Why slicing a cone gives an ellipse (beautiful proof)

[@rickeyward6540]

"You can often view glimpses of ingeniousness not as inexplicable miracles, but as the residue of experience. And when you do, the idea of genius goes from being mesmerizing to instead being actively inspirational" -Grant Sanderson (3blue1brown) phenomenal quote.

[@cranberry4860]

I was just procrastinating for finding quotes for a final essay about the future, and thanks to you, I found one that is both generally wise, and came from a topic that I'm actually interested in. Thank you, you might've just saved me some points.

[@markkennedy9767]

Im always so impressed with how Grant seeks to understand the processes behind mathematical thinking in his videos. Not many videos have the guts to do that. I like how he second guesses the famous mathematician's quote about genius being a light bulb moment with his better summation of it being more like a residue of experience (a mathematical maturity, I suppose) which for me is more on the money. Grant's one of a kind.

[@laurv8370]

+1!

[@oOPPHOo]

And it works similarly for all kinds of ingenuity from music to business and beyond. Phenomenal is the right description.

[@sugarandbones6272]

I'm crying it's truly so beautiful

[@Fermion.]

Me: Just going about my daily life. 3Blue1Brown: Ever wondered why slicing a cone gives an ellipse? Me: I wonder why slicing a cone gives an ellipse.

[@josephstalin7458]

Now I do

[@nihongojozu121]

@@josephstalin7458 you are everywhere. why? prove mathematically

[@subhrajitroy1477]

@@nihongojozu121 lmao...nice one mate hehehe

[@subhrajitroy1477]

Lit one bro haha

[@binhturtle179]

I wonder why they mentioned cone in the definition when slicing a cylinder also produced an elipse

[@m_riatik]

regular mathematician: QED 3Blue1Brown: *BADA-BOOM BADA-BING*

[@maurocamaraescudero1062]

MatrixWolf BADA BOOM BADA BANG

[@willking4512]

We should add a "bada-boom bada-bing" symbol to LaTeX for use at the end of proofs...

[@YourMJK]

BBBB.

[@m_riatik]

Will King haha yes! what do you suppose that would be?

[@michaelleue7594]

Is that 3 B's followed by 1 B?

[@Chicabaduk]

No, I've never wondered why slicing a cone gives an ellipse. But surely was wonderful to see why.

[@이준연-e6r]

팩트폭행 나쁜거임

[@bimalpandey9736]

0:04 "Suppose you love math ..." I clicked on a 12 min video about an ellipse, come on dude.

[@ShlokParab]

It's _just_ 12 minutes. I'm gonna be having to watch a 6 hour long video on conice after this..... Mmaybe after the next video in the recommendation list..

[@dolichakraborty6020]

​@@ShlokParabSelf study bruuuhh

[@AustiuNoMatterWho]

Idk why I love squishification so much as a word

[@TheNinthGeneration1]

Austin Thompson because it’s perfectly ridiculous

[@Jasper_4444]

And ridiculously perfect.

[@pet3590]

SQUISH THAT CAT

[@squeakybunny2776]

Wait till you Learn about "spaghettification" It's when an object gets stretched while getting closer and closer to the center of a Black hole

[@Jasper_4444]

Isn't that more or less the same thing?

[@laradimello5791]

While watching your videos I often wonder how can a teacher teach these topics without the softwares and animations you use. Like this videos are truly a form of art!

[@coopergates9680]

Do you make software, too?

[@drawingtheways3601]

You didn't get her commentary.

[@origamiandcats6873]

They've been doing it for thousands of years. You can slice a party hat on an angle or cut a canteloupe on an angle or cut a canteloupe into a cone shape and then cut it on an angle. Yummy.

[@nowonmetube]

@@origamiandcats6873 😒 Not everything is about eating you know!

[@softdorothy]

I would suggest you use their animations? Otherwise head out for some styrofoam balls, string and construction paper to make your cone.... I think the point you make is that visualizing these problems is what makes them graspable. So by all means you are going to have to do something visual as well.

[@JackFlashTech]

I like what you said about genius being a product of experience. Many people look at Einstein's Relativity, for example, as being an idea out of nowhere, but in its day, and for those in the field, it was not as strange as you would think. Einstein didn't just see something that no one was looking for; there were known inconsistencies between the theories of Maxwell's equations of how electromagnetism worked, and Newton's equations an how people thought electromagnetism SHOULD work. Einstein was trying to figure that out, and the Theory of Relativity was the result.

[@nonitathomas4620]

Yeah agreed. Most of Da Vinci's engineering ideas had already been looked at by other people. E. g. Brunaleschi.

[@scottsound4711]

Bahhh

[@person8064]

And he built on ideas put forth by so many, such as the Michelson-Morley experiment, which showed that, strangely, light travels at the same speed relative to you, no matter how fast you yourself travel.

[@qtheplatypus]

“Let’s focus our attention” well played sir.

[@ewthmatth]

Well there's two foci. I must have been on the wrong one :(

[@MrShmazoo]

Just returned to this video and was reminded of what a wonderful production it is. You are a gem, 3b1b. You have an absolute knack for mathematical communication, and I admire you passion for making the art more accessible. I’ve shown this video to many friends who ask me “why do you think pure math is beautiful?” or “Why should I care if it doesn’t have an application?” and I think you answer those questions better than anyone else I’ve ever seen. Bravo!

