# Feynman's Lost Lecture (ft. 3Blue1Brown)

You. May be aware that I'm a huge fan of the YouTube channel three blue one brown run by Grant Sanderson, grant makes excellent videos about math and mathy aspects, of other topics so I'm letting him take over my channel for today grant, take, it away a week. Ago I put out a tweet showing a peculiar, place where an ellipse arises, but, what I didn't mention is that this arbitrary, seeming construction, is actually, highly relevant to a once lost lecture by Richard Feynman on why, planets orbit, in ellipses, the. Construction, itself starts by drawing a circle and then choosing some point within that circle that's not at its center what, we'll call an eccentric, point and then, draw a whole bunch of lines from this eccentric point upto the circumference somewhere, and then, for each one of those lines rotate. It 90 degrees about, its midpoint and when, you do that for all of the lines an ellipse, emerges, in the middle out. Of context, this is a mildly, pleasing, curiosity, but, there's a much deeper form of satisfaction, on its way once you understand, the full story surrounding this front. And center of that story is Richard, Fineman who's, famous in a number of dimensions to, scientists, he's a giant of 20th century physics winner. Of the Nobel Prize for his foundational, insights in quantum, electrodynamics among. Many other things to, the public he's a refreshing, contradiction, to the stereotypes, about physicists, a safecracking. Bongo, playing mildly, philanderers, nonconformist. Who's heavily Brooklyn accented, voice you've probably heard either relaying, some bit of no-nonsense pragmatic. Wisdom about the only sensible way to view the world or, else some wry joke told through a crooked smile but. Two physics students he was an exceptionally, skillful teacher both, for his charisma and his uncanny ability to make complicated, topics feel natural, and approachable, many. Of the lectures he gave to a Caltech, freshman course are immortalized, in the now famous Fineman, lectures whose three volumes you can find for free online, but. Not all of the lectures he gave made, it into this collection one. In particular, a guest lecture given on march 13th 1964. Entitled. The motion of planets around the Sun survived. Only as an unpublished partial, transcript, with a smattering of notes buried in the office of one of fineman's colleagues until it was eventually dug up by Caltech archivist, Judith Goodstein despite. The absence of some crucial blackboard drawings to follow what Fineman was actually saying her, husband David eventually reconstructed. The argument of the lecture which, the two of them published in a book titled fineman's, lost lecture conveying, both the lecture itself and the surrounding story, in a really beautiful way. Here. What I'd like to do is give a more animated and more simplified, retelling, of the argument, that Fineman was presenting, the, lecture. Itself is about why planets and other astronomical, objects, orbit, the Sun in ellipses. It, altom utley has to do with the inverse square law the fact that the gravitational, force pulling an object towards, the Sun is inversely. Proportional to the square of the distance between that object, and the Sun but. Why, how. Exactly does that law give rise to an, ellipse of all shapes of. Course the gravitational, attractions, between different, planets and moons and comments and all of that means that no orbit is a perfect, ellipse but come on to a very good approximation this. Is the shape of an orbit, you. Could solve, this analytically setting. Up the appropriate differential, equation, and seeing the formula, for an ellipse pop out but. Fineman's, goal with this lecture was to do something special and to not rely on any heavy mathematical, machinery in fact, let's listen to him articulate, his own goal I am. Going to give what I will call an elementary, demonstration. My, elementary, does not mean easy to understand. Elementary. Means. That. Nothing very, little is required, to know ahead of time in order to understand, it except to have an infinite amount of intelligence. There. May. Be a large number of steps that are very hard to follow but. Each step does not require, already knowing calculus, already knowing Fourier, transforms. And so on. Yeah. That's, all just a little infinite intelligence I think you're up to that don't you I've. Done what I can to simplify, things down further from his original lecture, but that's not to say that a good deal of focus won't still be required, first. Things first we, need some definition, of an ellipse otherwise, there's just no hope of proving that that's the shape of an orbit some. Of you might be familiar with the classic way of constructing, an ellipse using two thumbtacks, and a piece of string use.

The Thumbtacks to fix the ends of a small string in place and then, pull that string taut with a pencil and try, to trace out a curve while keeping, that string taut it's. Similar to how you might use a single, thumbtack to construct a circle where, the fixed length of the string guarantees. That every point you trace out is a constant, distance from the thumbtack but. In this case with, two thumbtacks. What. Property, are you guaranteeing about. Each point that you trace out, well. At, every point the, sum of the distances, from that point to each of the two thumbtacks, will, be the full length of the string right, so. The defining, property, of this curve is that when you draw lines from any point on the curve to these two special thumbtack, locations, the, sum of the lengths of those lines is a constant, namely, the length of the string, each. Of these points is called a focus, of your ellipse collectively, called foci and fun. Fact the word focus comes, from the Latin for fireplace, since, one of the first places where ellipses were studied was for orbits around the Sun a sort, of fireplace. Of the solar system, sitting, at one of the foci for a planet's orbit making. Up a little bit of terminology for ourselves let's, call this constant, some of the distances, from any point on the ellipse to the two foci the. Focal, sum of that ellipse will. Get to the orbital mechanics in, just a moment but first I want you to turn back to that construction that I showed at the very beginning which, is going to come up again later in the story, remember. We take all of these lines from an eccentric, point of the circle to its circumference, and rotate, each of them 90 degrees about its center but what. On earth does this have to do with the constant, focus on property, I just described you. Could just take my word for it that this emergent shape is an ellipse but I think you'll be much more satisfied in the end if we take a little time right now for, a brief sidestep, into geometry, proof land, first. Off there are really only two special. Points in this diagram, there's the eccentric, point from which all the lines emerge and the, center of the circle so a, reasonable, guess would be that each of these is a focus. Of the ellipse given. The defining, property of an ellipse you know that you're gonna want to look at the sum of the distances, from these two points to well. Something, also. If you're doing any geometry problem, involving a circle you'll, very, likely want to draw a radius, of that circle at some point and to, use the fact that this radius has a constant, length no matter where you draw it I mean, that's what defines a circle so you're probably going to need to incorporate that fact somewhere, with. Those two thoughts in the back of your mind let's, limit our attention, to just one of these lines touching, some point P on the circle remember. What happens in our construction you, rotate this line from the eccentric point 90. Degrees about its center and the, geometry enthusiasts. In the room might fancifully, call this a perpendicular, bisector of. The, original line now. Take a moment to think about the sum of the distances, from, our proposed focus points to, any point Q, along. This perpendicular bisector. The. Key insight, here is that you can find two congruent, triangles, and use, them to conclude that the distance, from Q to the eccentric, point is the same as the distance from Q to P, so. That means that adding, the distances, to each focus, is the, same as adding, the distance from the center to Q then Q to P and there. Are two key things that I want you to notice here first. At the, point where this perpendicular, bisector. Intersects. The radius that. Sum is clearly. Just the radius of the circle and since. That radius is constant no matter where we draw it. The focal sum at that intersection point stays. Constant, which, by definition means. That it traces out an ellipse specifically. An ellipse whose focal, sum is equal, to the radius of the circle isn't.

