Final block collision video! (Early view)

Thanks so much! I had a lot of fun making it.

3blue1brown

By the way, this whole miniseries was fantastic. I’m psyched to see what you come up with next. I saw people Tweeting it on Pi Day.

Great work again. Re thoughts on "What ending should go here?" there's more maths in this one than usual, so I suggest a simple visual wrap up is needed. Perhaps just re-work through a few example digits of pi (as you did with the blocks) but using the new-fangled "method of pseudo-reflections".

Chris Jennings

So am I! It's a really big value of having a selective inner audience like this.

3blue1brown

Evidently Patreon has a slight delay right now in processing payments, but just by a few days. All should go through properly (at least as I've been told), so no need to worry. Thanks for asking!

3blue1brown

Paying for these excellent videos - did you get your Patreon payments on time? Do you have alternate ways of getting donations in the works?

seerpea

The final video is much improved over the preview. I'm glad the Patreon community can help boost the quality to where we all want it to be :)

Max Goldstein

Yeah, I ultimately opted for a less is more approach on this one.

3blue1brown

In theory I like the idea of talking about mirror reflections / mirror geometry in general (I know this idea from the problem of computing 'early reflections' for audio reverb--computing when echoes of a sound source in a rectangular room reach a sound receiver, where it's a very simple and powerful solution to the problem), but I also think it's probably too much of a tangent to actually be worth trying to squeeze into this video.

nothings

I will try it in "my" lab too, inspired by your video. By the way I'm looking forward to send the third video to a spectroscopist, who was working on a paper on light propagating in a wedge. Couldin't find his paper after 3min search, maybe he didn't publish it, but the video will make him happy, I think.

3:07 The light/block bounces off Y=0 instead of the mirror. 5:00 The bottom arrow is off center. At the very end, I would extend a dashed Y=0 line through the leftmost mirror slice to show the chain of reflections has passed 180 degrees because without it, it's a little hard to tell. Maybe toggle that last slice on and off as the value switches between 3.142 > pi and 3.141 < pi, though I know those values do not reflect the 12 bounces in the visualization above, so that may end up more confusing than helpful. I feel like your idea for reflections in a square room would make for another full video, but I was assuming the video would also end up building a model for cubes too. Without that, it may make sense to include it here. Not sure. I’m not sure how to end the video either. This video is a good example of one of the things your channel is best at: explaining systems by building models that map the parameters of that system in clever ways to make their connections obvious. This problem in particular contrasts well with how my physics class modeled elastic collisions which was to draw free body diagrams and leave all interactions up to the algebraic equations. This type of mathematical system to model a contrived scenario is usually seen on YouTube as the solution to arbitrary seeming riddles or thought experiments, but you’ve created one for a physics problem many of us learned the basics of in high school, making it much more accessible. I know that’s not really an answer to your question, but this is what I saw as the core message of the video with the physics/pi interaction used as an interesting example of that kind of approach to mathematics, so that it what I would expect an ending to capitalize on.

Yup, you're correct. Not sure what got in my head.

3blue1brown

I ended up having a shockingly hard time getting good shots of a laser bouncing between mirrors that didn't ultimately make things look more confusing than helpful. The issue was that you don't just see the "actual" beam bouncing and the "illusory" beam going straight, you see a combinatorial explosion of possibilities where at each bounce point the view of the beam splits into two. Also, the fog setup I was using to make the beam visible, namely dry ice in water, wasn't great, making for sort of a cloudy image. Time permitting, I may get a fog machine take another crack at it. We'll see.

3blue1brown

Thoughts on what ending: For me, the fact that three different things, or even more, collisions, calculus, geometry and light follow the pi pattern or better, emerge to one structure from the laws of conservation is mesmerizing, I mean these laws, and the scientific and mathematic tools have got mankind here and more importently, can help to solve problems. How many scientists have written poems of equasions, hardly knowing their impact, their potential. How mutch more is out there to be discovered. How amaizing it is to see in these videos the beauty the connected puzzle pieces of the fractal world and to keep the flame of this spirit, from the discovery of pi and geometry, to functions, calculus, and algebra for generations to come, and have fun with it, as we do. I would maybe show an actual lightbeam bouncing on and of in a acrylic wedge, or a laser between mirrors in a for, or maybe even a wedge with polichromatic rainbow(?), but it's not common for the videoformat - but you have the mirrors already....