[@shoebmoin10]

I don't know what's the magic in your voice that i cant close the video until i watch it completely.

[@9308323]

Mathemagic. Okay, I'm out.

[@zes7215]

wrg, no such thing

[@cubethesquid3919]

DUDE ME TOO!! I'M EVEN GOING TO BE LATE FOR WORK BECAUSE OF THIS AND I DON'T EVEN CARE

[@joaquinel]

The voice of a fascinated man, in love with Math. Now that you mention it, it is not the standar imposed youtube documentary accent, it is hipnotic like Carl Sagan or David Attenborough. Passion.

[@RyanLynch1]

Skyscraper me too

[@kishanthakkar304]

The concluding idea was interesting too... “You can often view glimpses of ingeniousness not as inexplicable miracles, but as the residue of experience.... And when you do, the idea of genius goes from being mesmerizing to instead being actively inspirational...”

[@tetsi0815]

Can we have a t-shirt? :-)

[@ThreesixnineGF]

I want this on a T-shirt!

[@mika1998125]

I competely agree with that statement, this has inspired me many times i love math

[@kainotachi]

He could have made it full-circle and connected that idea back to art. While things like paintings and novels may also seem to be born from some mysterious and miraculous inspiration, in actuality the painter and author are drawing on various experiences from making previous works, as well as from studying the world and the works that others have created.

[@JorgetePanete]

Kishan Thakkar you used four dots instead of three

[@drone_better7757]

12:20 Only you would make sure the hands of a clock that spends 3 seconds on the screen are perfectly synchronised.

[@drone_better7757]

Sorry, it was supposed to be 12:20. Corrected.

[@IrrelevantPlease]

Swoopy _1 3Blue1Brown would have made sure the time stamp matched perfectly.

[@erik9376]

If you look at the GitHub repository for manim, you will see that he has a pre-made animation called ClockPassesTime that is configurable with the "run_time", the "hours_passed" and a "rate_func". With this template it is super easy to insert a clock anywhere in one of his videos that advances x amount of hours over y amount of run-time, and it will always be perfectly in sync. github.com/3b1b/manim/blob/db649d6576daab97b1e21312c4161b5526638bdc/mobject/svg/drawings.py#L378

[@edgepixel8467]

Erik .py? What sort of devilry is this?

[@erik9376]

That's a Python file. Not sure what you mean.

[@TioiraMusic]

I chose to study arts, languages and history just to getaway from Math due to a horrible teacher I had in high school. That is the way the education system works is Spain regretfully. You are forced to make a choice on branches of study when you are anything more than a teenager. Now I'm 37 and Math has been ever since growing on me. I found this video so beautiful I almost cried. Thanks a mill!

[@carlosraventosprieto2065]

Nunca es tarde! Yo estoy estudiando matemáticas por la uned y hay mucha mas gente mayor de lo que me esperaba. Algunos se cogen 1 asignatura por cuatrimestre y poquito a poquito disfrutando. Es una carrera preciosa!

[@jfbarbosaboro]

The genius is who explain things like this in such an EASY way for everyone!! Congratulations!! You are an AMAZING teacher!!

[@8Papa1]

10:22 Henri Poincaré said : "Mathematics is the art of giving the same name to different things." (La mathématique est l'art de donner le même nom à des choses différentes.)

[@realbignoob1886]

8Papa1 69th like reeee

[@PavanKumar-xv1hg]

lmao meme

[@jesussanchezherrero5659]

Je suis un grand admirateur de Poincaré. La science et l'hypothèse et tout ces oeuvres vulgarisateurs sont une vraie pépite.

[@chandramauligupta6787]

I asked my mathematics teacher for proof of those two definitions of ellipse yielding same thing (i.e. ellipse) and she asked from some other people who called the question stupid. But now video of 3B1B came and it's proof is his one of his favorites, I wish I could throw it in face of them. An amazing video, as always.

[@itays7774]

Your teacher sounds uninspiring and awful, it's a shame to see how little appreciation mathematics gets nowadays after all of the great things we've achieved thanks to it. I don't mean it in a "I was born in the wrong generation" way, I'm just quite disappointed by how mathematics is treated, especially by students.

[@chandramauligupta6787]

In India there's no creativity while teaching math and only those proofs are taught which are in book. Students are NEVER encouraged to work our something new, lateral thinking is nearly zero. Now of course this isn't the case everywhere but in most school this happens. Most people just think it's all about applying algorithm and nothing else.

[@rumfordc]

+DrNawMai Maybe back when Pythagoras was the only math teacher? lol

[@chandramauligupta6787]

Well there are movies on many social issues but people never change from a movie. You will see people supporting the movie but still staying trapped in stupid ways of teaching

[@itays7774]

DrNawMai well... yeah, there was a time when being a mathematician was at least considered honorable, and not a nerdy nerd nobody likes. It's sad to see how schools butchered the reputation of math

[@cil7ea]

"You can often view glimpses of ingeniousness not as inexplicable miracles but as the residue of experience. And when you do, the idea of genius goes from being mesmerizing to instead being actively inspirational." -3b1b ❤️

[@ItsJJOLO]

On the other hand, the animation quality is well explained and simplified in a mesmerizing way; and not only the animation but your voice. Very comforting and makes what ever topic you talk about comforting.