That Me and. Second. Because the sum of these two lengths at every, other point, on that perpendicular bisector. Is larger. Than the radius meaning. The sum of the distances, to the foci from those points is bigger, than, the ellipses focal sum all, other. Points of this line have, to lie outside the ellipse, what. That means and this is going to be important is that, this perpendicular, bisector, the line that we got after our special 90-degree rotation, is tangent. To the ellipse, so. The reason that all of the lines that we drew earlier make this ellipse appear from nothingness is because we're basically drawing a whole bunch of the tangent, lines to that ellipse the. Reason, that that's going to be important as you'll see later is that this tangency, direction, is going to correspond, to the velocity, of an orbiting object, okay. Geometry. Poofiness done on to some actual physics and orbital mechanics. The. First fact is to use Kepler's, very beautiful, second, law which. Says that as an object orbits, around the Sun the, area, it sweeps out during, a given amount of time like one day is going, to be constant no matter where you are in the orbit for. Example maybe you think about a comet, whose orbit is really skewed close. To the Sun it's getting whipped around really quickly so it covers a larger arc length during a given time interval but. Farther away it's moving slower so, it covers a shorter, arc length during, that same time and, this. Trade-off between the radius, and the arc length balances. Out in just such a way that the swept out area is the same a quick way to see why this is true is to leverage conservation. Of angular momentum. For. Any tiny little time step delta-t, the. Area, swept out is, basically. A triangle right in. Principle you should think of this as a very small sliver, for a tiny time step but I'm going to draw it nice and thick so that we can better see all of its parts the. Area, of this triangle is one-half, base times height right that, base is the distance to this and what, about the height this little linked here how do you find that well. It's going to be the component of the object's velocity perpendicular. To, the line of the Sun what I'll call V perp multiplied. By the small duration of time, so. The full area is one-half. Times the radius times, V perp times, delta T now. Conservation, of angular momentum with, respect to a given origin, point like this son tells. Us that this radius times, the component of velocity perpendicular, to, it remains, constant so long as all the forces acting on the object are directed, towards that origin, well. Specifically, it says that this quantity times, the mass of the object stays, constant, but I mean the mass of the orbiting object isn't going to be changing, so. Our, expression, for the area swept out depends. Only on, the amount of time that has passed delta. T, historically. By the way this went the other way around, Kepler's, second law is one of those empirical, facts that led us to an understanding, of angular momentum and I, should emphasize this. Law does, not assume that orbits, are ellipses, heck, it doesn't even assume the inverse square law the, only thing needed for this equal area property, to hold is that the only force acting on an object is, directed, straight towards the Sun this.

Is A fact that Fineman spent a lot more time showing recounting, an argument, by Newton from his Principia but. It kind of distracts, from our main target so I figure assuming, conservation, of angular momentum, is good enough for our purposes here, I'll be it at some loss of element arity. At. This point despite my suggestive, drawings we don't know the shape of an orbit for all we know it's some wonky non elliptical, egg shape the. Inverse square law is going to help us pin down that shape precisely, but, the strategy is a little indirect, before. Showing the shape of the path traced out by the orbiting, object what, we're gonna show is the shape traced out by the velocity, vectors, of that, object, here. Let me show you what I mean by that as the. Object orbits it's velocity, will be changing, right it's, rotating, always tangent, to the curve of the orbit and it's, longer at points where the object is moving quickly and shorter. At points, where it's moving more slowly what. Will show is that if you take all these velocity, vectors and collect. Them together so, that their tails sit, at a single point their. Tips actually. Trace out a perfect, circle now. This is an awesome fact if you ask me the, velocity spins, around getting faster and slower and various angles but, evidently the, laws of physics cook things up just right so, that these trace out a perfect, circle and the. Astute among you might have a little internal light bulb starting to turn on at the sight of this circle with an off-center point. But. Again we have to ask why. On earth should this be true, Fineman. Describes being unable to easily, follow Newton at this point so instead, he comes up with his own elegant line of reasoning to explain where this circle comes from he. Starts by looking at the orbit whose shape we don't know and slicing. It into little pieces which. All cover the same angle, with respect to the Sun all. Right now think about the amount of time that it takes for the orbiting object to traverse one of these equal angle slices, and how that time changes, as you go to a bigger slice, well.

By Kepler's second law that, time is proportional to the area of the slice right, and, because these slices have the same angle, as, you get farther away from the Sun not, only does the radius increase but, the component. Of arc length perpendicular. To that radio line goes, up in proportion. To the radius so. The, area, of one of these slices and hence the time that it takes the object to traverse it is proportional. To the distance away from the Sun squared. In. Principle, by the way we're only going to considering, very small slices so there won't be any ambiguity, in what I mean by the, radius, and the little. Arc length will basically be a little straight line all. Right now think, about how the inverse, square law comes into play at. Any given point the force that the Sun imparts, on the object, is, proportional to one divided by the radius squared but. What does that really mean what. Force is is, the, mass of an object times, its acceleration the. Amount, that its velocity changes, per unit, time this. Is enough to give us a super useful bit of information about, how, the velocity of the orbiting object changes, as it goes from the start of one slice to the start of the next that. Change in velocity is, the acceleration, times, the change in time right, what. That means is that this change to the velocity is proportional, to the change in time divided. By the radius squared, but. Since, the time that it takes to traverse one slice is proportional. To the radius squared these, terms cancel out so. The change in velocity as, it traverses a given slice is actually, some constant, that doesn't, depend on the slice at all here. Unpacking, what I mean by that if you, look at the velocity at the start of a slice and, then you look at the velocity at the end of that slice and directly. Compare, those two vectors by joining their tails and you look, at the difference, between them the little vector joining, their tips this. Difference has the same length no matter which slice of the orbit you were looking at, so. As you, compare these velocity, vectors at the start of each slice they'll, be forming some kind of polygon, whose, side lengths, are all the same, also. Since, the force vector is always, pointing, towards the Sun as, you go from the start of one slice to the next that, force vector and hence the acceleration, vector is turning. By a constant, angle in geometry. Lingo, what, this implies is, that all the external, angles, of our polygon, are, going to be equal to each other. I know. That, this is a little tricky but hang in there remember, all you need to follow along is infinite.