I could be totally wrong here, but what you're describing at 3:33 is reflection. It's only at 10:45 that something like refraction comes into play, when light passes through a boundary, although it doesn't bend in a Snell's law way.

Max Goldstein

That's awesome! Glad to hear it :)

3blue1brown

It is not the spoon that bends, it is only yourself!

André Mello

My intuition (which should be seriously doubted) makes me inclined to agree with you. I think it has the potential to be harder to calculate, because you might not always get the nice geometric argument to facilitate counting, but I'd love to find out if there's anything significant that comes up. In particular, I'd be curious if there's significance to things like how far up the light gets on the y-axis. And I also would really like to know if there's a parallel to distributed Bragg reflectors (or Bragg mirrors), which are applications of light interacting with alternating layers of material. If you showed a cool property, people might actually make these.

Evan Miyazono

Tying it all together by going back to the simulation from the previous video is my favorite ending, personally. For the "is arctan always good enough" - I'm surprised that's unknown. I would think it would converge in the limit. Some discussion of that might be interesting (but I'm not sure I'd close on it). As far as "other examples of using this technique on reflection puzzles" - I think some discussion of the general principle would be good, but if you give an actual concrete puzzle, it may distract rather than inform. But that's just my opinion.

Kevin

Funny you should mention this! TED-Ed did a light-bouncing riddle recently ( <a href="https://youtu.be/P4-n0IMQSrQ" rel="nofollow noopener" target="_blank">https://youtu.be/P4-n0IMQSrQ</a> ), and though they use a cute trick to solve it for their particular case, you can use these "reflected worlds" to solve the general case. It's pretty neat!

Hey Grant, great video as always. Sorry if this comment gets posted twice, but my original comments seems to have been deleted... I don't have any corrections (I see people have already mentioned the refraction vs reflection), but I wanted to share something this video helped me do. TED-Ed recently released a riddle ( <a href="https://youtu.be/P4-n0IMQSrQ" rel="nofollow noopener" target="_blank">https://youtu.be/P4-n0IMQSrQ</a> ) about a bouncing beam of light in a rectangular room with mirror walls, asking in which corner will the beam of light will finish. There's a very simple and clever answer provided in the video, but it only applies to the particular case of the light at a 45 deg angle. I attempted to solve a more general case and started sketching these "reflected rooms," without a formal understanding of the optics theory. I got pretty lost and gave up. But while watching your video, the idea of the reflected rooms clicked and became better solidified in my mind. I went back to the problem this morning and solved it! For any angle (that corresponds to a rational slope), I found a general solution/algorithm. It's short and sweet and pretty, and it feels awesome to have worked it out. So thanks! =D

Ooh, that's a great quote. In fact, let's go ahead and make sure that's in the video.

3blue1brown

Personally I'd like to see reflections applied to other puzzles. I like the shift in perspective. I was curious how you "cheated" when modeling the collisions to avoid issues with truncation errors. And Max and Peter are correct: it should be "angle of reflection" not "angle of refraction"

It supports this "sometimes half circles also show up in math, but let's not quibble of which constants are more meaningful and instead have fun solving real problems" case :)

3blue1brown

You're right, it never hurts to be clear about this.

3blue1brown

Perhaps Gauss would have chosen a lower number if he felt the need to animate each one :)

3blue1brown

I mean if you look at the first 2n digits, for some number n, is it the case that the (n+1)st through (2n)th digits are all 9's

3blue1brown

Ah! Silly me, you're very right. This is why I early release, I guess the wrong word just got stuck in my head while recording.

3blue1brown

Angle of Incidence = Angle of Refraction? You mean Reflection? OR I am missing a point here?

Never mind: I rewatched as saw that the x and y coordinates represent the locations of the right and left edges of the left and right blocks, respectively

Kevin Iga

On a less substantive note, might this support pi in the pi vs. tau debate?

Kevin Iga

If the small block bouncing off the wall is not the x axis, but the width of the block above it, then shouldn't the collision of the two blocks be not y=x but y-x=1/2(width of first block)+1/2(width of second block)?

Kevin Iga

I thinks it can be helpful to make the audience remember that the full circle is 2pi, then just half of the circle is pi, so it can be better understand where pi comes from in this analogy.

I think what he meant was that it never happens that after k normal digits the next k digits are all 9. So in none of the approximations of pi we get a situation where the last half is all 9.