[@jamespotter1436]

Congrats! U got a 6th grader who knows essentially nothing about stuff like spheres, cones, and ellipses to understand the concept

[@Oxygenationatom]

Hows 8th grade bud

[@cat-des650]

i do not thing you understand it well🤡

[@Oxygenationatom]

@@cat-des650 i don’t thing you spelled all the words correctly

[@brightblackhole2442]

@@Oxygenationatom i do not thing he new what he talking about

[@Oxygenationatom]

@@brightblackhole2442 i do not thing the meth is mathing

[@AustinGarrett777]

Hi, physics major here. If you turn the cone upside down, and imagine it to be a graph of force as a function of (x,y) position, what you have are two decoupled simple harmonic oscillators. Thus any free trajectory along the surface of the cone has two components that oscillate with simple harmonic motion, which we know describes an ellipse. More insights into math from physics!

[@joaoluizmoraesgomes7778]

Nice

[@qizhengli5685]

Austin Garrett What’s the direction of the force you mentioned?

[@reubenadams7054]

Despite the likes, I'm not convinced this is right. It sounds legit at first, but I can't get it to work. Here's what I think you're saying: 1) 'Turn the cone upside down.' If we put the apex at the origin, that means the height of the cone at (x, y) is sqrt(x^2 + y^2) if the angle is 45 degrees (if it's not 45 degrees the height is proportional to sqrt(x^2+y^2)). 2) 'Imagine the height of the surface is equal to force.' Okay, so f(x, y) = sqrt(x^2 + y^2), but in which direction? 3) 'You have two decoupled simple harmonic oscillators.' For a simple harmonic oscillator, the force must be in the opposite direction to the displacement. If we assume you mean the force points towards the origin, then we would have f(x, y) = -xi - yj, since this is in the opposite direction to displacement (xi+yj) and indeed has magnitude sqrt(x^2+y^2). Now f(x, y) = -xi - yj does indeed describe two decoupled simple harmonic oscillators in the (x, y) plane, the solution of which is an ellipse in the (x, y) plane. But this ellipse would be symmetrical about the x-axis and the y-axis, whereas the ellipse on the cone is clearly offset. Perhaps you are talking about a particle travelling on the *surface* under the influence of gravity. This does indeed describe an ellipse, but the motion is not SHM. This can be seen by noticing that the potential energy is lower at the low end of the ellipse, meaning the kinetic energy and therefore the speed is higher. But if it were SHM then it would have the same speed at opposite points in the motion. Perhaps you are saying the particle is constrained to the surface of the cone and is subjected to the force f(x, y) = -xi - yj. But this would cause reaction forces from the surface of the cone, meaning the total force is not -xi - yj anymore but something different, the solution to which might not be SHM. Clearly there seems to be a connection somewhere, but I think you were too quick!

[@Yui-ee9mw]

That's why few hundred years ago, an astronomy physicist thought planetry motion is harmonic.

[@cl0p38]

So this is very the Sin comes from!

[@JoelDowdell]

When you talked about how people get inspiration based on their immersion in a subject leading to a certain thought process reminded me of Vihart's recent video on the utilities on a mug problem. (and how to ruin a bagel) Towards the end, she explains her though process and mentions that it all happened in a short amount of time. Specifically because she thinks about it a lot, meaning it was near the surface, ready to present itself.

[@becausescience]

Awesome video. Worked on me! -- KH

[@theanthropocentric272]

I hate the fact that you're wandering in utterly desultory manner, please stay focused. I've been waiting for follow-up videos of vector fields for 2 months!

[@samuelrodriguez8613]

Because Science Wow Kyle you're on here too? Makes sense, considering what you do haha love your videos too

[@vladtepes1753]

No TAKE on me

[@Akyo_Az]

Hi show, love the Kyle!

[@pcwave]

Loved the demo, but loved even more your discussion after it. This is what gives deeper meaning to what we do. Thank you!!

[@alexanderherbertkurz]

"Ingeniousness is the residue of experience" --- What a beautiful, deep statement. And thank you so much for all your beautiful maths videos. I hope they are going to transform maths teaching for ever. My children (15 and 18) do respond very well to them. While they are not so fond of churning for hours through Khan academy skill drill, they are struck by the beauty of the mathematics that shines through your videos.

[@MrRayeef]

This question have always been pondering in my mind since I took the first conics class. Thank you very much for such a concise and insightful explanation.😊😊😊

[@AdityaPrasad007]

There are no "non math" lovers. There are only 2 types of people on Earth, people who love math, and people who are yet to actually learn math properly.

[@fabulator2779]

Aditya Prasad and in India very less people Know what is maths.. no even the teachers..haha

[@AdityaPrasad007]

1/0 We can change that! I can see that is about to change due to channels like this.

[@muhammedberkonder7802]

So there are 10 people ?

[@cubethesquid3919]

Actually, I think this is true

[@endlessentropy9107]

"non math lovers"

[@Tsiferblat]

The concept of these moments of creativity being the residue of experience resonates with what was referred to in my undergraduate education as "developing an intuition," which I know I found comforting when I found myself stumped by proofs. I didn't feel as though I was just missing some talent or genius and therefore unable to do math meaningfully. Thanks for your videos. They've revived my curiosity and interest in pure math.

