Extraordinary Conics: The Most Difficult Math Problem I Ever Solved

[@feynman6756]

"I'm not 3b1b" He said it to hide the truth.

[@prinssdgunofficial2400]

Fhtt

[@shoam2103]

3b1b is how I found this video

[@asailijhijr]

He's secretly TL;DR.

[@Chiken1]

2b2t

[@jockbw]

but when their powers combine they art captain planet

[@Yogfan800]

i didn't understand like 90% of this video but yeah shapes are cool.

[@CodeParade]

Thank you! I know the actual material is dense, but I was hoping the visuals still make it fun and interesting to watch.

[@conlangnovids4974]

@@CodeParade I loved this I didn't understand 10% of it but It was cool I Love math and coding so this was really good

[@firSound]

Watching advanced material well beyond one's current knowledge and comprehension of a subject, strengthens diffuse mode learning. So even if you don't know it, you're actually smarter.

[@EtanMarlin]

I agree 😂

[@kkTeaz]

@@firSound yes my brain is better now

[@Jellylamps]

My favorite thing in math is “oh you can just do this simple and seemingly unrelated thing to figure out the problem and it always works”

[@chriskrofchak]

ALL OF MY PROOFS CLASS...

[@ferociousfeind8538]

It's like shooting a duck to get winter to come and go. Like, what? What?? But it works, and some guy 500 years ago proved it works with like... wait, how did he know any of that? Q... quantum theory and general relativity? What does THAT have to do with a DUCK and WINTER? Ahem. Math might not be for me lmao

[@Ssacred_]

@@ferociousfeind8538 you ever thinked how fucking blown mind is the 2 grade ecuation, just that simple thing, like how the fuck did they figure out, srry for my english eksdi

[@RagbagMcShag]

@@Ssacred_ A part of my brain melted inmidst of this comment chain

[@Ssacred_]

@@RagbagMcShag xdd

[@sireevictineerivero342]

"There is a conic that passes through any 5 points." Yeah. "Parabolas are halfway between an ellipse and a hyperbola." Mhmm... "The equation can be simplified by this matrix." Uh...Right. Sure. "AcosTheta + B....." ...I guess? "Frobenius product." Now you're just making up words.

[@scptime1188]

I get the conic stuff and tangents and all that, but everything in the written proof section about the matricies and stuff, i was completely lost.

[@TheMajorpickle01]

@@arnehurnik If you don't understand matrices, it's an entire topic in a physics undergrad. Not to discourage you from looking it up but don't be mistaken into thinking it's a minor undertaking. If you are used to linear math non linear math is a headache

[@Temeliak]

@@arnehurnik 3blue1brown made a quite nice and relatively easy to follow series on linear algebra, a good way I think to wrap your head around matrices

[@gamma-bv6ty]

@@TheMajorpickle01 Matrices are part of linear algebra so I don't really see what's nonlinear about them. Also, the actual theory of matrices would be covered more in a math undergrad than a physics undergrad.

[@TurkishLoserInc]

@@gamma-bv6ty Any reputable Physics, math, and comp sci dept is going to be sticking you into a sophomore-junior level linear algebra class that will essentially focus on matrices. All engineers were also required to take it at my school, as any FEA(finite element analysis) is likely going to be done with either calculus or simpler linear algebra.

[@abigailmcdowell4248]

I now really REALLY want 3b1b to prove all the assumptions in this video 😅

[@conlangnovids4974]

why not we all ask 3blue1brown (:

[@adammoussa7295]

yeah good idea, let's start bombarding his videos comment section!!