Imre Polik

I haven't worked out the details, but I'm fairly sure we could do this with refraction instead of reflection having alternating denser/lighter mediums. Then we wouldn't have to rescale the picture with the mass ratios.

Imre Polik

In the spirit of Gauss, he never accepted anything as fact unless it could be proven 7 different ways. So can we expect 5 more videos?

What exactly is the 'last half of pi digits'? Can infinite digits have a half-way point?

Hmmm. If Pi suddenly ended in all 9's could it still be irrational? Or any 'irrational' number for that matter. If the ending is predictable doesn't that somehow violated the premise of irrationality? Just a gut-feeling. (Note: since I 'air-quote' with single quotation marks, I have revealed myself as a Pythonista)

Great video! Personally, I think the digits of pi part is already a bit heavy, so I would save that non-proof for another video. Since this is the last video, I would finish with a summary of the videos to motivate how there are often many different ways to solve math and physics problems, each with their own interesting perspective and advantages. I like Alan Kay's quote "A change in perspective is worth 80 IQ points." It is perfectly illustrated here.

Gabe

Both the light and block go past the "wall"

Brian Matthews

Extraordinary, as usual, but a couple of suggestions: - 3:07: shouldn´ t the light reflect in the line showing the small block´ s width rather than on the x axis? - Shouldn´´ t we talk of angle of relection rather than angle of refraction? - Finally, at the end of the video it is not shown in a graphica nd intuitive form how the digits of pi are approached. But , overall and as usual, apart from that the idea and teh video are awesome!

Daniel Armesto

Perhaps labeling the blocks in the graphic would further identify them (if they are not called Alice and Bob).

Gregor Shapiro

There's a little bit of hand-waviness with arctan(something small) ≈ something small in terms of the model. But they're not *quite* equal, and that seems to make the model of "reflecting the universe" off slightly. Intuitively, where does that small amount of difference play into things? Are there any examples where that small difference is *just* enough to throw the total number of bounces/collisions/reflections off by one?

David Henderson

True about the "reflection" vs. "refraction" distinction, mentioned above... Technicality, maybe, but in science (and also math! ;-) precision of language matters... Since you do not consider any "medium" (material) with a refractive index different from one, there is no "refraction" here... Just REFLECTION!

Reflecting the world, instead of the beam, is just genius! Did you come up with this? Or is it an "established trick"???

I’m definitely interested in hearing more about how you used this beam of light analogy to simulate the large mass ratios accurately. However, I’m not sure if it is the best possible ending to wrap up this topic. I think the discussion of the arctan approximation would make the whole mini-series feel a bit more rigorous. When that approximation came up in the previous video, I personally felt it was a little hand-wavey. Even seeing some brief discussion of the topic on reddit gave me a better feel about it, and I think that could wrap things up nicely. Nevertheless, I think any of the proposed endings sound very interesting.

This may not matter, but the video defines the angle of incidence as the angle between the incoming light ray and the vector tangent to the reflecting surface. But in optics, the angle of incidence is always defined from the angle between the incoming light ray and the vector *normal* to the reflecting surface. The video also calls the angle the reflected ray makes with the surface the "angle of refraction", when it is properly called the angle of *reflection* (and also commonly defined from the surface normal vector, not the surface tangent vector).

Max Fagin

I'd vote for bullet 1, namely how the analogy facilitated the simulation for large mass ratios. It's nice because it gives the audience a small glimpse into your process, and that may cause them to feel more personally connected to you and what you do. I want more people to feel personally, emotionally connected because, well, math! On a personal level, I would find that the most interesting of the bullets. I'd also personally rank the connection to other puzzles second.

I agree the colliding simulation was interesting. Was it actually counting the millions of collisions in the sub-second time frame, or did you manually do the on-screen counter animation seperately, Grant?

I love it! And I would find it very interesting to see how you coded the simulation (full disclosure, I'm also about to start studying CS so maybe I'm the exception). No matter what ending you settle on, I'm sure it'll be insightful; keep up the awesome work!

As of 2016 π had been calculated to 12.1 × 10¹² digits [<a href="http://www.numberworld.org/misc_runs/pi-12t/]" rel="nofollow noopener" target="_blank">http://www.numberworld.org/misc_runs/pi-12t/]</a> As of 05:50 UTC friday The video was still being processed... As of 07:00 UTC it plays... As far as the previous proof goes I would have liked even more graphical emphasis on the circle that arises from the conservation equations.

Gregor Shapiro