Why do colliding blocks compute pi?

An equally valuable form of support is to simply share some of the videos.

Many of you shared solutions, attempts, and simulations with me this last week. I loved it! You all are the best. Here are just two of my favorites.

NY Times blog post about this problem:

The original paper by Gregory Galperin:

For anyone curious about if the tan(x) ≈ x approximation, being off by only a cubic error term, is actually close enough not to affect the final count, take a look at sections 9 and 10 of Galperin's paper. In short, it could break if there were some point where among the first 2N digits of pi, the last N of them were all 9's. This seems exceedingly unlikely, but it quite hard to disprove.

Although I found the approach shown in this video independently, after the fact I found that Gary Antonick, who wrote the Numberplay blog referenced above, was the first to solve it this way. In some ways, I think this is the most natural approach one might take given the problem statement, as corroborated by the fact that many solutions people sent my way in this last week had this flavor. The Galperin solution you will see in the next video, though, involves a wonderfully creative perspective.

If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then "add subtitles/cc". I really appreciate those who do this, as it helps make the lessons accessible to more people.

Music by Vincent Rubinetti.
Download the music on Bandcamp:

Stream the music on Spotify:

Timestamps
0:00 - Recap on the puzzle
1:10 - Using conservation laws
6:55 - Counting hops in our diagram
11:55 - Small angle approximations
13:04 - Summary

Thanks to these viewers for their contributions to translations
German: Greenst0ne
Hebrew: Omer Tuchfeld

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Various social media stuffs:

[bluebrown]

Some added notes (copied from the pinned commend to the next video):

1) Some people have asked about if the tan(x) ≈ x approximation, being off by only a cubic error term, is actually close enough not to affect the final count. It's actually a very interesting answer! I really went back and forth on whether or not to include this in the video but decided to leave it out to better keep things to the point. This difference between arctan(x) and x could be problematic for our final count if, at some point when you're looking at the first 2n digits of pi, the last n of them are all 9's. It seems exceedingly unlikely that this should be true. For example, among the first 100 million digits of pi, the maximal sequence of consecutive 9's has length 8, whereas you'd need a sequence of 50 million for things to break our count! Nevertheless, this is quite difficult to prove, related to the question of whether or not pi is a "normal" number, roughly meaning that it's digits behave like a random sequence. It was left as a conjecture in Galperin's paper on the topic. See sections 9 and 10 of that paper (linked in the description) for more details.

2) A word on terminology: I tend to use the word “phase space” to describe any space like the ones described in this video and the last, encoding some state of some system. You should know, though, that often in the context of mechanics, this term is reserved for the special case of a space which encodes both the positions and the momenta of all the objects involved. For example, in that setting, the “phase space” here would be four-dimensional, where the four coordinates represent the position and momentum of each pair of blocks. The term “configuration space”, in contrast, just refers to one where the coordinates describe the positions of all the objects involved, which is what we do next video.

[SimonClark]

This is one of the most beautiful educational videos I've ever seen.

[mgsquared]

It’s interesting to note that in the 64 kg example at 0:28 it actually computes the first 5 digits of pi in binary. 2^(6-1) gives the first 6 digits of pi in base two! This method works for all bases!

[______]

This video has lived rent free in my head for 4 years now. I literally think about this every 2-3 weeks. Congratulations

[vampy_noah]

Tbh I watched this vid like 2 years ago and did not quite understand the equation. Now that I am older and almost done with high school, I understand the conservation of energy and momentum and we literally just had it in our recent physics class. This is why I love physics, it always has a relation to something. Whether it is mathematical or irrational and just a fun fact. It blows my mind everyday

[gogll]

Grant I don't know if you're aware of this, but you're actually changing the world. The next generation of mathematicians will be a bunch of people inspired by you. You're absolutely a master of presenting and visualisinh beautiful proofs without the need of advanced mathematics.

[blakebodycote]

if there's anything I've learnt from my maths, statistics, computer science and data science courses is that visualisations of the relationships between numbers is most definitely the future of teaching mathematics. It's videos like this that show how describing concepts with visual elements make teaching mathematics exponentially easier and I cannot wait to see how much more it becomes integrated with learning.

[SpiritmanProductions]

I'm not a maths expert but I do love these videos because I get an intuitive sense of understanding, even when I don't quite grasp all of the proof's finer points. And the animations are amazing. Great work, and thanks for posting.

[akshaytiwari]

You experience ultimate happiness when you see the links between two completely unrelated topics.

[jaiyank]

Didn't understand most of it but loved the sound of that collisionsss.

[jacktheguy]

I love just listening to this in the background not hearing a single word just enjoying the frequencies of your voice

[krishgarg]

I watched this video a year ago, when I didn't know anything about, equations of circle and straight line, conservation of momentum and energy, and collisions. Now I have covered all the above topics, so I can finally say that I understood the video.

[welovfree]

11:51 That sound is satisfying for some reason.

[rivertaig]

What I really appreciate about this channel - and it's so well exemplified in these last two videos - is that through the creative use of animation, geometry, and well thought out naration, you can spark in a non-math major like myself not just understanding of what would otherwise be esoteric and unapproachable concepts, but genuine excitement. I'm turning 50 this week, and I'm finding myself wanting to go back to college and get a degree in mathematics. This was superb. Thank you 3, 141, 592, 653 times.

[dasburstling]

I am not at all interested in maths, but this is definitely one of the coolest videos I’ve ever watched

[초]

Hello! I'm a Korean student who loves your videos! I don't know how many times I've seen this video of you. When I was in the 3rd grade of middle school, I memorized it without any understanding. It was fun even though I didn't understand it at the time, but now that I'm in my second year of high school, I think I understand the principle a little bit, so I'm very happy If I learn a little more math, one day I can understand all the math videos in your videos, right? Your video is healing for me. Thank you for posting this video

There may be misinterpretation using a translator I'm sorry😢

[adude]

As Archimedes once said, "give me an infinite mass object and an immovable wall and I can compute all of pi"

Wait...

[omooba]

i always though i loved math,
college taught me that i didn't,
videos like this remind me that i did

[iamkiubi]

I understood precisely none of this, but it got me so interested that it makes me want to learn maths beyond what i did in high school. Great video!

[sonicd]

I remember the humble beginnings of this channel vividly, with Euler's identity being the spark that first guided me here many years ago. Your adorable pi mascot has continually played an indispensable role, often unexpectedly appearing in the most surprising places. I'm incredibly grateful for the consistently stunning visual content you produce, which has succeeded in shedding light on complex ideas in an extraordinary way. Thank you for your tireless efforts in bringing such high-quality content to your viewers.