[@peckdec]

First of all I truly love your videos. Secondly, they are extremely useful for me at pinpointing exactly how and why I learn certain things so strongly, something which I have been trying to figure out for years. The short answer to this problem is that (whereas in contrast isolated facts do nothing for me) I have to have a certain amount of logically connected facts presented clearly enough, which then somehow click and produce a sense of true understanding. At this you are doing a phenomenal job. For those who are interested, I want to add finally that I find it very helpful to decompose the final problem into it’s logical parts and kind of learn the parts separately as isolated facts, which are then connected to each other. As an example from this video, I ”isolated” facts such as ”a sphere inscribed inside a cone touches the cone tangentially around the cone” and ”travelling any length of line tangentially from a sphere, the lines from that end point and tangent to the sphere are of equal length”, which when combined together mentally produce a pleasurable sense of coherence. Ok, sorry for this rant, and thanks again!

[@stevenelliott216]

Nice explanation. I really like the graphics. That must have taken a while.

[@vicrobot]

Its a well developed software by him. See:- github.com/3b1b/manim

[@abiuniverse]

I totally agree what you just said about being Genius is not a miracle/special .. It is the bi-product of lots of hard work and passion for hard and new problems.

[@abiuniverse]

Well how do you define that threshold ? :)

[@abiuniverse]

Ben Jackbag Mank X- axis as IQ and Y-axis as number of humans . Then I would suppose ( intelligent guess) it would look like a Guassian so what 'drop' you are talking about?

[@abiuniverse]

interesting point indeed. So my next question is what do you think about, How IQ works? I mean if u are really interested in some deep insights(mathematical) into certain problems and you are willing to work hard for it then your IQ would grow gradually, Is this trait a product of certain environment (e.g. certain TV show maybe inspired when you were child) or Is it genetic? or maybe both ? or certain other factors... Are you suggesting only certain people in the general population have that kind of unique combination of ENVIRONMENT and PARENTS and "other factors". Thats the reason there are so few numbers of such Geniuses exist..

[@avy0010]

Abhishek Sharma I'm about to cry right now great speech

[@abiuniverse]

:) :)

[@pi17]

Please prove a similar one for a Parabola.... It's a little difficult to imagine how a point is always equidistant from a line(directrix) and a fixed point (focus).... Thanks....

[@SimpleStory95]

How would you find the largest tangent sphere?

[@rishiagarwal926]

Yeah, I have the same doubt

[@yanxishan6575]

I believe that the answer is like this (I may be wrong): A parabola is an infinitely tall conic section. The larger tangent sphere would be both infinite in size and infinitely far away. Thus the second focus would be infinitely far away. You cannot directly prove it for a parabola because of this.

[@squibble311]

standup maths did this, go check it out

[@Hi-6969]

i wanted one for the hyperbola

[@thomasrosebrough9062]

What a beautiful sentiment at the end. We tend to glorify impressive acts of thought or creation or even physical ability to such a degree that we impede ourselves in doing so. Not that we shouldn't hold up people who do great works, but that we should remember to demystify them a little and not count ourselves out before we even try. It's silly but it reminds me of that Ratatouille quote: "Not everyone can be a great chef, but a great chef can come from anyone".

[@FadiAkil]

10:30 This's the essence of the intuitionism of Poincaré & Brouwer👌

[@aflower6955]

Everywhere it follows me KZbin School House EVERYWHERE I think maths and physics knows that maths and physics are my passion

[@kirenireves]

"Ingeniousness is the residue of experiments". Love it! I'm gonna use that one. This fits with another of my favorite ideas: "Curiosity is the cure for boredom". And so, since experiments are how you learn about things, putting it all together gives you: "Being bored leads to ingenious discoveries". ^_^ (boredom -> curiosity -> experiments -> Genius!) Love your channel !

[@gibbogle]

This phrase supports my belief that memory is a crucial component - the crucial component? - of intelligence. Without a good memory there is no residue. By the way, he said "residue of experience".

[@tonifasth]

I wish you could be cloned to every school in the world and then used as a maths and physics teacher. We would then have a lot more people interested in basic math and they would actually understand a lot more of what they are doing with all those equations and formulas. Your representations of this topic was beautiful and very easy to understand.

[@origamiandcats6873]

You could do a demonstration with fruit. Yum.

[@chiranjitray760]

While the demonstration mode is really captivating crisp and clear , application scenarios really is another ball game---it really brings forth the question whether your understanding has developed from within yourself or has just played along the the understanding of another person. Self realisations of topics are the crux to learning physics or maths. No amount of crystal clear demos can bring you the joy that you find by understanding something on your own for the first time, regardless of how small it is

[@AbhishekPatelab9]

Chiranjit Ray totally agreeing with you. The best teachers are the ones who provoke self realization and providing just the perfect amount of knowledge as a push.

[@nabeelhasan81]

Your videos remind me why math is important and beautiful at the same time. Thanks for making these videos

[@ADstaycool]

This guy is the best teacher. He genuinely wants to teach and equip his students with skills.

[@nirakshara]

Oh God!!! This is insanely beautiful...!! When you brought the two spheres into the picture, my heartbeats almost stopped!!!