[@ck88777]

3b1b kinda just demonstrates other people's proofs and theorems idk if hes capable of proving all this in a timely manner

[@abigailmcdowell4248]

@@ck88777 doesn't need to be timely, and I think it'll be an interesting enough exercise for him to want to show

[@shadiester]

Another commenter, Rishabh Dhiman, included this relevant information which I thought might be of interest: """I was really delighted to see a relatively large youtuber talk about point-line duality and projective geometry. If you want a proof of these properties and a lot of other cool properties I would highly recommend AV Akopyan's book Geometry of Conics. [1] Also, the line formed by the three collinear midpoints is called the Newton-Gauss line. [2] The proof for the case of the tangent ellipse being a circle is called Newton's Theorem. [3] The fact that the centres are collinear comes from a more general fact about the locus of pole of a fixed line with respect to the the inconics of a given quadrilateral being collinear. This is Theorem 3.16 on page 88 of Geometry of Conics. When the fixed line is moved to infinity, we get centre of ellipses and hyperbolas. [1] AV Akopyan's Geometry of Conics geometry.ru/books/conic_e.pdf you can also buy a physical copy on Amazon [2] Newton - Gauss Line en.wikipedia.org/wiki/Newton%E2%80%93Gauss_line [3] Newton's Theorem - www.cut-the-knot.org/Curriculum/Geometry/NewtonTheorem.shtml """

[@dumbeh]

me having no idea what any of this means. “ah yes of course... the... matrix.”

[@nixel1324]

Don't forget to... invert it?

[@zadejoh]

@@nixel1324 in case you're serious, a matrix is basically a grid of numbers. Inverting a matrix is the equivalent of finding the reciprocal of a number (let's say 8 and 1/8). Multiplying 8 and 1/8 gives 1; for matrices A multiplied by its inverse A^-1 gives back the identity matrix which is the matrix equivalent of the number 1. Of course finding the inverse of a matrix is not as easy as the reciprocal of a number at times, but this is the gist of it.

[@MrTtawesome]

@@miso-ge1gz When you switch the numerator and the denominator. Say you have 5/2, the reciprocal is 2/5. Or 3, which can be written as 3/1, it's reciprocal is just 1/3. Multiply a number by its reciprocal and you always get 1, which is pretty cool

[@vikaskalsariya9425]

@@miso-ge1gz You haven't seen the Neutron style.

[@michalgolonka832]

Same. F*ing same. Matrixes, tangents, sinh, cosinh. I vaguely understand sin and cosin

[@johnerickson8160]

A mathmatician: Aw yes a very satisfying math problem Me: Whoa look at the cool lines on the screen

[@Nasrul260]

Math with text: **boring** Math visually: *_"let's get funky!"_*

[@EsperantistoVolulo]

A random taxicab with the number 1729: Am I a joke to you?

[@olivervan7265]

@@EsperantistoVolulo what

[@olivervan7265]

I won’t like you sense I know your secret

[@John-hz8xy]

He makes Desmos look like a children's toy.

[@telleahuman5585]

THIS IS DESMOSSS? omg

[@BambinaSaldana]

​@@telleahuman5585 Doesn't look like Desmos

[@skj983]

Kids today are lucky to have these kinds of visualizations for geometry. This type of stuff works wonders for the young mind in developing a very valuable sense of intuition for mathematics. This is really great work. Keep it up!

[@Otori6386]

I know enough to know I don't know enough to fully appreciate this hehe pretty lines and shapes

[@Vit-Pokorny]

programmers be like: "Just knowing it works was good enough for me"

[@dreckneck]

This makes a mathematician cry 😂😭

[@jakehate]

"you might have seen a comic section represented like this before" Me: hmmmm yes go on

[@pauld8747]

Conic

[@kornsuwin]

yea

[@kebman]

I think you'll _love_ POV-Ray. It's an old raytracer. You have to program the inputs. Modellers exist for it, but the true joy of using this program is wading through the pleasurably well-made documentation, and the complicated yet fully logical mathematical models used to trace the forms. You can make some very complex forms with it, including quartic objects, and objects modelled with various forms of "noise" algorithms, and of course fractals. I don't know any other raytracer that is so comprehensive, and yet logically set up. It might be old, but it still has it's uses.