Intelligence, It's. Worth reiterating just, to make sure it's clear what's happening, with this velocity diagram, the. Change from one vector to the next this little difference vector joining. The tip of one to the tip of the next always. Has the same length that was the consequence, of the perfect cancellation, between mixing Kepler's, second law with the inverse square law and because. Those constant, length change vectors rotate. By a constant, angle each time it. Means that they form a regular polygon, and as. We consider finer and finer slices, of the original orbit based, on smaller and smaller angles, for those slices the. Relevant regular polygon defining. The tips of these vectors in the velocity diagram approaches. A perfect, circle isn't, that neat. Hopefully. At this point you're, looking at the circle you're looking at the eccentric point and you're just itchin to see how this gives rise to an ellipse the way that we saw earlier but it's. A little weird right I mean, we're looking at this diagram in velocity. Space so. How do we use that to make conclusions about the actual orbit, what. Follows is tricky. But clever, step. Back and consider what we know we. Don't know the specific shape of the orbit only the shape that the velocity vectors, trace but. More, specifically, than that we, know that once the planet has turned an angle theta degrees, off the horizontal, with respect to the Sun that. Corresponds. To walking theta degrees, around our circle in the velocity diagram since. The acceleration, vectors rotate, just as much as the radius vectors, this. Tells us the tangency, direction, for each point of the orbit, whichever. Vector, from that velocity, diagram touches. The point theta degrees around the circle that's, the velocity vector, of our orbiting, object, and hence, the tangency direction, of the curve in fact. Let me just start drawing all these vectors as lines, since. All we're going to need to use is the information, they carry about the slope of the orbit curve the, specific, magnitude, of each velocity, will not be as important. Notice. What, I'm not saying is that the angle, of the velocity, vector at this point is Theta degrees, off, vertical, no no no the angle I'm referencing in the velocity diagram is, with, respect to the circles, Center, which, is almost certainly a little different from where the velocity, vectors are all rooted so. The question is what. Special, curve satisfies. The property that, the tangency, Direction the slope for, a point theta, radians, off the horizontal, is given.

By This vector, from, a special eccentric, point of the circle to, a point theta degrees, around. That circle from the vertical, okay. Is the question clear well. Here's, the trick first. Rotate. The whole circle set up 90, degrees and then, take, each of those individual. Velocity, directions, and rotate, them 90 degrees, back the other way that way they're oriented just, like they were before it's. Just that they're rooted in a different spot ah-ha, we've, spotted our ellipse but. We still have a little bit of thinking ahead of us to really understand, how this little, emergent, ellipse is related. To the astronomical, orbit. Importantly. I didn't just rotate these lines about any point I rotated, each one of them about its center, which, means we can leverage the geometric, proof we saw a few minutes ago and, this. Is probably the moment where you kind of have to furrow your brow and think back okay wait a minute what was going on in that proof again well. One, of the key points was that when you have two lines one, from the center of the circle and one from the eccentric, point both, to a common, point on the circles circumference, the. Perpendicular. Bisector, to, the eccentric. Line is tangent. To the ellipse and, what's. More the, point of tangency is where, that perpendicular bisector. Intersects. The, radial, line from the center what. That means is, that the point of our little, ellipse which is theta degrees off the horizontal with, respect to the circle center has, a tangent, slope perpendicular. To the eccentric, line and because. Of the whole 90 degree rotation, thing this, means that it's parallel to the velocity vector, that we need it to be so. This. Little emergent curve inside, the velocity, diagram has, exactly, the tangency property, that we need the orbit to have, and hence, the shape of the orbit must be an ellipse QED. Okay. Pat yourself on the back because, there is no small amount of cleverness required, to follow this first. There was this peculiar, way of constructing, an ellipse which requires some geometry savviness, to properly prove and then, there's the pretty clever step of even, thinking to ask the question, about what, shape the velocity, vectors trace out when you move their tails to the same spot and showing. That this as a circle requires, mixing, together the inverse square law with, Kepler's second law in another sly move but. The cleverness doesn't end there showing, how this velocity diagram with, vectors rooted at an off-center point, implies. An elliptical, orbit brings. In this very neat 90 degree rotation, trick I just. Love this watching. Fineman do physics even elementary physics is like watching Bobby Fischer play chess. Thanks. Again to grant and you should definitely go check out his videos on three blue one Brown.

*2018-07-25 23:58*

__Comments:__

*2018-07-28 14:01*

At 18:53 the rotation is to the left for the angles BUT for the shape the rotation snould be to the right!!

*2018-07-29 13:45*

Outside of of the first ellipse, there are what look like two circles that intersect each other where the lens encompasses the ellipse. What are those? Is there any significance to them?

*2018-07-30 04:11*

That link to the feynman lectures doesn't appear to work :'( It's just a cool drawing of a canary

*2018-07-31 12:49*

would the moon still orbit in a elliptical pattern if the hot iron core cools below curie temperature inside earths magnetic field

*2018-07-31 15:41*

Raise your hand if your IQ is too low to understand

*2018-08-01 17:52*

I have watched Feynman Lectures, MinutePhysics, and 3B1B Calc series. Youtube was dying to recommend this to me.

*2018-08-01 23:21*

That explanation was SUPER cool:-) I loved it!

*2018-08-03 00:39*

is that Robert Hookes inverse square gravitational law

*2018-08-04 04:02*

It’s a pity I cant like twice

*2018-08-05 17:56*

You make me to feel that i should study math, but i guess i will be down under in this courses... Boy this is hard

*2018-08-06 05:13*

@ 5:07 6.61+1.38 = 7.99 not 8

*2018-08-06 08:24*

Just some constructive criticism 3 blue 1 brown but all of these “isn’t that neat” things can really detract from what we are trying to learn. If it isn’t necessary try not to mention it, at least not at first because we are trying to learn a difficult new concept and then throwing little “fun facts” that in the end don’t actually matter can make it more difficult to learn because of there being too many things to keep track of and try and remember. Keep it simple.

*2018-08-08 05:38*

Don't feel bad if you are having a hard time understanding this. Keplerian physics from first principals is very difficult. The first time you really are brutalized by it is as a first-year physics graduate student. As a grad student, you are required to be able to derive Kepler's laws on the fly and do so during a test; I was required to do it at any rate. By the way, if you think the geometric proof is bad, try the analytic. You end up with something that is, not surprisingly, called an elliptic integral. Most elliptic integrals do not have closed solutions. It just so happens that the one for Keplerian motion, i.e. an inverse square law, is solvable. However, as far as solvable integrals go it is a worst case scenario that is going to take like 10 pages of various substitutions and every other dirty trick in your anti-differentiation arsenal including trig sub. I still have nightmares.