[@xKuukkelix]

I have a latin test in 2 hours and im watching this for some reason

[@HeyKevinYT]

veni vidi vici

[@nowonmetube]

Good luck Jk

[@horatio3852]

you should find same video about latin))

[@aathish04]

How did it go?

[@biblebot3947]

Kevin vs. Gamingz is that classical or ecclesiastical pronunciation you’re speaking in

[@HarshadWarnekar]

I used to always think that cutting a cone, that way, would give an egg shape. And thought that it was not right to say it would make an ellipse. Thanks to this video for clarifying. :)

[@malignusvonbottershnike563]

Wow, I must've watched this video at least a year ago now, before it popped up in my recommendations today, and this time I managed to predict the proof before it was concluded. Whether that's the result of videos like these, or if it's a result of the massive amounts of wisdom I have obtained (unlikely...), I've become very slightly better at maths. Cheers Grant.

[@Kishblockpro]

You know, same, the last time i saw this was probably in 9th grade, i wasn't able to come up with it then, and although i remember seeing it back then now i was able to come up with it in 2 minutes, (of course with the introduction of the spheres), and since 9th grade i've done olympiad math, I'm older though, its hard to know if my brain has developed or if i really did become more astute, or whether i just had the details stored in my brain and i was secretly recalling them, you know what, thinking about this is a pain, why bother?

[@vrsgng]

I was watching the video as a break from studying Kant, and then came 10:50. goosebumps.

[@MATHalino1]

This video is a point of no going back. I will never see ellipse that boring again. The length of the major axis as the sum distances to tangent spheres is so elegant and mesmerizing.

[@NoriMori1992]

Oh man, I got nostalgic when you talked about curve constructions. I like to learn about things on Wikipedia and KZbin and the like, and every so often I find a topic that just clicks with me. This was ages ago; it might've been after watching a Vsauce video that mentioned cardioids and the like, I'm not sure. Either way, I ended up reading all about plane curves and how they're constructed. I was absolutely fascinated by how these curves are defined, and how similar they really are (especially the conic sections) once you understand their mathematical and geometric underpinnings (loci, foci, and directrices). Their constructions are satisfying to me in the same way that compass and straightedge constructions are satisfying to me: I love that it is possible to not only _create_ but actually _define_ these mathematical objects according to physical constructions that can be performed in the real world.

[@KrishnaPrasad-nq5xp]

Math is the language of the universe and you have mastered it!!! Great job and good video!!

[@dsdsspp7130]

krishnaprasad chandrashekar well yes but actually no

[@official-obama]

i still cannot find the exact perimeter of an ellipse

[@kushagra64]

12:30 only 3b1b would spend time syncing the end card with the music

[@MelvinDorkus]

i figured out the proof immediately because i had already watched a numberphile video with a similar sphere-tangents technique. thats one thing i love about math, one beautiful proof can give you a smart and simple technique for a completely different problem whether you expect it or not. i loved this video :)

[@ardaagn1105]

I had just started the video, and the visual proof of Gauss’s Formula just blew my mind! 0:18

[@cypher3612]

"Ingeniousness is the residue of experience" damn boy

[@sanaaluminum7272]

slicing cylinder diagonally also gives ellipse

[@romi9485]

I guess a cylinder is like an infinitely tall cone

[@markgerads975]

The left side of 10:04 can be seen as the exact same argument, with the note that [an infinitely tall cone is a cylinder].

[@brownie3454]

that’s like saying two infinitely long skewed lines are parallel

[@davidwilkie9551]

Orthographic Projection and Drawing, orthogonal cross-sectioning for Woodwork and Sheetmetal construction were once taught as part of a High School "Industrial" Course, which was where I became familiar with the techniques shown, and it's why Phys-Chem, and Geometrical Drawing and Perspective, are combined to provide intuitive visualization abilities to imagine assembly of the elemental components presented in KZbin video Science-Math lectures.., plus Radio and Electronics Hobbies.., and anything else interesting. "Genius", and pattern recognition, favor the prepared mind. Everybody has a capacity for some expression of the "Art", in their own way. Since watching this video, I've been reminded of exactly why I have the particular style of analytical intuitions I naturally use, because GD&P in conjunction with the hands-on skills, taught together, associate actual physical systems, slice by slice in the kind of cross-sectioning/perspective that is characteristic of the Superspin Quantum Operator, and the drawn symbolic representation of Polar-Cartesian Coordinates of this time-connection act. Learning by doing tends to allow practical people who "get actual knowledge and results", to be impatient with the complimentary skills of Artistic Representation, Math-Phys-Chem and Philosophy, of the observed Time Duration Timing Actuality, in symbolic short-hand. (Identification of relative infinities) So it's no surprise that some Theorists aren't allowed near experimental setups... ----- This particular video has become stuck in mind, the point of the Conics is aka Singularity Centre of Time Duration Timing Conception. Holographic Principle Imagery projection-drawing Actuality. Big thanks for the picture.

[@hareecionelson5875]

my favourite shape used to be the hexagon, but now it's neck and neck between ellipses and the unit circle

[@ChristophLaimer]

The video inspired me to think more about the ellipse: Are there ellipses with more than 2 focal points? Yes there are! See en.wikipedia.org/wiki/N-ellipse ... Now I'm wondering if this is just a math game, or if there are items of my daily life related with N-ellipes? How does a 3d-3-focal-ellipsokaedrion look like? ...

