1477]

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

[@o11k]

powerful outside in energy

[@davidben-zvi4684]

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.

[@onthemerits]

I took "Algebraic Geometry" in my Masters program and struggled to understand it. THIS makes it start to come together... Thank you!

[@aieousavren]

mindblowingly well made!!! Keep up the AMAZING work, thank you!

[@zaheercoovadia4745]

honestly incredible 🫢

[@lgooch]

Yesss, I’ve been wanting to learn about circular points for so long but I couldn’t understand, THANKSS

[@nathanhunt4448]

Excellent. More, please!

[@swapnilshrivastava116]

Wow.... Please make more math videos... I'm blown away....❤

[@poscat0x04]

Love the conversational format!

[@poscat0x04]

Although I think you should explain (or at least) mention quotion constructions when talking about "identifying" or "pasting" points etc.

[@DmitriyNE]

Really really superb!

[@dehilour_arauz.jr.]

Es increíble este video, me sorprende la calidad que tienes. Es sorprendente, ojalá llegues a ser un gran divulgador.

[@AllenKnutson]

¡Gracias!

[@kaloka521]

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.

[@juanluisclaure6485]

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

[@Mjrm-99]

Good video! I got a little lost on chapter 3 so I will have to rewatch it latter.

[@malicksoumare370]

Great video

[@acrommclain5233]

Loved it

[@kkanden]

so cool!

[@diogoandre756]

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!

[@user-hr8rc1of3x]

The conversation format reminded me of "Outside In"

[@usermlgbzzcnm]

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?

[@taranknutson175]

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!

[@rururu5877]

I love the so me2 series

[@nodrance]

I feel like this was inspired to some degree by outside in

[@TheGluemess]

gold

[@realcygnus]

nifty

[@miguelcerna7406]

Cool video. What did you use to make it?

[@AllenKnutson]

Desmos (see links in description) and DaVinci Resolve

[@alexmcdonough4973]

Great video! Were you inspired by the classic video about turning the sphere inside out?

[@taranknutson175]

Actually no

[@AllenKnutson]

While I definitely saw Outside In nigh 30 years ago kzbin.info/www/bejne/rYCZYndvrZufhLs 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.

[@NKY5223]

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 ❤)

[@AllenKnutson]

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.

[@columbus8myhw]

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.

[@columbus8myhw]

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.

[@AllenKnutson]

Yeah p_m = [1,m,0] now that you mention it. That would indeed have been good to make explicit. Sigh.

[@Klarpimier]

I’m getting “Sphere inside out” vibes

[@H_fromDiscord_real]

Desmos

[@morejpeg]

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

[@AllenKnutson]

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.

[@morejpeg]

@@AllenKnutson I like this explanation, thanks. Totally with you on this, I like saying infinite slope as opposed to undefined.

[@innokentiyromanchenko1450]

wait how... ok i get it. month later - wait how?? )))

[@tomkerruish2982]

By any chance, are you related to world-renowned juggler Allan Ivar Knutson?

[@taranknutson175]

How did you guess? Allen by the way

[@tomkerruish2982]

@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.

[@tomkerruish2982]

@@taranknutson175 Also, tell him major congratulations on writing a paper with Tao.

[@tomkerruish2982]

​@@taranknutson175One more thing: I'm still watching your video. I like what I've seen so far!

[@AllenKnutson]

Hey Tom. I'm "A:" in the subtitles. @@tomkerruish2982

[@wargreymon2024]

It's excellent introduction to projective geometry. The takeaway is that both of you aren't punctual at all which adds difficulty to understanding.

[@taranknutson175]

Hi and thanks. What do you mean by punctual here?

[@2hcobda2]

4:01

[@2hcobda2]

5:11 +/- ☆

[@2hcobda2]

6:26

[@2hcobda2]

6:50

[@deadman746]

That circle isn't so great. I've seen greater.

[@kayurbach5182]

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.

[@taranknutson175]

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.

[@aviralsood8141]

Amazing presentation style, I saw a lot of the thoughts in my head being echoed by the two speakers, that's good writing!