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Awesome! I’m glad you ran so far with it.

3blue1brown

I would like to offer some constructive criticism but I can't because this video is too perfect

might do a writeup of my approach

Thanks for making these videos! Here is my solution: <a href="https://www.wolframcloud.com/objects/lnemzer/Published/pi.nb" rel="nofollow noopener" target="_blank">https://www.wolframcloud.com/objects/lnemzer/Published/pi.nb</a>

I applaud the determination!

3blue1brown

Big fan of this new type of format. I think that after giving the puzzle some thought, and unsderstanding its difficulty will make the solution much more interesting. I'm far from understanding why this happens, but at least I was able to play a bit with the problem and to see by my own eyes that it works. Made small animation here: <a href="https://www.youtube.com/watch?v=XniVFpXEX3A&amp;feature=youtu.be" rel="nofollow noopener" target="_blank">https://www.youtube.com/watch?v=XniVFpXEX3A&amp;feature=youtu.be</a>

<a href="https://m.imgur.com/a/OVC2jLY" rel="nofollow noopener" target="_blank">https://m.imgur.com/a/OVC2jLY</a> aaah

Digits after that, if series is 801, do we count 81

Saketh Vns

That's fine, it just means the number of collisions ends in a 0.

3blue1brown

If the number of collisions indicate the pi, what about the point where the digit becomes zero in pi and how do we interpret digits after that

Saketh Vns

With our idealizing assumptions, it will never actually be 0 on both sides.

3blue1brown

I'm glad to hear that! It also makes for slightly more regular content, and videos which are not intimidatingly long.

3blue1brown

I appreciate having a few days to think about it, maybe with others. Sometimes It's hard not to be lazy and pause the video when you know the solution is right there and others have probably watched it.

Question: Assuming the larger mass is approaching infinite mass, is there ever a moment when the separation on either side of the smaller block is zero and if so, what is the accumulated clicks: 1/2 or 1/2 + 1? (Zeno’s paradox is leading me to question the behavior of the physics here)

Chase Turner

So instead of 1 infinite series, you sum 4 infinite series (I assume the = is exact, since there are only 4 terms on the RHS and no ...). How is that efficient? (I know, they converge really quickly. Just being a SNARK, wink, wink) In comments below you see same approaches using sin or cos. I assume those Taylor series expansions are slow. So the arctan is really efficient? Who knew.

I also used the geometric approximation to find pi, (n denotes the number of sides, n goes towards to infinity.) I managed to get some formula via cosine rule. (360/a)*sin(a/2). Its results are wonderful but I believe there is something problematic in there so that it prevents it from being perfect. But I couldn't catch the problem.

Really interesting, I'll try to solve this until the deadline.

The sound was perfect! :)

Haha, unintended. Updated the video to let them complete.

3blue1brown

Great question! The idea is to use the Taylor expansion of arctan to explicitly compute digits. Formulas like the one shown on screen are called Machin-like formulas, and can be incredibly efficient ways for computing pi when put in conjunction with this Taylor series. Something like pi/4 = arctan(1) is of course true, and you could use it to make the computation, but having 1 as the argument will make for much slower convergence than if the argument was closer to 0.

3blue1brown

Very interesting. Nicely introduced to get us interested.

No way!

Daniel and Rebekah Slonim

Hey, I got a better solution for PI/4, than the one above. Why not PI/4 = arctan(1)? Seems using the arctan to calculate PI is a rather circular (no pun intended) algorithm. That is using a function that relies on knowing PI already.

Indeed, as I said, things very quickly depart from reality.

3blue1brown

Perhaps, part of the reason for my hesitation there is not that the "rough edges" would feel embarrassing, but that I know the act of making such a video would encourage many people to go play with it, and before its better documented/tested/robust/etc., that experience runs the risk of being more painful than I want it to be for people. Even without a video on it, though, everything is very open for those who want to see, and who are willing to look at the updates I make to the github repo.

3blue1brown

Yup, the same reasoning will apply to other bases.

3blue1brown

Actually, in that case its the first four digits, 1100. Animation here: <a href="https://youtu.be/_0LT6_aJTDw" rel="nofollow noopener" target="_blank">https://youtu.be/_0LT6_aJTDw</a> Really it might be better to think of the square root of the power of the base as giving the number of digits after the decimal place. In this case, 2^4 gives 2 bits after the decimal, just as 10^4 gives up to 3.14. Likewise, 2^6 would give the first 3 bits after the decimal, 11.001 -&gt; 11001 -&gt; 25 clacks.

3blue1brown

So if I set up the experiment between two cubes with mass ratio 1:16, and then wrote down the number of clicks in binary, would I get the first three digits of pi written in binary notation?

If you use a base other than 10, I assume the scale factor is that base? That is if calculating PI in base 8, then the multiplier is 8, and PI comes out as base 8 counts. Or does it work in other bases too?

Consider making a video showcasing how to use your software. Just as your videos appeal to those with patience, open the doors to more Brownies using your awesome software despite the rough edges.

Brian Matthews

I was looking for a reason to avoid prepping for classes. Now I've got to go figure this out.

The sound effects were superb at appealing to the general audience; the slow-mo sounds as well as the sped up sounds to emphasize scale. Well done.

Brian Matthews

Neat! As you take the mass ratio to be larger and larger, the smaller (now infinitesimal) mass starts functioning as an ideal spring, bringing in Hooke's law? I'll try and make that connection over the weekend. It certainly brings in the circular motion, as it smoothly rotates the trajectory through space-time. I had fun drawing a quick space-time plot of the masses to show the smooth curvature of the trajectory. Variations on the scenario I considered: two or more small masses bouncing in between in parallel, or in sequence. From there it's not hard to jump to a compressing gas cloud.

This is truly fascinating. So much mystery around Pi. Could you go into more detail on the graph of efficiency vs elegance in the future video, would like to understand how some of them got derived. Ramunjan’s expression to calculate Pi was so much more advanced ( no of digits and the complexity of the expression) than previous researchers. I am still mind blown by his Pi expression.

Rajesh

I think somewhere around the 12 digits of Pi scenario you need to start considering relativistic effects :-) And that frequency of collisions will start emitting ionizing radiation somewhere around that point too!

Great video. Very astonishing fact too.

Bernhard

Geez, I'm stumped! What a delightful problem.

Max Goldstein

So is it like, the conversation of energy being constant provides the equation for an ellipse, which in turn, assuming please collisions, provides the hidden circle?

Caleb Pheloung

thanks for providing a quality moment for hubby and me :-)

Edith Dubiner

brilliant teaser. love it. let's see if i can figure anything out off of the hints..

issa

Good to know, I'll quiet those down.

3blue1brown

Wait a second...you're telling me you don't have a physics engine laying around?

i really like the satisfying clacks with the 10,000kg example. however the clacks seem like the peak especially in the slowed version. you did a good job compressing the others however. just wanted to let you know it gets a little loud there

You're trolling us so hard right now by not letting those two simulations get to their final clack...

Matt Goldman

Something to do on the weekend or something

Timur Sultanov

Damn, now I need to remember the physics of collision... well I suppose time to put my Master's Degree in Technical Physics to the test or something...

Timur Sultanov

cool!