[@ganondorfchampin]

If something is in the shape of N-ellipse, there must be some reason for it. There is something that constrains the distance among multiple focal points.

[@truezikovoice]

Think of an elipse with infinite focal point all conjugated to where the magic happens!!

[@ganondorfchampin]

Zak El Aboudi that would just be a circle.

[@truezikovoice]

Yes, an incomplete circle. Infinite focal points does not mean each point is connected to its neighbor. You must zoom in and flip it 90 degrees to take a hold of the new dimention.

[@AnandKumar-ql1sv]

So cool...

[@davidappelgate320]

9:33 3B1B stumbling over his words!? I had to rewatch four times out of amusement!

[@canrex7540]

“Luck favors the prepared mind.” - Louis Pasteur

[@mmariokart231]

I watch this whenever I get really discouraged and it shifts things enough to make me excited to try again

[@pace_18]

this is my favourite 3Blue1Brown video

[@cinderwolf32]

i brought up ellipses at the dinner table earlier today, and the two definitions i knew... you're freaking me out.

[@mairisberzins8677]

Let's make this guy surpass T-Series. He deserves it.

[@abdelhakouinharoune7504]

T series is BS

[@harshranjan8526]

What?

[@fakefirstnamefakelastname8305]

Mairis Bērziņš 1) Not really 2) You came from pewdiepie vs t-series didn’t you? Pewdiepie himself didn’t want to be associated with t-series anymore, therefore the meme died. Your reference to t-series is invalid

[@mairisberzins8677]

@@fakefirstnamefakelastname8305 Are you dumb? My comment had nothing to do with Pewdiepie but rather than refferencing T-Series as the most subscribed channel on youtube. It seems to me like you are totally fucked up by the Pewdiepie vs tseries shit that happened that you can't think clearly and notice that everywhere. Go and rethink your life. This channel truly deserves a lot more subs than it already has.

[@squibble311]

you ruined my day by reminding me T-series is the biggest channel on the platform

[@elijahbuchanan2368]

Thank you so much for doing what you do. You have changed the entire way I think about math with videos like this, and you have opened my eyes to the workings of mathematical problem solving. I have always loved math, but you have shown me that it is an art. Math is no a whole bunch of distinct formulas and ideas, but a huge interconnected expression of imagination and logic. Thank you for what you do in spreading the message of math's beauty.

[@Tomyb15]

You and your channel are truly unique. You make this fantastic videos explaining a certain topic related to math and you *always* ask the question that pops into my head as if you are reading my mind. This ranges from a simple question in a video like "But why would these two seemingly different ways to define something result in the same geometrical thing (the ellipse)?" or a whole video by itself like "What does genius look like in math? Where does it come from?". Before this video was uploaded, I had a conversation with my cousin about how the nature of creative ideas is really mysterious and how it almost renders the merit of the person who had that idea completely nonexistent. Your timing is always so great!

[@ElysonPanolino-g5g]

when the topic of conic sections is taught to us, this is one of the questions that haunted me for years that proof is so beautiful for me

[@alexandrubutnaru8098]

The moment you realise there are 3 Blue pi-s and 1 Brown...

[@MAYONNAISEMOOSE]

Ive been decieved this whole time. Orwellianism isnt supposed to be in effect yet!!

[@dielaughing73]

Like on my t-shirt!

[@fakehesap1731]

@@Agvazela_Vega not everyone sees color everywhere they look

[@realbignoob1886]

3blue1brown , I didn’t know until someone told me lol

[@paulunga]

Apparently I'm a "non-math-lover" because my favorite part of this video was your use of an interrobang.

[@cereal_chick2515]

Wait, where??

[@yayouknow7]

So, I work with orbits (aka ellipses) daily and I never knew about the intersection curve of a cone. There is ALWAYS more to learn. I love math and your channel is great. Keep up the good work!

[@brinerustle]

I would love to see a second part to this video, showing how dandelin spheres also relate to hyperbolas and the parabola.

[@mattmonroe2979]

You know, @3Blue1Brown Videos are amazing... When I watch and learn, I really feel a connection with the content and the concepts underlying it. I can't help but remember the phrase, "The best way to learn anything is to teach others." I have a bit of a messy bookshelf and a bit of a messy mind, but occasionally some of you guys can break through and hold my attention long enough to share something really beautiful. Which I think is exceptional brilliance to say the least. Great Job to all those responsible for this content, and thank you to all those funding it, the spread of knowledge helps me in my day to day life, and occasionally has immediate functions that I hope will take me somewhere that I too may have something important and constructive to add to the conversation. Take Care Guys!

[@kariahola463]

Ever wondered why slicing a bread on an angle gives you bigger slices - however, as many as cutting straight ? Whoa - your bread just got bigger! :O Damn I'm hooked on 3Blue1Brown videos.

[@nl9570]

Lol. You are just going to have varied slice sizes though.

[@minutiaetcetera]

@Maselek Thanks for dressing up the trite sentiment in a creative way, but explaining what is not understood seems less cruel than capitalizing on it with a joke.

