New video (early view): But *why* is a sphere's surface area four times its shadow?

I'm not working on one right now, but maybe in the future. In the meantime, here's a nice illustration (sadly no explanation to go with it): <a href="https://www.youtube.com/watch?v=AKotMPGFJYk" rel="nofollow noopener" target="_blank">https://www.youtube.com/watch?v=AKotMPGFJYk</a>

3blue1brown

Hey Grant, big fan of your work. I would KILL for a video explaining Hopf Fibration. Any chance you're working on one?

Huh? You shouldn't need to do anything through reddit for early-released videos. You can find them here: <a href="https://www.patreon.com/posts/so-whats-deal-19845962">https://www.patreon.com/posts/so-whats-deal-19845962</a> The reddit thing is just for patrons to have some flare on the /r/3blue1brown subreddit if they want.

3blue1brown

"Weapons of Math Destruction" by Cathy O'Neil I understanding better the dangers of allowing web masters to aggregate data on users. Any chance you could post through another, more privacy-minded site that protects its users?

Dear Grant: Love your work! However, there is a big problem. I am trying to access the extra content through Reddit. They insist on linking my Facebook and Google accounts to access their site through the sign up process (they claim otherwise). After reading

Glad you got something out it! And sorry you I didn't catch the suggestion before the video went live :)

3blue1brown

I'm glad appreciate it! It's so tempting as a creator to cut out pauses for risk of thing being less exciting, but as a consumer I always like the opportunity to think more, and think the audience here does as well.

3blue1brown

Thanks for the pause and ponder moments. It encouraged me stop and think instead of just bask in the feeling of "knowing."

I appreciate you saying that Roger. I'd love to make problem-solving tips more of a regular habit.

3blue1brown

I absolutely LOVE that you were giving tips along the way. So helpful for getting better at solving math problems myself! Your animations are constantly improving. I remember the time when you first started using 3D animations (I’ve been watching your channel since the very beginning ;)), and now you’re doing animations like this! Absolutely amazing! Keep up the great work, Grant. What you’re doing here is so important! Thank you!!

Janik

great work again Grant. Amused by "a bit more archimedean" :-) had fun later capturing the sin(2_theta) hint, good thing I'm not epileptic :-) Minor suggestion: at 1:43 the "label" is a bit too tall (&gt;2R) so not quite aligned to north pole of sphere.

Chris Jennings

I love this video. Thanks so much!

Daniel and Rebekah Slonim

Yeah every video has an opportunity cost. You'll figure out a balance, but personally I think a quantum computing video would be fantastic.

Thanks so much George. I started writing the video for the average shadow of convex shapes, and might do something collaborative with it by measuring the shadows physically with a different YouTuber. We'll see. Part of me wonders if it's as important as, say, putting out a good video on Bayes' theorem, Laplace transforms, Quantum computation, etc. What do you think, worth it to make? And thanks for the typo catch! I'll pass that along to DFTBA.

3blue1brown

So awesome. Is the video idea at the end (the one about the average shadow of convex shapes) in the works or not really? Also I think someone misspelled hex-compliant on the sticker page of your DFTBA store. Not really important. Anyway, you should be really proud of what you're creating man.

Ah, I see the problem now and it *could* be okay! I was worried about the part near 4:45 where you commented that we were assuming the rectangle is small enough that doesn't matter which of the two possible lengths we call "d" Maybe if you use something between the two then you can get those vertical equal signs that you want at 9:04. But they are not equalities if you choose d to be the lower of the two possibilities! As an example of an extreme situation, consider an octahedron (as a very low resolution sphere) and consider wrapping it by a rectangular prism (a very low resolution cylinder). Wrapping it tightly corresponds to your choice of d being the lower of the two possibilities. The surface area of an octahedron of side length a is 2 sqrt(3) a^2. A tighly wrapped rectangular prism has surface area 4 sqrt(2) a^2. An image of what I have in mind: <a href="https://i.imgur.com/u6cDFY3.png" rel="nofollow noopener" target="_blank">https://i.imgur.com/u6cDFY3.png</a> What do you think?

Ebrahim Ebrahim

Great question, but that's actually the whole point. Since each rectangle from the sphere precisely equals the area of its projected rectangle, which is one of many approximating the area of the cylinder, the approximations are exactly the same. Admittedly, this could potentially be clearer, especially in spelling out that if you want to consider the projection of the rectangle on the sphere over to the cylinder to still be a rectangle, it wouldn't quite be *on* the cylinder (since its flat, not curved).

3blue1brown

Amazing work as usual! The two sequences shown at 9:04 have some vertical equal signs that I'm suspicious of. Equality holds in the limit only, and the whole point is that those vertical equal signs are "amost equalities" that get closer and closer to being true equalities.