[@yinq5384]

6:55 We consider the standard ellipse (x/a)^2 + (y/b)^2 = 1 as an example. (General cases are same after one rotation and translation.) All points on the ellipse have the parametric form P(a cos(s), b sin(s)). The obvious choice of vectors A and B are A = (a,0) and B = (b,0). In general, say we know one skew vector A = (a cos(t), b sin(t)), and we try to find out another vector B so that A sin(theta) + B cos(theta) + C representing the same ellipse. (C = the zero vector here since we assumed the center is the origin.) Assume B = (a cos(s), b sin(s)) A sin(theta) + B cos(theta) = (a cos(t) sin(theta) + a cos(s) sin(theta), b sin(t) sin(theta) + b sin(s) sin(theta)) For any angle theta, the above point is on the ellipse (x/a)^2 + (y/b)^2 = 1. Thus (cos(t) sin(theta) + cos(s) sin(theta))^2 + (sin(t) sin(theta) + sin(s) sin(theta))^2 = 1. Simplify and we get 0 = [cos(t) cos(s) + sin(t) sin(s)] sin(theta) cos(theta). Thus 0 = cos(t) cos(s) + sin(t) sin(s) = cos(s-t). We can choose s = t + pi/2. That is, B = (a cos(t+pi/2), b sin(t+pi/2)) = (-a sin(t), b cos(t)) To summarize, "Skew vectors" ARE still "Perpendicular" in the parametric sense. 7:10 Area: |A x B| = | (a cos(t), b sin(t)) x (-a sin(t), b cos(t)) | = a cos(t) b cos(t) - b sin(t) (-a sin(t)) = ab Thus pi |A x B| = pi ab = Area C^2 Invariant: |A|^2 + |B|^2 = (a cos(t))^2 + (b sin(t))^2 + (-a sin(t))^2 + (b cos(t))^2 = a^2 + b^2 Inside Test: Using the parametric form again, say P - C = P = k(a cos(s), b sin(s)). Point P is inside the ellipse if and only if |k| < 1. |(P - C) x A| = kab (cos(s) sin(t) - sin(s) cos(t)) = kab |sin(s-t)| |(P - C) x B| = kab (cos(s) cos(t) + sin(s) sin(t)) = kab |cos(s-t)| |A x B| = ab as we already calculated. Then |(P - C) x A|^2 + |(P - C) x B|^2 = (kab)^2 and |A x B|^2 = (ab)^2 Then Inside test formula is equivalent to k^2 < 1. Tangent Test: Necessity: Suppose there is a tangent line. P is any point on the line and R is the direction vector of the line. Denote the tangent point by T. Then (P-T) // R. Thus R x (P - C) = R x (T - C). Actually, we can use similar parametric form as above, say R = k(a cos(s), b sin(s)) and T - C = T = (a cos(t), b sin(t)) Then |R x A|^2 + |R x B|^2 = (kab)^2 as before, and |R x (T - C)|^2 = (kab)^2 |sin(s-t)|^2. The formula is equivalent to |sin(s-t)| = 1, i.e. the different between t and s should be pi/2. And the tangent line passing through T(a cos(t), b sin(t)) is indeed with the direction vector (a cos(t+pi/2), b sin(t+pi/2)) = (-a sin(t), b cos(t)). Sufficiency: For any line L, we can always find a tangent line TL parallel to L. Thus the two lines have the same direction vector R but different points P_1 and P_2. For TL, we know |R x A|^2 + |R x B|^2 = |R x (P_1 - C)|^2. If L satisfy the tangent test, then |R x A|^2 + |R x B|^2 = |R x (P_2 - C)|^2. Thus |R x (P_1 - C)|^2 = |R x (P_2 - C)|^2, |R x (P_1 - C)| = |R x (P_2 - C)| R x (P_1 - C) = R x (P_2 - C) (there are exactly two tangent lines out there, that's why there are two cross products with opposite directions, we can choose the one with the same direction) R x (P_1 - P_2) = 0 i.e. (P_1 - P_2) // R, meaning P_1 and P_2 are on the same line. That is, L is actually the same as TL.

