This pattern breaks, but for a good reason | Moser's circle problem @3blue1brown

3Blue1Brown | This pattern breaks, but for a good reason | Moser's circle problem @3blue1brown | Uploaded 1 year ago | Updated 2 hours ago
An apparent pattern that breaks, and the reason behind it.
Summer of math exposition: 3blue1brown.substack.com/p/some3-begins
Learn more at some.3b1b.co
Help fund future projects: patreon.com/3blue1brown
An equally valuable form of support is to simply share the videos.

For the long-time viewers among you, if this sounds familiar, it's because it's a remake of one of the earliest videos on the channel. It's such a wonderful problem, and the audio/pacing in earlier videos was really suboptimal, so I wanted to freshen it up a little here.

Timestamps
0:00 - The pattern
2:20 - Counting chords
4:03 - Counting intersection points
6:20 - Euler's characteristic formula
11:30 - Connection with Pascal's triangle
15:10 - Reflections

Correction at 8:56 - The number of the regions should of course be (1, 2, 3, 4, 5), instead of (0,1,2,3,5)

Thanks to these viewers for their contributions to translations
Arabic: mouhsiiin
French: Jonhfing
Hebrew: Omer Tuchfeld
Hungarian: Fabó Bence
Russian: fedor
Spanish: Paweł Cesar Sanjuan Szklarz
Thai: @korakot

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These animations are largely made using a custom python library, manim. See the FAQ comments here:
3blue1brown.com/faq#manim
github.com/3b1b/manim
github.com/ManimCommunity/manim

You can find code for specific videos and projects here:
github.com/3b1b/videos

Music by Vincent Rubinetti.
vincentrubinetti.com

Download the music on Bandcamp:
vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown

Stream the music on Spotify:
open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u

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3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on new videos, subscribe: http://3b1b.co/subscribe

Various social media stuffs:
Website: 3blue1brown.com
Twitter: twitter.com/3blue1brown
Reddit: reddit.com/r/3blue1brown
Instagram: instagram.com/3blue1brown
Patreon: patreon.com/3blue1brown
Facebook: facebook.com/3blue1brown

Why “probability of 0” does not mean “impossible” | Probabilities of probabilities, part 2

$What is Eulers formula actually saying? | Ep. 4 Lockdown live math What does it mean to compute e^{pi i}? Full playlist: https://www.youtube.com/playlist?list=PLZHQObOWTQDP5CVelJJ1bNDouqrAhVPev Home page: https://www.3blue1brown.com Brought to you by you: https://3b1b.co/ldm-thanks Beautiful pictorial summary by @ThuyNganVu: https://twitter.com/ThuyNganVu/status/1258220129327800320 https://twitter.com/ThuyNganVu/status/1258220541686628353 Not on the homework to show that exp(x + y) = exp(x) * exp(y). This gets a little more intricate if you start asking seriously about whether the series really converge, what they converge to, and how exactly you define a product with infinitely many terms. For anyone curious about the technical details, what you would want to show is that the Cauchy Product of the series for exp(x) and exp(y) converges to the product of the values exp(x) and exp(y) for any particular x and y. That requires the Mertens Theorem. Thanks to these viewers for their contributions to translations Hebrew: Omer Tuchfeld Video Timeline (Thanks to user Just TIEriffic) 0:00:00 Welcome 0:00:20 Ending Animation Preview 0:01:15 Reminders from previous lecture 0:03:30 Q1: Prompt (Relationship with e^iθ=…) 0:05:40 Q1: Results 0:07:15 WTF, Whats The Function 0:10:00 Exploring exp(x) 0:11:45 Exploring exp(x) in Python 0:14:45 Important exp(x) property 0:15:55 Q2: Prompt (Given f(a+b) = f(a)f(b)…) 0:17:30 Ask: Which is more interesting, special cases or the general case 0:20:00 Q2: Results 0:23:50 Will a zero break Q2? 0:25:40 The e^x convention 0:27:10 Q3: Prompt (i^2 = -1, i^n = -1) 0:27:45 Ask: Zero does not break Q2 0:30:20 Q3: Results 0:31:05 Comparison to Rotation 0:33:00 Visualizing this relationship 0:36:50 The special case of π 0:39:20 Periodic nature of this relationship 0:39:40 Q4: Prompt (e^3i) 0:41:35 Q4: Results 0:43:55 Explaining the celebrity equation 0:45:55 Homework / Things to think about 0:49:15 Ask: Zero does break Q2. 0:50:30 Closing Remarks The live question setup with stats on-screen is powered by Itempool. https://itempool.com/ Curious about other animations? https://www.3blue1brown.com/faq#manim Music by Vincent Rubinetti. Download the music on Bandcamp: https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown Stream the music on Spotify: https://open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u 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. 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on new videos, subscribe: http://3b1b.co/subscribe Various social media stuffs: Website: https://www.3blue1brown.com Twitter: https://twitter.com/3blue1brown Reddit: https://www.reddit.com/r/3blue1brown Instagram: https://www.instagram.com/3blue1brown_animations/ Patreon: https://patreon.com/3blue1brown Facebook: https://www.facebook.com/3blue1brown$

Why 5/3 is a fundamental constant for turbulence

$How wiggling charges give rise to light | Optics puzzles 2 Explaining the barber pole effect from the last video: https://youtu.be/QCX62YJCmGk Next video on the index of refraction: https://youtu.be/KTzGBJPuJwM Help fund future projects: https://www.patreon.com/3blue1brown An equally valuable form of support is to simply share the videos. Timestamps: 0:00 - Recap 0:44 - The radiation law 6:10 - Simulating the radiation law 11:11 - Why the diagonal stripes? 16:31 - Why does it twist? Thanks to these viewers for their contributions to translations Hebrew: Omer Tuchfeld Indonesian: akhyarr These animations are largely made using a custom Python library, manim. See the FAQ comments here: https://3b1b.co/faq#manim https://github.com/3b1b/manim https://github.com/ManimCommunity/manim/ All code for specific videos is visible here: https://github.com/3b1b/videos/ The music is by Vincent Rubinetti. https://www.vincentrubinetti.com https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown https://open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u 3blue1brown is a channel about animating math, in all senses of the word animate. If youre reading the bottom of a video description, Im guessing youre more interested than the average viewer in lessons here. It would mean a lot to me if you chose to stay up to date on new ones, either by subscribing here on YouTube or otherwise following on whichever platform below you check most regularly. Mailing list: https://3blue1brown.substack.com Twitter: https://twitter.com/3blue1brown Instagram: https://www.instagram.com/3blue1brown Reddit: https://www.reddit.com/r/3blue1brown Facebook: https://www.facebook.com/3blue1brown Patreon: https://patreon.com/3blue1brown Website: https://www.3blue1brown.com$

But what is a neural network? | Chapter 1, Deep learning

Hamming codes part 2: The one-line implementation

Olympiad level counting (Generating functions)