Extraordinary Conics: The Most Difficult Math Problem I Ever Solved
ฝัง
- เผยแพร่เมื่อ 4 มิ.ย. 2024
- This is a real math problem I faced, and the process I went through to solve it. Despite being a difficult problem, I tried to focus on the beautiful visuals and interesting discoveries I made about conic sections that I doubt anyone knows about. While this isn't my typical style of video, there was a ton of coding that went into this (all open source) and an interactive app of this entire video. Check them out below!
Source Code: github.com/HackerPoet/Conics
Download Windows (64-bit): github.com/HackerPoet/Conics/...
Chapters:
0:00 Introduction
0:23 5 Elements
3:47 Duality
5:49 Skew Axes
7:39 My Hardest Problem (Part 1)
10:14 My Hardest Problem (Part 2)
13:59 My Hardest Problem (Part 3)
In Homage To:
3Blue1Brown: / @3blue1brown
Mathologer: / @mathologer
Definitely subscribe to these channels if you like this video!
Six Umbrellas - The Psychedelic And (CC BY-SA 4.0)
Six Umbrellas - Monument (CC BY-SA 4.0)
www.sixumbrellas.de/
Meydän - The Beauty of Maths (CC BY 4.0)
/ meydansound
Yakov Golman - Dance (CC BY 4.0)
yakovgn.awardspace.info/
"I'm not 3b1b"
He said it to hide the truth.
Fhtt
3b1b is how I found this video
He's secretly TL;DR.
2b2t
but when their powers combine they art captain planet
i didn't understand like 90% of this video but yeah shapes are cool.
Thank you! I know the actual material is dense, but I was hoping the visuals still make it fun and interesting to watch.
@@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
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.
I agree 😂
@@firSound yes my brain is better now
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”
ALL OF MY PROOFS CLASS...
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
@@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
@@Ssacred_ A part of my brain melted inmidst of this comment chain
@@RagbagMcShag xdd
I now really REALLY want 3b1b to prove all the assumptions in this video 😅
why not we all ask 3blue1brown (:
yeah good idea, let's start bombarding his videos comment section!!
3b1b kinda just demonstrates other people's proofs and theorems idk if hes capable of proving all this in a timely manner
@@ck88777 doesn't need to be timely, and I think it'll be an interesting enough exercise for him to want to show
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
"""
"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.
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.
@@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
@@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
@@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.
@@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.
me having no idea what any of this means.
“ah yes of course... the... matrix.”
Don't forget to... invert it?
@@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.
@@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
@@miso-ge1gz You haven't seen the Neutron style.
Same. F*ing same. Matrixes, tangents, sinh, cosinh. I vaguely understand sin and cosin
A mathmatician: Aw yes a very satisfying math problem
Me: Whoa look at the cool lines on the screen
Math with text: **boring**
Math visually: *_"let's get funky!"_*
A random taxicab with the number 1729: Am I a joke to you?
@@EsperantistoVolulo what
I won’t like you sense I know your secret
programmers be like: "Just knowing it works was good enough for me"
This makes a mathematician cry 😂😭
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!
He makes Desmos look like a children's toy.
THIS IS DESMOSSS? omg
@@telleahuman5585 Doesn't look like Desmos
I know enough to know I don't know enough to fully appreciate this
hehe pretty lines and shapes
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.
"you might have seen a comic section represented like this before"
Me: hmmmm yes go on
Conic
yea
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.
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.
I'm an absolute sucker for clean, fluid math visuals. Instant subscription.
Oh my god, He's back.
ItsLogic my brain...
Owwe..... Cool
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!
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.
15:30: "And negative areas are hyperbolas." Correction: this is area squared, so negative 'area squared', or imaginary areas, are hyperbolas.
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.
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?
@@Keldor314 it is common to see diverging things have a connection to imaginary numbers.
@@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.
@@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
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
I only know gauss for the xyz problems :v
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.
I'd love to see more "hardcore maths" videos like this.
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.
Ofir David cool idea
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.
Ah yes..
I totally understand what you mean.
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!
Wow this is amazing. Really demonstrates the crazy interconnected nature of mathematics!
I loved this! Please keep making this kind of high quality hardcore math + code content :)
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.
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.
Mind blown confirmed. More hardcore math videos please.
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.
This was great! I enjoyed every second of it. You spent the right amount of time on every point to have me intrigued.
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!
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.
Hard core math is good for the brain, keep going!
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!