[@AgentMidnight]

I'm an absolute sucker for clean, fluid math visuals. Instant subscription.

[@lock_ray]

After taking a more advanced linear algebra course I came back to this video and actually understood it this time! Thanks for the motivation CodeParade!

[@columbus8myhw]

15:30: "And negative areas are hyperbolas." Correction: this is area squared, so negative 'area squared', or imaginary areas, are hyperbolas.

[@CodeParade]

You're correct. I was trying to say 'the areas of the curve below the x axis' but it was confusing because I'm also talking about literal area.

[@Keldor314]

This might be an interesting area for further investigation. Clearly any intuitive "area" for a hyperbola is infinite since it's an unbounded shape, but here we have a solution that assigns such an area to an imaginary number. So what's the deeper meaning here? Also, what about the duality between positive and negative area? Negative area is one of the two solutions to a square root, but is there a geometric meaning to negative area that's distinct from positive area? Maybe you could introduce some idea of handedness depending on whether the elipse goes around its center in a clockwise or counterclockwise direction according to the parameter theta? This makes sense in the context of a mirror image perhaps. Finally, is there some way to give meaning to area as a generalized complex number? What about instead of looking at a plane (being the cartesian product of two real lines), we look at a "hyperplane" (the cartesian product of two complex planes) instead? If we take the original problem to be looking at the planer cross section through real directions, is there meaning in looking at a complex area as a solution to where the complex conic section becomes an elipse in a different cross section? Could all this be related ultimately to the same structure that gives rise to the Fundamental Theorem of Algebra?

[@hybmnzz2658]

@@Keldor314 it is common to see diverging things have a connection to imaginary numbers.

[@samuelthecamel]

@@Keldor314 In abstract math, divergent sequences often "converge" to some negative or imaginary number. For example, 1+2+3+4... = -1/12. Although this isn't really an iterated sequence, it may be related in some way.

[@hiiistrex2838]

@@samuelthecamel how is 1+2+3+4... Supposed to equal -1/12 tho I feel like I've seen it before but it makes zero sense Or it makes -1/12 sense idk

[@ItsLogic]

Oh my god, He's back.

[@dantekiwi7926]

ItsLogic my brain...

[@jeffwillsea6757]

Owwe..... Cool

[@DeveloperDesmond]

CodeParade! This video is amazing! Here's my criticism: - When you have variables on screen, like A, B, or R1, it's really hard to keep track of *what* the variable represents. Salman Khan does a really good job in his videos of alleviating this problem in two ways: 1.) He keeps the diagram on screen when doing algebra. 2.) He color codes the variables to the diagram. If x represents a distance, he'll draw the distance in blue, and then use the same color blue whenever he writes x. If you pause your video at 10:34 or 10:25, you'll notice a block of text and a diagram, but no way for the viewer to quickly relate the diagram to the text. - You introduced the problem statement at 8:00, which is probably too late. I also don't think you explained the *why* well enough for this problem. 3Blue1Brown's video, "This problem seems hard, then it doesn't, but it really is ," is an example of Grant Sanderson's effort to tell an engaging narrative, even when the problem being solved isn't important.

[@phileiv]

That's really interesting. I came back to this video after a couple of days because i found it a bit confusing, and i had paused at exactly 10:34.

[@StNick119]

I'd love to see more "hardcore maths" videos like this.

[@abd.137]

9:00 The mid points lie on a line is called "Gauss line of a complete quadrilateral". Whose existence in proved in the Gauss Bodenmiller Theorem

[@Dekross]

I only know gauss for the xyz problems :v

[@Icenri]

Thanks! I came back to this video looking for this comment. I studied projective geometry but never got to that theorem and in the video it seems so obvious that it has to be connected to the complete quadrilateral and the harmonic conjugate somehow.