*2018-08-08 15:28*

Are we presupposing the collection of tangent lines uniquely determines a curve in this argument, or did I miss the justification in the video? I'm not seeing an easy way to justify this without the uniqueness theorem for solutions to differential equations.

*2018-08-09 20:13*

"Very little is required to know ahead of time in order to understand it, except to have an infinite amount of intelligence." Well, that pretty much rules me out.

*2018-08-11 04:32*

HEAD HURTSSSSSSSSSSSSSSSSSX

*2018-08-12 20:51*

"minutephysics"

*2018-08-14 15:24*

I don’t understand the area of the triangle, if someone could help me on that, I don’t understand how you get the area of the triangle

*2018-08-15 04:56*

Nope I can’t understand anything ;D

*2018-08-16 04:12*

excellent.... please let me know how to these animations...

*2018-08-18 02:02*

Oh my word this is beautiful

*2018-08-18 09:07*

I don’t see how any of this is possible when the sun vanished.

*2018-08-18 19:53*

Second year calculus:from celestial mechanics to special relativity, by David Bressoud covers this much more clearly for those with interest in a book. It's the first chapter or two, not much calculus needed, and he covers Newton's argument. After having read that book, then seeing this I'm spoiled by the former. Sorry Grant. Keep up the good work.

*2018-08-19 06:46*

Loving the content from the two of you! Amazing work guys.

*2018-08-19 17:43*

very nice

*2018-08-21 00:09*

Wow, fascinating. Concepts are much easier to understand with the diagrams. I have wondered, though, in an planetary orbit, I know the sun is always at one focus of the ellipse. I always thought there must be some significance to the other focus, but I have never heard anything about it.

*2018-08-21 05:39*

I made it to 11 min physics before nose picking become more interesting...

*2018-08-21 18:26*

Hi grant and Henry, great to see you guys together!

*2018-08-22 09:46*

Amazing !!! Thx for sharing

*2018-08-23 14:12*

Great video! Very easy to understand! Due to the narration.

*2018-08-24 08:00*

In 17:35 did you mean to say velocity vectors and not acceleration vectors?

*2018-08-25 09:19*

i feel so stupid now.

*2018-08-25 13:38*

obviously your talking about eccentric circles, but it sounded like you were pronouncing eccentric as in not quite normal. #1 is "normal behavior, circular? as in "its normal to go around in circles?" or use circular reasoning? #2 for readers here, do you pronounce the two usages differently?

*2018-08-26 16:32*

So what s the conclusion then?

*2018-08-27 03:14*

Besides elliptical orbits not really requiring the inverse square law, Feynman also mentions that gravity is not intrinsically an inverse square law. Elliptical orbits result from any centrally directed force. The force doesn't have to vary inversely as the square. Feynman figured out what the force of gravity would be for a plane of constant thickness stretching to infinity. (Instead of limited and spherical.) The force of gravity would then be constant regardless of the distance from the plane. In other words, the apparent law of gravity is really due to the shape and finiteness of masses. That is, it is the result of geometry. More well known is the peculiarity that the force of gravity is zero everywhere inside a spherical shell of constant thickness. And everywhere outside of the sphere, the force is equivalent to some mass located at its center. Gravity becomes less underground, and is zero at the center of the Earth. Newton concealed what became known as calculus (and the fact that he used it) until long after the Principia was published. (Calculus was tacked onto the end of a later book not on gravity.) Instead he gave geometry style demonstrations, similar to the ancient Greeks, a form which Newton thought was better by far. Principia was published in Latin. (Newton's early education was in Latin.) According to people that know Latin, the Latin in Principia is practically incomprehensible. And the approach was also alien to what a normal physicist (a natural philosopher) of the time would know. The first readers only knew that the man had to be some kind of genius to be that difficult. (Newton actually said he changed his mind from at first making it simple to making it difficult.) If Feynman gave up deciphering the original, raw Newton version, he was following in the footsteps of many.

*2018-08-27 05:29*

do you know how long did i wait for you guys to talk about feynmann in a video . finally

*2018-08-27 09:57*

Kepler's 3rd law was known to Mesopotamian astronomers, raising the period by the fraction 2/3 gives you the distance from the sun in AU ( to a high degree of accuracy ) You should look into Naram-Sin's metrological reformation and his royal gur ( cube ) of water

*2018-08-28 20:14*

THIS IS SO BEAUTIFUL!!!

*2018-08-29 11:30*

this is soo amazing. I'm trying to get my 10 year old daughter to really, really comprehend the basics of fractions. And the more I give her examples and explain fundamental principles to her, the more I understand why she never has been exposed to the true fundamental principles. Sure, the teacher showed some stuff and explained some tricks on how to solve them, but never, ever was she exposed to what it actually MEANS. Basic, fundamental understanding of things in as many ways as possible, is the root to REAL understanding. I am a failed maths student. But I try to teach my children that the ONLY path to REAL knowledge is truly understanding the things you are investigating. Feynman is a real hero imho. His level of REAL understanding surpasses most of our's. And his ability to teach us his understandings, is a blessing to the world of science. Although I've long since left the world of science (I'm now a humble web developer - pls don't judge) I still find these insights very fascinating at the very least.

*2018-08-30 20:15*

Great work! What did use for the simulation at 3:10 please ?

*2018-09-01 19:46*

is the spin of an object furthest away from the gravitational pull of another object faster than when it is its closest to the objects gravitational pull?

*2018-09-03 09:52*

Fucking BRILLIANT! Thank you!

*2018-09-04 11:52*

@Buzzfeed needs to make a video called "Everyday People React To 3b1b Videos"

*2018-09-04 15:56*

Aaah,... Geometry proof land, my favorite place!

*2018-09-05 20:46*

I probably wouldn't ever get the time to sit down and do these thing, much less understand it (the math, mostly) in a timely fashion. Yet I found the video extremely fascinating to watch.

*2018-09-06 05:36*

wow thats math.

*2018-09-08 03:21*

I guess I don't have Infinite intelligence.

