New video (Early view)! Dandelin Spheres

Grant you may have lived a sheltered life if you haven't met people who aren't at all readers. I know almost all people can read to a functional level, but a lot of people have zero interest beyond that and are suspicious of those of us who do..

Toby Archer

awesome, as usual :) also, you saying "please do pause, and just try to carry on without me" really encouraged me to try and finish the proof by myself. i admit that i very probably wouldn't have tried if you hadn't said that. (but maybe i am more inclined to laziness while watching your videos because i consider myself a CS guy and not a Math guy.)

Oren Milman

Grant - loved it ... I hope someday in your essence of linear algebra or even abstract algebra you will help me understand why quotient spaces are so important. I know they partition spaces. I know that they obey the rules of algebra. But I have a hard time understanding why people turn to them in the topology, homology and cohomology world. What is the motivation to ​use these constructs???

I'd just like to note that the way you color code the line segments and wiggle them in sync with the script around 7:37 is just perfect. Bravo.

John Rauser

I'm not sure, but I think you can argue that by varying the slope of the plane as it intersects the cone, you can attain ellipses of any eccentricity, thus implying that every sort of ellipse is covered, since eccentricity is what characterizes ellipses up to scaling.

At 0:17, the formula and the picture of the squares are suddently enlarged for no apparent reason. At 2:53 (and for the follwing seconds), I think the diameter of the ellipse should stay constant (or else the horizontal bracket should vary with it).

There is just one nitpick I have to make - you claim you’ve shown equivalence, but haven’t you really only shown inclusion of the set of all conic ellipses in the set of all thumbtack-ellipses? I don’t see how the other inclusion follows from this proof and I think you should at least comment on it.

Sascha Baer

Dandelin's spheres theorem is also one of my favourites! Elegance and beauty all together. Thanks a lot for the pleasure of seeing such a nice video! It's also very interesting the part of the theorem involving the other conic sections: parabola and hyperbola. It's a good homework to try to prove the focal properties of these conics using the ideas shown in the video. Here, nice artwork of Dandelin spheres involving the sea and fishballs! <a href="http://clowder.net/hop/Dandelin/Dandelin.html" rel="nofollow noopener" target="_blank">http://clowder.net/hop/Dandelin/Dandelin.html</a>

Nèstor Abad Viñas

home run, man :)

Delightful, beginning to end.

Awesome, thanks for the feedback. Both will be left in! What do you think of "How to convert a non-math-lover (Dandelin spheres)" as a title?

3blue1brown

“One of my all-time favorite proofs” it really shows. Practically every screen in the video is animated and it all builds on itself really well. After reading your comments, I would keep the full philosophizing bit. It turns an interesting video into a truly meaningful one. It provides the best explanation I have seen for why un-applied math is an important and interesting subject. This video would serve well as something a teacher could show the first day of geometry class to show there subject matters. I would also keep the eccentricity tangent. I got through an engineering degree without anyone explaining it, so it’s definitely useful information that fits with the narrative of the video.

It's the interrobang :) <a href="https://en.wikipedia.org/wiki/Interrobang" rel="nofollow noopener" target="_blank">https://en.wikipedia.org/wiki/Interrobang</a>

3blue1brown

Hmm, thanks for the feedback. I'll give it a second look. Part of my wants the thumbtack-definition to feel very visceral and hands-on, but I see what you mean about stylistic inconsistency.

3blue1brown

Yes! Isn't that super pretty?

3blue1brown

Thanks!

3blue1brown

If the plane slant is more than the cone slant, you won’t be able to draw two tangent (Dandelin) spheres in the manner described. He could have mentioned it, but it’s also reasonable to let curious viewers figure it out. I actually am a little peeved that hyperbolas were never mentioned in this video or the Feynman lecture, but intellectual overload is always a concern. I’m sure Grant considered it.

Jacob Mirra

You're missing an opportunity to show even more elegance. When you show the equation for the elipse as the 2 radii added together, you're missing the focal length added in. I realize that it is a constant, but the layman wants to count the entire length of the string. By accounting for this constant length in the equation, you can show that by dividing by it on both sides of the equation spits out this eccentricity.

Brian Matthews

Awesome video!! I've been wondering how the hell the conic section bananza has anything to do with the other definitions for a long time and couldn't get a clue. Now I wanna try look for a proof for hyperbolas. Thank you!

Ofir Kedar

Great video! Thanks! and about the point you made at the end - I couldn't agree more, I see it in several fields that people tend to use the "miracle" stamp too easily, avoiding the "hard work" required to acquire the intuitions, thus can't understand how it happens.. one tiny thing: at 10:26 - the title has both "!" and "?" on top of each other - it's nice, but is it on purpose?

Very very nice!

Edith Dubiner

Great video! After some fiddling around, I noticed that one can also derive the focus-directrix property of the ellipse using Dandelin spheres. The directrices are the lines at the intersections of the plane of the ellipse with the planes of the circles of tangency of the Dandelin spheres with the cone.

Beautifully done.

Charles Weaver

3:40 - I think that the comment about your surprised that the cross section isn't lopsided is spot on as I feel it's very relatable to students and I've seen friends have a similar confusion when playing with a 3-D conic sections demo in the classroom. Also, at 3:50 to 4:00 or so and on, I find it very hard to see the "thumbtack demo" in the top left and it seems out of place compared to the other two animated examples. However, the 2D animation around 4:17 does a really good job at showing this. Edit: Also around 10:30.

does the cone slant mater?

Dylan T

0.17 - Beautiful so far, but (and I know its a little petty) the sum of integers proof has strange snap and becomes wider around 0:17 in the video that could be made smoother.