[@eofirdavid]

I think that many of the phenomena that you mentioned follow from the fact that an ellipse is simply the image of a circle under a linear transformation (multiplication by a matrix where you columns are your vectors A and B). I think that your cross product which measures the area is (up to a constant) the determinant of the matrix. When you rotate the vectors, you multiply by a rotation matrix, and since it has determinant 1, and det(XY)=det(X)det(Y), then you know that it should not change the determinant, so the new crossed product should still compute the same area. For the |A|^2+|B|^2, this computes the Frobenius norm of a matrix. Unlike the determinant, this norm in general is only submultiplicative, but luckily for us it is multiplicative when you multiply by rotation matrices.

[@pianojay5146]

Ofir David cool idea

[@uganasilverhand]

I've considered versions of the ellipse formula since high school such as: (x-a)^2/sin^2(theta)+(y-b)^2/cos^2(theta)=r^2 -- no need to calculate eccentricity, it's actually built in now and describes any simple 1 or 2 focii solution as a projection from a spheroid or cone as theta is similarly a projection of the angle from the plane or the "light" source.

[@lj8324]

Ah yes..

[@thegamehouse4245]

I totally understand what you mean.

[@7s1gma]

Mind blown confirmed. More hardcore math videos please.

[@kikivoorburg]

Wow this is amazing. Really demonstrates the crazy interconnected nature of mathematics!

[@adamschultz7127]

"iT tUrNs OuT yOu JuSt InVeRt ThE mAtRiX" like that means anything in the world to anyone but Lawrence fishburn

[@debblez]

correction: anyone who passed 10th grade

[@martinbrink6711]

KZbin desperately needs more hardcore math videos! I'll be looking forward to your next masterpiece!

[@DerLibertin]

The visuals of this video have some serious Windows XP screensaver vibes. I love it.

[@ddiva1973]

Hard core math is good for the brain, keep going!

[@johan7999]

You've fed the curiosity within me. I'm enjoying your source code, your math and you're fascination for these mathematical discoveries! You sound like a child when he first are a candy, absolutely wonderful!

[@modus_ponens]

Whoah what animations and effects! On top of that using c++. Also interesting findings indeed. Enjoyed the math content, particularly the matrix derivation, as it showed quite some many tricks.

[@DeGandalf]

I have NO idea about this math stuff, but with the nice visuals it was still entertaining; I enjoyed it.

[@younlok1081]

yes satisfying

[@fish8622]

I understood half of it. So I knew what he was talking about, what he was trying to do, and what he did. I have entirely no clue as to how he did it.

[@dexstevens5993]

Fish same

[@Sciencedoneright]

11:17 Even though honestly, I didn't understand the concept, that simplification was *BEAUTIFUL!*

[@MusicEngineeer]

it is so satisfying, being faced with a challenging math problem, sitting down for many hours or even days (or more), researching, thinking, finally arriving at a solution, implementing it, testing it - and seeing it WORK ...and then harnessing the so found solution to do all the cool stuff that one wanted to do with it! thanks for the video and the code. should i ever be facing a similar problem, i now know, where to look. yes - i would definitely like to see more videos of this sort.

[@Magnogen]

I'm intrigued to see a collaboration between 3b1b and cp. It would make a cool watch.

[@R23874]

Mind is definitely blown. Stumbled upon your channel today and I'm so glad I did, all of your content is incredible. Will be eagerly watching your github as well.

[@TuddYT]

I loved this! Please keep making this kind of high quality hardcore math + code content :)

[@hyperspaceadventures1416]

I loved this! Please make more hardcore math videos! KZbin really needs more beautifully visualized math stuff.

[@Tehom1]

Definitely more hardcore math videos if they're going to be this good.

[@Banarann]

This was a very cool watch! It makes me think of how 3b1b quoted, "Math tends to reward you when you respect its symmetries"

[@NovaWarrior77]

16:15 yes, more like this if you can please! This was awesome! I'm sure that if you're consistent, you will blow up!

[@servantking1519]

That "hardcore math" warning scared me... I was already somewhat confused and had just understood one of the things you said right before it happened

[@joaogabrielneto697]

I'm a lawyer, why am i seeing this and why its so interesting?