*2018-09-08 12:24*

Knowledge-gasm indeed, Sebastian. Re the origin of the word "focus": I thought it was because the fireplace was always in the middle or in some significant spot of the cave/tent/yurt/room/cabin... Let's re-view the dictionary: 1644, from Latin focus "hearth, fireplace", of unknown origin, used in post-classical times for "fire" itself, taken by Kepler (1604) in a mathematical sense for *"point of convergence", perhaps on analogy of the burning point of a lens* [:-)] (the purely optical sense of the word may have existed before 1604, but it is not recorded). Introduced into Eng. 1656 by Hobbes. Sense transfer to "center of activity or energy" is first recorded 1796. The verb is first attested 1814 in the literal sense; the fig. sense is recorded earlier (1807).

*2018-09-08 22:21*

I wonder if the precession of an orbit over thousands of years doesn't scribe a spiral wearing the ellipse eventually down to a circular orbit over time. (The original eccentricity, ellipse , being a result of the planetary formation dynamics.) Also, I can't help wondering what would happen in a binary system. If the conditions were right could a figure eight orbit emerge? Thanks so much for making such complex topics visually accessible to the general public! You are awesome Grant!

*2018-09-10 13:18*

phew

*2018-09-11 11:51*

Yo, Feynman, Minute Physics, and 3Blue1Brown...? Yeah, I’m not gonna watch this video at all or anything... ❤️

*2018-09-12 03:20*

Why craks form in soils when it is exposed to water?#minutephysics

*2018-09-12 14:54*

Loved it...

*2018-09-14 05:49*

I have learn that I don't have infinite intelligence

*2018-09-15 02:32*

That "QED"

*2018-09-15 08:43*

Great topic... but damn, was this a hard lecture. I'll have to rewatch it again and slowly.

*2018-09-15 11:09*

In 2030 all youtube videos will be made by 3Blue1Brown XD

*2018-09-16 12:21*

and if alive today, he'd be an adversary of the #metoo movement.

*2018-09-17 08:00*

so what was the question? why planets orbit in ellipses or prove that all orbits are ellipses? I could just prove that circle is a special case of an ellipse, so naturally it will occur less often.

*2018-09-18 01:42*

ps: Amazing to see these relations! I wonder what kind of physical properties this virtual second point has if any. Regarding orbital mechanics ...

*2018-09-18 23:55*

Darn! That was way too many minutes for this channel :D

*2018-09-20 23:25*

just kidding! I love it!

*2018-09-21 23:45*

I'm stupid when it comes to math. But this guy makes me want to learn, and already did it a few times. Just have to watch it again and again until you get it, and practice... 3Bue1Brown, I wish you were my teacher... You are the best!!!!

*2018-09-23 15:15*

Youtube compression makes some REAL NEAT patterns out of the lines at 0:47 (I assume it's the compression doing moiree-like stuff)

*2018-09-25 08:34*

22 minutes physics. Love it

*2018-09-25 12:01*

I wonder about the other focus point (for example at 19:50) I mean if the first is the sun then where is the second one?

*2018-09-27 11:13*

Great watch - thank you

*2018-09-28 11:22*

Half an hour physics.. Love it!

*2018-09-29 00:30*

3blue1brown is my favourite channel.

*2018-09-30 04:32*

It's four in the morning, I barely have any knowledge of physics, and I've worked all day, and I'm tired. I only catch every other word, but your voice is beautiful

*2018-09-30 19:38*

This is brilliant! a prove using just geometry! wow

*2018-09-30 23:07*

Spectacular!

*2018-10-01 20:14*

So that why the earth is round.

*2018-10-03 02:46*

Why draw the line at the midpoint, instead of the golden ratio?

*2018-10-04 08:03*

omg that team; 3Blue1Brown and minutephysics

*2018-10-05 02:09*

20:30 Like 10 seconds of "aaaaaaaahhhh coooool that makes seeeenseeee"

*2018-10-06 04:13*

I am in tears, I really am. This video made my day!

*2018-10-06 15:41*

Bobby Fischer reference- A+

*2018-10-07 23:36*

3m16s I have done the solution of the differential equations. There is some error in my work which implies I'm getting an empty set of real values (x,y). Nevertheless, I get an ellipse-like equation (but, the sign is wrong on one of the terms). But, I (and I believe, Newton) never actually get an explicit function for x(t) and y(t). You get t = (an elliptic function) integral of r*dr/sqrt(quadratic in r). But, one would need to compute the integral explicitly (which is do-able) then use Lagrange Inversion to get r as a function of t. That is the challenge that every physicist & mathematician stops short of. But, that's where I begin my work.

*2018-10-08 07:12*

AAAARRRRGGHHHH MY BRAIN FEELS LIKE ITS GETTING ALL CRUNCHED UP!

*2018-10-08 11:15*

Wow, hes amazing, also this channel is amazing

*2018-10-10 03:37*

158 people had no »infinite intelligence«

*2018-10-11 13:59*

This made total sense. Super easy to follow. Thank you!

*2018-10-12 22:23*

You must be the Modern Feynman, 3B1B. I am watching your works and definitely loving it! Make more videos. :)))

*2018-10-14 03:50*

*How does he make the animations with the changing numbers?* I doubt he does the math for each frame and types it in. Is there a computer program that does this?

*2018-10-15 04:59*

I’m here because of Feynman

*2018-10-15 09:08*

I would like to know who animated this and how plus really good job on explaining this

*2018-10-16 10:55*

Well I completely forgot what tangent means so I got lost in the first minute

*2018-10-16 14:41*

Theta degrees? Or radians?...

*2018-10-16 21:52*

great video!

*2018-10-17 00:46*

Oh man, this is difficult! Infinite intelligence? More like infinite attention span. Fun though.

*2018-10-18 15:15*

I think I understand all the steps but I don't understand exactly what it is proving? Really! (I understand the different relations/correlations but not the point.) Is it that the mass, speed and direction/vector of the object determines the path?

*2018-10-19 08:11*

Infinite inteligence. He is talking about the emergence of the universe from nothing. The universe is self contained. By definition. You understand because you are standing in the midst of it. Who could you be?

*2018-10-20 06:54*

Speaking of science, planets don't revolve around the sun , sun and planets move together on relative motion. Sun moves too due to the Planet. So if you keep the sun stable and observe the motion of would be and ellipse.

*2018-10-22 01:52*

I discovered this on accident when creating a level in geometry dash

*2018-10-23 15:59*

This proof is a good one, except for the fact: everything would work out the same way if had just picked the center point at the very beginning and instead of a ellipse got a smaller circle. Try it)))

*2018-10-25 03:00*

Yup. All that makes sense.

*2018-10-25 12:12*

I always get annoyed and irritated when I watch 3blue, because I don't understand anything... I guess I dont have an infinite intelligence

*2018-10-25 15:10*

2 of the best

*2018-10-26 10:45*

I can draw elipse by single thumbtack !!