Ebrahim Ebrahim

I'm just done with the exercise part and I must confess I found it quite harder than what I expected at first glance. First problem I found was that the aproximation of the sphere's ring's area didn't seem quite right when applied to the area of the shadow's ring (in my mind one was somehow closer to a cilinder, for which the aproximation would be true, but the other one was just a flat disc), so I went on to check both, and they turned to be right (of course).. Next I got stuck with the sin(x)cos(x) part, so after finding the trig identity in the comment section I reattempted. Even then the process wasn't as obvious as I expected. Now I think I've come to understand it, even if I wasn't able to make a rigorous prove for the last step, I think I have the feeling for why and how it works. One last observation is that when I think in terms of the shadow rings, it seems clear that the whole set of them represent half the area of just half of the rings on the sphere. However, if I think in terms of the rings of the sphere and try to find which of them are not represented by any ring on the shadow discs, I can't find them.

Very nice video!

Great video! I enjoyed the problem and animations :) Some feedback on the experimental part: I did really like the exercises since actively struggling with and tackling problems is part of what makes math fun for me, and I imagine others would feel the same. Although, I'm not sure if it's for everyone. Some viewers probably caught onto the proof pretty quickly, while others may not know where to start (e.g. anyone who doesn't happen to recognize the trig identity). Maybe you can try some middle ground - e.g. list the exercise problems as an outline/overview of the proof, tell viewers to pause and try to solve them, and then proceed with an explanation of the solutions.

Antzen

Thanks for the catch, but I think it should actually be rings’. Am I wrong?

3blue1brown

This is beautiful, Grant. Cheers :D

J

That "re-assembling" of rotated the projections at 8:29 was fantastic; worth the price of admission alone.

Don Sanderson

At 13:00, it should read „one of the rings‘s“ (plural)

slzb

The finite surface elements are a priori just trapezoids. Treating them as rectangles is an approximation that warrants a mention.

slzb

Usually, if you take a rectangle, and multiply one side by c, and the other side by 1/c, it's not at all the same as a rotation. But for the particular parameterization I'm using for this sphere, the ratio of the rectangle's width to height starts off as d / R. So when you multiply the width by R / d, and the height by d / R, only because of it's initial proportions does it happen to be a 90 degree rotation.

3blue1brown

Good catch, thanks!

3blue1brown

Can you elaborate on exactly which claim is "not true in general"? I don't get it, to me, everything that is said around 8:17 seems pretty solid.

Petr Čertík

The speech bubble at ~2:52 is cut off on the left, which looks odd. All of the other speech bubbles looked fine to me.

David Henderson

On my cell phone, there is no way to slow down the video, so even after double-tapping my way through that section of the video (to go forward a fraction of a second at a time) about a dozen times, I still couldn't make it stop on the presumably single frame with the hint message. I'd really appreciate making that at least a little bit longer.

David Henderson

Strongly agree

Also, I personally really appreciated your "exercises" approach in the second half. I didn't expect to spend three hours this evening getting sucked into a math problem, but it was incredibly satisfying when I finally figured it out. It was very helpful to have the hints as a sort of safety net for when I got stuck.

I assume the "Extra hint" frame that blinked at 13:17 was intentionally shortlived... I had to slow the video down to 1/4 speed and wear out my spacebar to see it!

Gregor Shapiro

I wondered this a great deal in middle school and early high school, but once they taught us Calculus, I immediately realized the surface area was the derivative of the volume (which is intuitively obvious, just visualize a balloon inflating and think about its rate of expansion). Shortly after, they taught us disk integration, which fully dispelled the mystery for me. So it's really interesting to see a different perspective on the same geometry.

Kevin

Beautiful, thought-provoking video as always! My only suggestion is that it might help to zoom in more on some of the animations. I found it a little difficult to see what was going on at some points during the cylinder section, for example around 3:45 and 5:06.

Just a personal opinion here - there are already a lot of guided maths channels that talk one through solving problems etc, the core love of your channel that I have comes from how you can explain things visually to me. Relying on my to have that level of visual understanding myself, and being able to visualise a solution myself effectively removes the only function your channel serves - for me. I will turn back to just algebra, trig and calculus to solve the problem and remain without the visual understanding you present. That is - the second half of your video gave me no benefit at all.

Christopher Burke

Nice elegant exploration. I must have been distracted for a brief moment at the end as I thought *aha!*, but what if the solid has a concave area!, Then I had to go back to hear you did explicitly specify it only works for convex solids. :)

PeetieGonzalez

The best place to share things in a way that you can know I will check is to put it onto this reddit thread. <a href="https://www.reddit.com/r/3Blue1Brown/comments/8u46tt/3blue1brown_video_suggestions/" rel="nofollow noopener" target="_blank">https://www.reddit.com/r/3Blue1Brown/comments/8u46tt/3blue1brown_video_suggestions/</a> Otherwise, I just get way too inundated with requests for content.

3blue1brown

Nice Vid! I really enjoy your work. Is there a way I can share a video Idea with you? I know you will illustrate it very elegantly.