[@maxwellsequation4887]

Too bad for you Now you are just a lawyer Always remember Fermat, one of the greatest mathematicians ever was a lawyer

[@AngrySkyBandit]

I have been curious about this very question for years. Never took the time to figure it out, and I stumble upon this video on yet another youtube bender. Many thanks for the ride!

[@sofia.eris.bauhaus]

okay, i understood roughly half of the non-hardcore part an none of the hardcore bit. still learned come cool things in a short time. thanks, will rewatch! B)

[@DavidScherfgen]

Great video. Coincidentally, it helped me understand a paper about fitting ellipses to images using gradients at the pixels as tangents. It makes use of the dual conic. The paper was so complicated to understand, but when I saw your video I instantly got it. Great work!

[@yuryeuceda8590]

The way it changes from parable to hyperbola is like when a star converts to a black hole. Interesting

[@joshuacole5608]

I love how you left time in the beginning just to play around with the shapes

[@chaimlukasmaier335]

I heard a really good lecture series on harmonised coordinate systems this semester... So there was not that much new stuff, but you animated it really well. For all who speak German, i can recommend "Geometriekalküle" by Jürgen Richter-Gebert

[@stefanamg63]

This was stupendously mind-blowing. I wish you had made this video 2 years ago when I was writing code which solved a 'tangents between 2 ellipses' problem. I ended up brute-forcing it after struggling for almost a year.

[@Francis-ce1qb]

I have no idea what I’m watching but i still find it interesting listening to it

[@MinhTran-wn1ri]

Refreshing video. The music and visuals were captivating. I wish mathematical concepts were taught this way when I was in grade school -- with visuals, animation, perhaps with code that students can play with. Of course back then, 3B1B wasn't a thing.

[@Galbex21]

I dont understand almost anythung but its so beautifully represented and edited that its still a pleasure to watch.

[@debblez]

this is still one of my favorite videos of all time

[@positivefingers1321]

Yay code parade!

[@larrywestenberg7839]

This was awesome! I don't "do" this sort of math - but you made it completely "followable" for me. What a cool trip that was!! The animations brought the equations to life very well. Bravo, buddy!!

[@atimholt]

A lot of what you’ve shown is *exactly* the math I need for my own project. Thank you!

[@bobmcbob8044]

Yes, I would definitely like to see some more hardcore maths stuff. It's very interesting, and great how so many patterns arise in unexpected places.

[@ChrisDjangoConcerts]

You should do more math videos! These are really awesome !

[@keithmacalam5443]

I dont understand lyk many things, but I feel peace when I watch these kinds of videos... I love shapes too! Specially some complex geometrical shapes!

[@jucom756]

"I'm not 3 blue 1 brown" My brain: the f*** yes you are Also my brain: oh wait yeah he isn't I really thought i was watching a 3blue 1brown video, this could make for a collab

[@marcelosantos5683]

this must make for a collab hahahah

[@aepokkvulpex]

i hope he sees this video tbh

[@johanrojassoderman5590]

Really interesting and thoroughly explained. I'm nowhere close to the mathematical prerequisites but still managed to grasp it thanks to the theoretical and visual explanations. Would definitely not have anything against seeing more hardcore math videos, but i think most of your videos are extremely interesting. Definitely one of the more unique math/coding channels on youtube, and far too underappreciated if you ask me. Keep up the good work!

[@caps_lock]

5:54 BRUH

[@flick2040]

The animation and motion in this video is so incredibly pleasing.

[@Menaiya]

My brain is fried. This reminded me of a lot of math I've forgotten.

[@hvok99]

Oh man, this sketch of your intuitive process was wonderful. This problem feels like it would belong to a whole family of problems where you are given a set of n points in k dimensional space and ask to find a curve that is uniquely defined by n+1 points in such a way that some aspect of the curve is maximized, or rational complexes on the curve.

[@TheIhavenot2]

Have you heard of Desargues' Theorem? It closely resembles the finding you mentioned at about 9:00 and has an elegant proof in projective geometry. It seems to be the theorem you couldn't find.