*2018-10-27 07:06*

i like 'gravity

*2018-10-27 10:26*

what if you pick the second po int outside the circle? AFAIU not every orbit around the sun is elliptical.

*2018-10-29 04:38*

3blue1brown and Feynman belong in one set regarding their ability to explain.

*2018-10-29 16:59*

Nice use of QED ;)

*2018-10-31 13:53*

So if the sun is one of the earths foci then what is the other focus?

*2018-11-01 18:44*

Sorry. Between the F#%^ commercials and your intro I lost interest. But not enough to let you know

*2018-11-02 07:55*

I am crying with tears of joy at the beauty...

*2018-11-02 21:01*

Wow, I would love it if you would please DESTROY flat-earthers!

*2018-11-03 05:33*

I think I'll have to watch this one a couple more times. But one things for certain, I wouldn't stand a chance of wrapping my head around it without your excellent animations.

*2018-11-04 13:02*

nice

*2018-11-04 23:26*

it's not lost anymore, right? :)

*2018-11-06 21:54*

joli!

*2018-11-06 23:59*

My infinite intelligence is not on point today

*2018-11-07 07:23*

21minutephysics

*2018-11-08 22:22*

Weird recommendation, luckily I was smart enough to kinda understand that someone is trying to explain why our planet orbits perfectly on an eclipse somehow despite how eccentric it is. Also I am appreciative of another way to make perfect circles. Well I guess I might just keep this fact in mind when I'm somehow meeting a Physics nerd and could actually start a conversation with this. Thanks.

*2018-11-09 09:42*

I subscribed to his channel a few years ago hoping to learn something and get some knowledge but all I'm realizing is I understand nothing and I think I'm getting dummer

*2018-11-11 04:37*

The bestway to watch 3Blue1Brown videos if you lack infinite intelligence is to keep your thumb on the spacebar and pause when you don't understand what's being said. Or rewind the entire video I don't know.

*2018-11-12 00:01*

PLEASE CAN YOU DO VIDEOS ON-1.WHY INTERNAL CONSTITUENTS OF A SYSTEM CANT AFFECT IT EXTERNALLY?( ie a person inside a box cant cause the box to move by hitting against the wall of the box from inside...I think this is also reason why we doubt emdrive cant work)2.WHY WE NEED FORCE CARRIER PARTICLES TO DESCRIBE FORCE?3.CAN WE EVER CHANGE THE LAWS OF PHYSICS INTO WHAT WE LIKE?

*2018-11-13 23:04*

This is some analytical geometry bullshit! J/K, great video.

*2018-11-14 04:56*

"To a very good approximation [an ellipse] is the shape of an orbit." Mr. Sanderson, I am a big fan of yours. This is one of the first times I feel compelled to point out that you're wrong (because you usually aren't). Planetary orbits are not elliptical, they are a non-symmetrical ellipse-like shape called a hypotrochoid (looks like a flower -- think of a spirograph). I thought this was common knowledge at this point but I stand corrected. https://malagabay.wordpress.com/2013/02/28/the-flowering-of-celestial-mechanics/

*2018-11-15 19:53*

By 6:18 I suddenly understood all of it! It was a great feeling

*2018-11-17 10:08*

This basically a normal 3blue1Brown video, so what was the point of hosting it on the Minutephysics channel?

*2018-11-17 18:49*

So does this mean that any orbit that is circular (put in that orbit due to, for example, rocket propulsion) will eventually work its way to becoming elliptical? If this is the natural way objects orbit due to gravity without adding any energy to the system, that means the only way to maintain a circular orbit is to constantly use some form of propulsion. Right?

*2018-11-18 07:24*

That's sheer genius and the visual explanation is brilliant! The graphics are beautiful and made the understanding much easier, wish I had that back in college! If I may, for clarity, I think the sentence at the 7:34 mark could be changed (perhaps in the subtitles) from: "So, that means adding the distance to each focus is the same as adding the distances from the center to Q, then Q to P" to: "So, that means adding the distance *from Q* to each focus is the same as adding the distances from the center to Q, then Q to P". I cannot believe I did not know the @3Blue1Brown channel until today, thanks @minutephysics!

*2018-11-19 03:09*

Question: in the geometry proof, will the rotated line from the eccentric point always cross the radial line from the center? If so (which I assume), how do you prove this?

*2018-11-20 03:44*

Do u ever cry cuz ur so stupid?

*2018-11-21 08:56*

You remind me of a spirograph!

*2018-11-21 19:38*

3b1b não é muito humilde, deveria ensinar a fazer animações

*2018-11-22 06:55*

Great video. and the snapshot from the game of the century is spot on

*2018-11-23 08:07*

i think we need another channel to analyze 3b1b videos to make it more detailed

*2018-11-23 15:17*

I wonder how this could describe the complex magnetic soup of ions we live in and their connection to the sun.

*2018-11-25 08:42*

Reminds me of lens law in electrical engineering. Stars and planet's gain and lose mass so their orbits would change as well. Consider the expanding earth theory, multiple suns theory as well.

*2018-11-26 10:26*

the eccentric point at 00:40 is the top of an inclined cone coming out of the screen,the radiating lines are along the direction of gradient ,after the 90 degree rotation they become vertical to their original direction and what is vertical to the gradient?...the contours of the inclined cone which are all ellipses and we are back to the old conic sections geometry

*2018-11-26 14:06*

I have once asked my physics teacher this question, why do planets orbits around sun elliptically

*2018-11-27 08:40*

PLEASE can u make one about this principle explanation of 2nd Kepplers Law? please, please, please

*2018-11-29 07:17*

Wow, how did this NOT show up in my subscriptions?

*2018-11-29 14:24*

Basically the base is the Radius right then in order to figure out the height of the triangle he took one of the components of the velocity. Now you may ask where does the component of the velocity arise from well think about this way the velocity of the object is moving in two dimensions the x and y dimension. So from this we know that the main velocity is a combination of these two x and y component velocity. So he took one of the component but now you may think where does the change in time come from well that's easy the formula for the magnitude of the velocity is v = change in distance / change in time . Since we want to known how much distance the velocity component traveled in given amount of time we just multiply v with change of time which gives us the distance it travel hence leading to the height of the triangle. Hope that helps

*2018-12-01 03:22*

I can follow the walk through but I never could initiate any of this. I do not see patterns this easily. I just marvel at Feynman’s insights.