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.
11:17 Even though honestly, I didn't understand the concept, that simplification was *BEAUTIFUL!*
TH-cam desperately needs more hardcore math videos! I'll be looking forward to your next masterpiece!
I loved this!
Please make more hardcore math videos! TH-cam really needs more beautifully visualized math stuff.
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!!
I have NO idea about this math stuff, but with the nice visuals it was still entertaining; I enjoyed it.
yes satisfying
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.
Fish same
I'm intrigued to see a collaboration between 3b1b and cp. It would make a cool watch.
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.
I love this style of video. It really shows just how beautiful math can be. Keep it up!
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!
You should do more math videos! These are really awesome !
I love your "usual" contente, but this video was amazing. Looking foward for more of the kind.
this is absolutely fantastic. The visualization are beautiful and everything is so clear
A lot of what you’ve shown is *exactly* the math I need for my own project. Thank you!
Definitely more hardcore math videos if they're going to be this good.
having a look into your mind and your experimental math approach was awesome. Very cool video !
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.
I dont understand almost anythung but its so beautifully represented and edited that its still a pleasure to watch.
"iT tUrNs OuT yOu JuSt InVeRt ThE mAtRiX" like that means anything in the world to anyone but Lawrence fishburn
correction: anyone who passed 10th grade
Thanks for this explanation with such a good visualization. I normally gloss over when watching math videos, but this one was really engaging.
This is the first video I’ve see of your channel and wow that was great. Such a great application of linear algebra in geometry
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)
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
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!
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
Yay code parade!
I'm a lawyer, why am i seeing this and why its so interesting?
Too bad for you
Now you are just a lawyer
Always remember Fermat, one of the greatest mathematicians ever was a lawyer
Thanks for the time you spent to make this beautiful video. I enjoyed it
Loved this video! Well above my understanding, but everything was really well explained and the visualizations were a great addition.
I don’t know any linear algebra, but this definitely seems like a cool problem!
5:54 BRUH
This is great! In the past a few mathematicians tried to show me the wonders of projective geometry and I wasn't that thrilled, but seeing some more complex uses without all the annoying rigor is much more interesting (also your plots are a bit better than the whiteboard we had)
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!
The way it changes from parable to hyperbola is like when a star converts to a black hole. Interesting
I have no idea what I’m watching but i still find it interesting listening to it
Once again, I'm blown away by the quality of your content.
I would love to see more hard core math videos from you. Really appreciate your work. 👏👏👏
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.
Seems he had known it, as it's listed during 5:43
For anyone interested, the "Insights into Mathematics" channel has a few videos covering this and other related concepts. I recommend Cromogeometry.
Good recommendation!!
Oh my god this is flippin' awesome! Now I'm even more interested in linear algebra, please do more hardcore math videos!
i'm so glad i found your channel, you are amazing man
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
"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
this must make for a collab hahahah
i hope he sees this video tbh
This was great! I would certainly watch if made more videos of this sort!
One of the best things I have seen in a while
Please make more videos of this kind
Me and my passing grades in post-secondary maths: Ah, yes. _Of course!_
more hardcore math videos plsssss
This is jam packed with mind blowing facts. It just keeps going. Love it.
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.
Apparently software engineers also fall into the trope of “engineers can’t do proofs.”
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.
My brain is fried. This reminded me of a lot of math I've forgotten.
I would love to see more mathematical content on this channel.
Wonderful video by the way.
The animation and motion in this video is so incredibly pleasing.
I have no idea of whatever this video was about, but I still want to someday understand it all
I should probably come back to this video 10 years later because I'm too young to understand anything but this stuff sounds interesting
Please, more! This was truly excellent
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
"Thanks for watching"
Ah, yes. I understood that entire sentence.
You know, i came to watch this video taking Calculus AB this semester and felt confident i cpuld understand this.
Boy was I wrong
It's good to be humbled sometimes. It makes you try harder
lol the video talks about projective geometry, how is calculus directly related?
@@mmukulkhedekar4752 because people taking calculus think they are taking the hardest math class and can understand any math
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.
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.
"I hope your mind has been blown like mine"
Me, who is still in pre-calculus: Y-yeah...
This is mostly precalc stuff tho
I guess maybe not the matrices
5:40
Everyone else: *theorem*
Desargues: Well, you see, I find it more fitting to call it a "converse"
Converse is just the inverse of a theorem..
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.
Loved this! Would love to see more of this type of content