[@NonTwinBrothers]

Seems he had known it, as it's listed during 5:43

[@ProjectPhysX]

Congrats for solving it! The hardest math problem I ever solved is the plane - unit cube intersection. You have given the orientation of a plane which intersects a unit cube centered at the origin, you know the volume of the truncated cube, and have to figure out how far off the plane is from the origin.

[@jafizzle95]

"Thanks for watching" Ah, yes. I understood that entire sentence.

[@Dezomm]

I love this channel so much. Gets me excited about all the complex stuff out there I don't know about yet. Really great stuff dude.

[@stirrcrazy2704]

Apparently software engineers also fall into the trope of “engineers can’t do proofs.”

[@Kittoes0124]

Can confirm. One has the ability to implement most algorithms in software; through rigorous experimentation and validation against credible sources. Explaining how any of the maths actually works would be damn near impossible however... For example, I wrote all of this: dev.azure.com/byteterrace/CSharp/_git/ByteTerrace.Maths.BitwiseHelpers?path=%2FProject%2FBitwiseHelpers.cs, and yet it still feels like sorcery every time I make a function call! The fact that unit tests pass and pretty results appear on my screen is enough for me.

[@GermanTopGameTV]

This was super satisfying to see. The way the minimum and maximum solutions boil down to a simple minimum on a quadratic equation was just beautiful. This made my headache go away it was so pretty.

[@Lord_Bon]

I have no idea of whatever this video was about, but I still want to someday understand it all

[@XIIJaguar]

This was great! I enjoyed every second of it. You spent the right amount of time on every point to have me intrigued.

[@mistycremo9301]

I don’t know any linear algebra, but this definitely seems like a cool problem!

[@nathanielscreativecollecti6392]

I freaking love it. This was beautiful. Thanks for scratching my math itch.

[@haph2087]

Wow. This was lovely. I am not at the point in math to understand all of this, but I understood most of it and learned a lot. Those visuals are amazing too.

[@cboniefbr]

I love your "usual" contente, but this video was amazing. Looking foward for more of the kind.

[@TheBcoolGuy]

Me and my passing grades in post-secondary maths: Ah, yes. _Of course!_

[@admiralhyperspace0015]

Dude, this is awesome. I can't tell you how much ny mind is blown even though I only know about conic sections and don't know the calculations that you did. I want more. I just subscribed for this. I envy that I have don't have the same amount of math and coding knowledge as you.

[@irisinthedarkworld]

"I hope your mind has been blown like mine" Me, who is still in pre-calculus: Y-yeah...

[@hiiistrex2838]

This is mostly precalc stuff tho I guess maybe not the matrices

[@tessisaturtle5217]

i'm looking forward to coming back to this video in a few years and actually understanding what he's talking about geometry is so cool

[@OrangeC7]

5:40 Everyone else: *theorem* Desargues: Well, you see, I find it more fitting to call it a "converse"

[@cfgcfh6350]

Converse is just the inverse of a theorem..

[@RomanNumural9]

I love these kinds of hardcore math videos. This is why I love math

[@leo.maglanoc]

more hardcore math videos plsssss

[@Scrum-Master]

Once again, I'm blown away by the quality of your content.

[@pranavlimaye]

I should probably come back to this video 10 years later because I'm too young to understand anything but this stuff sounds interesting

[@desidudes78]

Thanks for the time you spent to make this beautiful video. I enjoyed it

[@confusioned2249]

My brain: **literally melting** Also my brain: ooo pretty shapes

[@lenardvandermaas6893]

I love that you made this video! I'm a big fan of implementing math into code and making cool stuff like this (or actually using it in a game or something). I'd love to see more videos like this!

[@joygodwinwilliamhenry406]

How do u get these ridiculously awesome insights though you didn't solve the problem completely these sort of intuitions are really useful to make useful hypotheses which simplifies a really complex problem

[@TomFowkes]

I like the funky circles! I like how they move on the dots

[@mypetblackie108]

You know, i came to watch this video taking Calculus AB this semester and felt confident i cpuld understand this. Boy was I wrong

[@Fermion.]