*2018-12-01 23:31*

5:14 btw, the german name for the fokus in an ellipse is "Brennpunkt" (burning point in english)

*2018-12-03 23:24*

Nice droste effect at 1:01

*2018-12-05 20:04*

Hm, I see a problem: you proved that both ellipse and the trace of the planet have the same property (will have same direction of the speed vector for any angle theta). That doesn't mean thought that these two shapes are the same.

*2018-12-06 19:54*

How did you get the velocity vectors for the orbit when you don't have to orbit to calculate the vector?

*2018-12-07 19:01*

Hi, what software do you use to create such brilliant videos? Would be great if you could share. They are absolutely fantastic.

*2018-12-09 09:09*

I'm going to have to be a little odd here and say the "surprise ellipse" isn't really a surprise, it's pretty obvious...

*2018-12-11 01:18*

The velocity bit though, that might be one of the coolest things I've seen, that's amazing

*2018-12-12 10:00*

Wouw I thought lol I’ll never get that stuff it sounds too complicated

*2018-12-13 13:15*

THIS IS ONE OF THE BEST PHYSICS VIDEO I'VE EVER SEEN

*2018-12-15 04:32*

Feels good when all that high school physics finally make sense.

*2018-12-16 03:51*

"To a very good approximation [an ellipse] is the shape of an orbit." Wanted to point out to those who want to scratch their heads even further over this.. planetary orbits are strangely not elliptical in observation, they are a non-symmetrical ellipse-like shape called a hypotrochoid (looks like a flower -- think of a spirograph). It's super fascinating why and this guy did a decent job at explaining some of the reasoning. https://malagabay.wordpress.com/2013/02/28/the-flowering-of-celestial-mechanics/

*2018-12-17 11:00*

Does this mean we fully understand the dynamics between gravity, velocity and mass? And there is no such a thing as curved space-time, but only simple mechanical relations?

*2018-12-19 02:27*

原来是这样，太棒了！！

*2018-12-19 15:10*

Not to sound ignorant, but... Can someone please explain how having this knowledge is beneficial? Please?

*2018-12-19 18:12*

Holy shit my head hurts Why I even watch these. It just physicaly hurts me.

*2018-12-20 17:34*

This assumes space is flat, whereas the space is curved!

*2018-12-22 01:36*

All I can say is ... WOW Who are the over 200 idiots that gave this a thumbs down ? I notice that none of them had the courage to post comments, instead they just do a drive by thumbs down, oh well, the loser idiot at the back of the class who is failing has to make his mark someway ..LOL ...

*2018-12-22 19:25*

Hun?

*2018-12-24 16:28*

This is beautiful! What software did you use to animate this video?

*2018-12-25 07:04*

there is a myth going around that says 3Blue1Brown is keeping Feynman in the basement after this video, I believe it

*2018-12-25 16:15*

Relating the unknown [the ellipse] to a well known [the circle], is a process of seeding that effectively adds a 2nd deck of cards. It inserts a wedge between some related variables & adds relations to other known ones. Its the same problem we face when we have 2 eqns & 3 unknowns. Additional relations must be added. From there we follow the questions that arise while always having the target in focus. In this case here, Feynman added a twist of sorcery, at a crucial point of discovery he retraced his initial steps endeavoring to find a problem that yields his solution. The genius makes a crucial stop, backs up & asks how did I get here, answering that question reveals the dynamics of going from a to b, inside the new problem he has created. I remember seeing a vid on this strategy in some advanced Math course. Whats crucial to solving a stubborn problem is recognizing that there is not enough info, & the problem itself must be changed to a form that has enough info. There are multiple possible cause to an effect & thats usually the source of extra info

*2018-12-26 01:27*

no i'm not crying :')

*2018-12-27 09:27*

brain.exe stopped working,am I right

*2018-12-28 04:32*

MIND BLOWN

*2018-12-28 21:16*

and why the aceleration space for very excentric orbits draw a cardioid?

*2018-12-30 15:25*

I actually understood this to some degree. Neat

*2019-01-01 04:57*

I have infinite intelligence.

*2019-01-02 08:14*

Its like our “ellipse” is just a perfect circle written over the spacetime warping caused by effect and our physical laws are a differential gradient supported by our point of view. Heady.. Shooting in the dark isnt what i like but this is evidence of mathematical homeostasis despite some disruption. Blown away as usual!

*2019-01-03 17:43*

do perfect circles exist in nature?

*2019-01-04 18:42*

*NICE AND THICC*

*2019-01-06 10:29*

Grant is the only guy I would go gay for

*2019-01-07 11:29*

I just realised again why I hate maths.

*2019-01-08 03:10*

thank you 3Blue1Brown... now i have to sweep my brain off the floor cause it leaked through my ears

*2019-01-09 07:32*

The one thing that still isn't clear to me is why you're allowed to rotate all the velocity vectors about their center to conclude that the path is an ellipse. One could just as easily rotate the vectors about any other point and come to a different conclusion.

*2019-01-11 06:38*

A big like for Grant presenting this beautiful marriage of maths and physics.

*2019-01-12 13:36*

Perhaps the circle is a pictorial representation of the power displacement of the system and the ellipse is the pictorial representation of the displacement of the planet in the system. Used for beings in higher dimensions to understand whats happening with the sun and that planet. just a thought haha.

*2019-01-13 05:04*

Who else tried to wipe off the black spot at 1:43

*2019-01-14 08:36*

This was incredible... elementary but the way it was explained was just great...

*2019-01-16 01:59*

This video shows that the tangential line at the intersection of the ray starting from the center of the velocity circle and the ellipse at the velocity diagram is parallel to the velocity direction at the intersection of the ray and the orbit of the star. This indicates that the ellipse can simulate the orbit, or the orbit may be an ellipse, but it does not mean that the orbit must be an ellipse. My question is: when we know the tangential direction at the intersection of each ray starting from a fixed point in the graph and this graph,is there a theorem that can determine the shape of the graph ?

*2019-01-16 16:53*

4:14 omg 3b1b hand reveal!!

*2019-01-18 09:05*

Day 3: I have used the concepts in this video to clear up 4 of my doubts, again.

*2019-01-18 15:50*

In fact there is a generalization of the ellipse 'problem ' to 3 , 4,5, or n dimensions. And by the time you work in 5 or more dimensions you are solving the {approximate} gravity wave equation --- Poisson's Wave Equation. I am working on this problem in n dimensions. By the way it {this problem -- Kepler Feynman problem?} has some connection with the factorizing of integers. Can you imagine !

*2019-01-19 12:29*

Henry only agreed to this so he could watch a new 3Blue1Brown video before the rest of us! D:

*2019-01-20 13:54*

Awesome presentation!