[@GoronRob]

Yo thanks for the Kant shout out

[@nullifier_]

I got clickbaited by a mathematical nipple

[@thanostitan.infinity]

Nihilistic Depths lmao

[@rogerab1792]

dude you need a girlfriend or something equivalent

[@wilddogspam]

Roger I wonder what's the mathematical equivalent to a girlfriend...

[@ionutradulazar8984]

Autumn Zuleeg god dammit

[@spamspamspambot]

So what you're saying is that rather than getting married, I should just buy a coffee mug with 2 handles on it. Topologically equivalent, but costs much less.

[@ANAND02120]

I really like the end lines of the video. " We can view a glimpse of genius as not an explicable miracle but as a residue of experience. When we do, then the idea of being mesmerizing turn to instead being actively inspirational. "

[@emilyscloset2648]

0:25 A proof showing that a specific set is a normal subgroup. I love how elegant the logic is

[@jamespfp]

12:23 -- "...not as inexplicable miracles but as the *residue of experience* ..." -- In other words (from my studies in philosophy, *LULZ* ) this is where intuition and science intersect. To be utterly pedantic about it, it is necessary to observe something in Nature in order to develop an Intuition for what is happening, or why it is the way it is. I particularly like Feynman's TV-ready "We can't know WHY, we do know HOW" philosophical approach, in this regard.

[@jamespfp]

"We cannot know WHY," is Feynman speaking about the measurement problem, Heisenberg Uncertainty. The manner in which we make measurements will sometimes result in "lost information" which we know must be there. The HOW, according to Feynman, is repeat scientific testing and experimentation, as well as geometric proofs to help decipher the curved data-sets. The benefit of the geometrical approach was a *astoundingly* rigorous formal mathematical definition for quantum electro-dynamics which won Feynman and others a Nobel Prize. Since this video presents an excellent reason for using geometry as a foundational educational component for mathematics, I think Feynman diagrams would make an excellent follow-up video topic! :D

[@jamespfp]

Also, as a physics-related aside, the concept of the "Light Cone" makes this all the more interesting. "Gravity Wells" could be another linked concept.

[@iskda1585]

I beat cataclysm In geometry dash while watching this video 😈

[@merpie1017]

Do tidal wave next

[@HariEaswaran98]

Is this channel beautiful or what?

[@rogerab1792]

it is what

[@piman7319]

De rigeur insult attempt from vapid dullard: "What?"

[@cleitonoliveira932]

It's pure love.

[@riverpearson5559]

As soon as you brought in the spheres my jaw dropped and I figured it out. Awesome visualization!

[@steelshade]

I can trace my life-long love of math to 9th grade when my algebra teacher, as an aside to the lesson, but with enthusiasm, demonstrated the equivalence of repeating decimals and fractions. It seems so simple now, but at the time the question had never occurred to me, so to see how simple it is was so delightfully surprising. It left me hungry for more.

[@DxBALLxD]

Did you know of the origin of the word genius? How rather than to describe someone as a genius it was used to say someone *had* a genius. Kind of like a muse, some otherworldly spirit visiting people in a single moment, like lightning striking down. Real works of genius often has their author say as much; "it created itself". In any case. I still don't see much in maths after this. I feel obligated to learn as much as possible due to it's utility, but I see no way to have fun with that.

[@windflier1684]

"Ever wondered why slicing a cone gives an ellipse?" Me:"Why wouldn't?"

[@anselmschueler]

Math isn't about why, it's about why not! *throws combustible polynomial at 90° angle*

[@isaacmiles-watt2758]

A cylinder is just one of the limits of a cone

[@neopalm2050]

I feel like mentioning that this is done by fixing a circular cross section then pushing the vertex infinitely far away. Also, that it needs to be in infinitely long cylinder.

[@AwesomepianoTURTLES]

neopalm2050 Yeah but you can just take a subsection of that and it would be a cylinder. Damn we need to start calling infinitely long cylinders just cylinders and call what we know cylinders now as cylinder segments.

[@neopalm2050]

Terminology doesn't easily change.

[@AjayKumar-fb3gx]

neopalm2050 thanks, I was just thinking how? And u gave the answer.

[@zikomo8913]

I don't get it, do we pull one end of cylinder(pipe) infinitely far away? But how would that make it approach the shape of a cone, technically it would still be a cylinder, right? Or is it that if we look at it from the other end the radius of the circular cross section of the other end of the cylinder would approach (apparently) to zero, thus looking like a cone? I haven't really studied any other, so bear with me.

[@davidjones5319]

Absolutely incredible and beautiful. I came across the Dandelin proof a free years ago! But took me a full day to understand it. Your animation made it so much more clear. The ellipse one is all that is needed. Though the parabola and hyperbole Dandelin proofs would also be a great animation some time in the future. But again, just a stunning animation. Thank you

[@alivateRocket]

Thanks for demystifying "genius" - you summed it up very well. It's feasible for practically everyone.

[@pr3nzlb3rg3r]

Good luck explaining the proof without the 3d animation.

[@Lonech]

lol just draw the cones and spheres duh

[@DreadKyller]

I know this is late, but this animation was based on a written proof, the written proof already exists, it's just more mathematically defined. showing it as a 3D animation just helps to get an intuition for why the math checks out.