It's good to be humbled sometimes. It makes you try harder

[@mmukulkhedekar4752]

lol the video talks about projective geometry, how is calculus directly related?

[@athelstanrex]

@@mmukulkhedekar4752 because people taking calculus think they are taking the hardest math class and can understand any math

[@death1weller]

this is so relaxing to watch. it's like listening to music or a relaxed podcast on another language. I love listening to this while working, so my mind doesn't get distracted by it but can't drift off to unrelated things since it keeps my interest on trying to understand it ksksk

[@jbritain]

ok I know I'm out of my depth because idk what a conic is

[@ilonachan]

Okay, you know what a cone is? The idea of conic sections is, if you draw a plane straight through a cone (basically imagine cutting it in half), the cut will have the shape of an ellipse. The exact shape of the ellipse depends on the angle at which you cut. That's just a rough outline, since I don't really know anything about conic sections myself: we never discussed them in my school. But 3b1b did a proof about them once, so do check that out

[@fernandossmm]

To add to the previous answer: when cutting a cone you always get a conic section. The edge of it is a conic curve. These can be: A circle, if you cut parallel to the cone's base. A parabola, if you cut parallel to the cone's side. An ellipse, if you cut "slightly angled" from the circle. This would mean that your cutting plane crosses the line reaching down from the center of your cone. A hyperbola (or half of it), if your angle is so big that you don't cut through the line going down from the center. Or, more precisely, you cut it above the center or at infinity (being parallel to the ground)

[@SlipperyTeeth]

Adding to what others have said, a conic is also (in the algebra of the reals) the graph of any polynomial equation of degree 2 on 2 variables So, you have 2 variables x and y, this is what makes the graph a 2-dimensional picture. Then you make a polynomial equation of degree 2 with those variables, meaning you can multiply at most 2 variables together per term and add those terms together and set the whole thing equal to 0. For example you might have 3x^2 + 5xy + y^2 + 2x + 4y + 9 =0. Notice: the first 3 terms have 2 variables multiplied together, sometimes the same variable like x^2, sometimes different variables like xy; the next 2 terms have 1 variable multiplied together; and the last term has 0 variables multiplied together. Normally, you exclude degenerate cases, which are whenever you graph 1 or 2 lines or a single point.

[@mjcard]

Take a 3 dimensional cone shape and cut it across and the cross section is a circle. Cut it at a slant and the cross section is an ellipse. Cut it on one side, not through side to side but side to bottom, and you have a parabola shape cross section. Those are conics.

[@xatnu]

@Stale Bagelz It's like two infinite ice-cream cones stuck to each other facing the opposite diections. Imagine a lighthouse with two lights on opposite sides, the shape of the light coming out is a conic

[@santiagocalvo]

even tough i didn't understand pretty much anything i loved the video, don't even know why, i struggle enough with simple math when coding and watching this is such an inspiration to keep learning and trying to be better, thnx a lot!!!

[@undefined12]

Nice vid. That line with midpoints is called "Euler Line"

[@maxwelljiang4729]

no, it the newton-gauss line

[@philippkusterer9230]

Not 100% sure, but isnt the euler line the line where almost all centers of a triangles lie?

[@maxwelljiang4729]

yes. the euler line contains the orthocenter, circumcenter, nine-point center, and centroid of a triangle. not to be confused with the newton gauss line, which connects the midpoints of the diagonals of a complete quadrilateral. :)

[@insightfool]

Thanks for this explanation with such a good visualization. I normally gloss over when watching math videos, but this one was really engaging.

[@joshuadelacour1106]

For anyone interested, the "Insights into Mathematics" channel has a few videos covering this and other related concepts. I recommend Cromogeometry.

[@Icenri]

Good recommendation!!

[@Concentrum]

i'm so glad i found your channel, you are amazing man