*2019-01-22 02:14*

This also explains why large masses in any system such as the solar system are round...dosent it?

*2019-01-23 10:35*

If the ellipse satisfy the tangentsy it means the orbit could be an ellipse, that I understood (more or less). But why dose the orbit MUST be an ellipse? Couldn't the orbit be a different curve that satisfies the same conditions ?

*2019-01-24 19:21*

Amazing! I just love 3Blue1Brown! And of course I love minutephysics, too. ;)

*2019-01-26 16:03*

There is nothing ignorant about your question it is an understandable one. Aside from the obvious and its implications It teaches you profound truths if you can see them. The ellipse is an Archetypal pattern that engenders forces that manifest the nature of reality. So it's not JUST allot of spacetime geometry - if you can see it , there are keys to understanding life and death within it. For instance the forces generated towards a point in space exhibit how forms become manifested out of an unmanifested aspect of a unified duality. It shows the relations between creation and destruction and how those forces are joined together in a symbiotic relationship in an eternal state. So seeing that you can therefore apply it to every form in the manifested Universe, including your own individual existence.

*2019-01-27 08:55*

I see a great amount of work was put into this video! What software did you use to create all these animations?

*2019-01-28 06:53*

Imo this is not a very elegant proof, it requires way more steps and is much more convoluted than the algebraic derivation. Also, the whole thing is backwards, the observational evidence already gave us the fact that orbits are ellipses and kepler's second law. The real trick was figuring out from these a) conservation of angular momentum and b) and most importantly, the inverse square law. You're supposed to start at ellipses at arrive at inverse square law not the other way around. The other way around is easy with a bit of algebra.

*2019-01-28 19:07*

So amazing, so incredible!

*2019-01-29 16:21*

Please, don't try to speak Latin. You mispronounce every single word. Stick to English.

*2019-01-30 07:41*

I would immediately commend all viewers to enjoy "The Mechanical Universe" where Dr. David Goodstein is the presenting introductory lecturer. All of the series to be found on Youtube, and is a great touchstone of entertaining physics learning, featuring what was (for the time) cutting edge 3D graphics, decent even today, decades later.

*2019-01-30 12:15*

KSP will never be the same :)

*2019-01-31 15:38*

I obviously do NOT have infinite intelligence

*2019-02-01 07:07*

I always thought it was bc a force is always felt most 90 degrees in the orbit ahead

*2019-02-02 16:45*

Something tells me calculus (using small slices of curves) and Fourier transform (velocity and position diagrams) we're used in this proof somehow, but I am not sharp enough to unveil it.

*2019-02-03 08:56*

Did anyone else get bothered by the multiplication sign he used and kept thinking it was a cross-product?

*2019-02-04 23:39*

The entire time in the end I was just waiting for them to move the right ellipse into the left hole, but they never did ;_;

*2019-02-06 02:32*

Thanks for this excellent video, really enjoyed it, and actually used it as an aid whilst discussing orbits with my kids. Much appreciated

*2019-02-06 16:03*

I'll have to rewatch that several times lol

*2019-02-07 00:07*

He really meant by "infinite intelligence" an infinite tenaciousness in applying a small unit of intelligence along a program that over time will solve the problem.

*2019-02-08 03:15*

13:57 why is the area proportional to the radius squared?

*2019-02-08 20:54*

QED!!! LOL Good pun

*2019-02-09 09:11*

i guess everyone sees he resembles Cornell Wilde (most handsome hollywood star).

*2019-02-10 04:55*

Well THAT was clear as mud. Picture some mud. What is it? It's tiny particulate entities, usually towards the opaquence(?!?) end of the transparence/translucence/opaquence(?!?) continuum, suspended in some volume of water. So, I've got a circle of mud on my living room floor. Don't ask why it's a circle. It has to do with surface tension, and the second law of thermodynamics, as applied to, well, mud on my living room floor. The COOL thing is, the more of these videos, in this channel, that I watch, the more WATER the authors are pouring into the circle of mud on my living room floor. And what does THAT do? INCREASES the SIZE of my mudcircle, (that's my 'collection of knowledge'), and CLARIFIES the overall APPEARANCE of my mudcircle. To put it another way, if I keep watching these things, I might finally get the education in Mathematics that I somehow failed to absorb back in my school days. I THINK I CAN LIVE WITH THAT!!! Thanx guys...

*2019-02-11 03:20*

This is so much more convoluted and difficult to understand than just deriving using calculus. You even went most of the way to “inventing” calculus with your differential velocities.

*2019-02-13 00:56*

Personally, I think solving the differential equations is a far more satisfying experience.

*2019-02-14 12:27*

The ellipse has all the properties we want but is it the only shape that satisfies those properties ?

*2019-02-16 07:00*

Please, don't try to speak Latin. You mispronounce every single word. Stick to English. Also, you just proved that the tangents to the curve must have the same orientation as those of an ellipse. You (as well as the lowest-IQ Nobel-prize winner) still need to prove that the ellipse is the only curve with such property, which is intuitively true but not elementary.

*2019-02-17 22:16*

There is always someone who treats great explanations like this as an opportunity to pretend they have a superior intellect. I recommend you take a chill pill.

*2019-02-19 03:56*

Feynman didn't claim it was an elegant proof nor that it was an easy proof. He claimed it was an _elementary_ proof. Elementary means that you don't have to know very much information about the subject to follow it. An elementary proof pretty much only relies on very basic facts. Often, elementary proofs are inelegant since there's much to say. Often, elementary proofs can be difficult, since you need to connect the basic facts. (Also the whole inverse square law thing is _also_ mentioned in the video.)

*2019-02-20 12:11*

fosi

*2019-02-20 20:11*

Hey 3Blue1Brown!!! You said mass of sun wouldn't change at 11:01 in angular momentum conservation But it is changing as sun is constantly using hydrogen atoms to make helium atoms and some heat and light energy which escapes the sun therefore mass escapes the sun as E =mc*c So that means the mass changes. Can you explain on this query

*2019-02-21 04:46*

@18:55 What shape do you get for this *if you had chosen a point actually on the edge of the circle as the second focal point?* I know that might sound ridiculous, because "how could an object orbit about a point that it would collide with?" but don't forget that neutrinos can pass through the Earth. Lastly, compare that shape to an electromagnetic field. Do electromagnetic fields arise from electromagnetism or are they the orbit of very small particles that can pass through the object they orbit? i.e. are they caused by gravity??? http://vixra.org/abs/1807.0364

2018-07-27 07:02Did anyone get the video? Like, actually understand the entire thing? Cause that was some real tricky mathematics right there.