[@alejrandom6592]

good luck understanding it

[@kapilg89]

Wow I wish we were taught the same way in school

[@Phanthomas8910]

This vid was as ingenious as your other ones and there are few that can explain complicated things as interestingly and as intuitively as you can, but seriously, I think you are overestimating people if you thing that people with no clue about maths can understand this. I'm pretty sure the most people who think maths are dumb and boring but produce the usual "Ugh?" after they see this. Nonetheless, amazing video, I wish my math teachers were as good at explaining things

[@WaldoRedelinghuys]

"residue of experience" is such a beautiful way to put it

[@macronencer]

Thank you! I've never seen this proved before, and it was just beautiful. I was fortunate, at the age of 11, to be faced with the problem of what shape would go into a cone to make a plane (I was making it out of card for a game involving marbles). I can't remember how I found out the answer now, but it must have been the first time I learned that an ellipse is a conic section - and it surprised me, for the same reason it surprised you - I couldn't understand where the symmetry could have come from.

[@jake_runs_the_world]

I love maths

[@jojojorisjhjosef]

I love colours

[@Farside-mk9ri]

I love maths to

[@DarkCloud7]

Does that mean they released a Math II? O_O

[@vishalshinde5252]

It's "math" without an s (totally not trying to be a snobbish a-hole but it really bugs me when someone says "maths")

[@kebman]

I love mats. I have one that is checkered, and another with dodecahedra. I used to have one with tesseracts, but it slipped away into the fourth dimension. :'(

[@ursoouindio]

i dont think non-math-lovers care about elipses

[@Evan490BC]

You could sell it to them as an "ice-cream cone packing problem"...

[@PerthScienceClinic]

Only because they don't know them. It is my experience that lots of mathematics can be sold as interesting if you don't introduce algebra :)

[@subhrajitroy1477]

@@Evan490BC Lmao.....or rather, a story called THE MYSTERY OF THE ODD-EGG SHAPE

[@ThapeloMKT]

👏🏾👏🏾👏🏾 beautiful 😔 most of my friends wouldn't care though

[@oceannuclear]

Move to a different place, get better friends.

[@joaquinel]

Ocean Wong No. No need. The place is here.

[@manishmayank4199]

I care :) I always do.

[@tategeiger5317]

I have noted the use of an interrobang at 4:00 and appreciated it

[@MatthewHanson-nf3xr]

I love this proof a lot. And your idea at the end that genius isn’t miraculous but inspiration is very similar to the definition of genius given by a favorite math professor of mine: “genius is the ability to see more things as interconnected than other people.” And the best way to improve one’s genius, then, is lord of hard work gaining experience.

[@MrsMars-kf5lb]

Me: vehemently denies being a math nerd Also me: gets goo goo eyes over this entire video

[@jimo8649]

I disagree with both you and Lockhart: I think the key step involved in coming up with a proof like this, which is not evident in the final "magic" proof, is an enormous amount of experimentation, trying out different things to solve the problem and very often failing or going down blind alleys that appear to help but don't quite get you to a solution. Trying this helps you build up an intuition, not just for geometry in general but for this particular situation. When you finally come up with a proof, it's probably far more complicated than it needs to be, involving steps that can be grouped together, or aren't even needed at all if you look carefully enough. In this problem, I bet the first proof someone came up with involved a lot of lines, seemingly unrelated, with their properties laboriously proved with angle identities. Then author noticed that some of these lines would be better explained as being the sides of right angled triangles, simplifying the proof from using angle identities to using triangle properties. Then they realised these infinite collections of right angled triangles could be even better explained as sections through those two spheres, making the proof simpler still. This is still an "ah ha" moment at the end, but a lot less magic than it seems. And at that point the proof already exists - the magic was just a simplification, not fundamental to creating the proof. Sorry, I know it's less romantic, but I think this is often the case in practice. It's also perhaps less inspirational: even when you're an advanced college professor, "ah ha" moments don't just appear out of your years of experience; you still need to spend effort poking and prodding every problem.

[@yourlordandsaviouryeesusbe2998]

You are absolutely correct. While solving many Olympiad math problems, I often come up with solutions where I needlessly make things to complicated, while the official solutions provided are much simpler and involve one or two crux moves. It seems as if someone with a very deep level of intuition can provide such a lucid proof, but in actuality it it's never the case. The crux move for any difficult problem can be discovered only through a lot of experience in previous problems or by spending time experimenting the various aspects of the particular problem. When writing the solution, one does not include the failed attempts and inelegant approaches. So it may seem to the reader that it is all one single stroke of genius.

[@Huntracony]

Today I learned (again) that heatwaves are not conducive to math(s).

[@wilddogspam]

Huntracony you need to first solve the heat equations with an appropriate heat pump.

[@jebbush3130]

*conducive

[@Huntracony]

Jeb Bush, Thanks, damn it. At least I already have an excuse.

[@pulseworks1663]

it's not a heat wave, it's summer time bro

[@Huntracony]

Dom Polsinelli, a few days ago was the hottest night ever recorded where I live, possible fellow tim.

[@James_Dolensky]

You actually blew my mind with the string and 2 thumb tacks. I always imagined both ends of the string tied to each thumbtack. I never thought of tying a loop and looping it around the thumbtacks. If I ever need to draw an ellipse for some reason that will be way easier

[@jarediannudalo6074]

I'm not very good at math but I love it when theres visualization like this .. this just blew my mind .. its so beautiful it makes me love math