3Blue1Brown | The Brachistochrone, with Steven Strogatz @3blue1brown | Uploaded 8 years ago | Updated 1 hour ago
Steven Strogatz and I talk about a famous historical math problem, a clever solution, and a modern twist.

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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 about new videos, subscribe, and click the bell to receive notifications (if you're into that).

If you are new to this channel and want to see more, a good place to start is this playlist: youtube.com/playlist?list=PLZHQObOWTQDPHP40bzkb0TKLRPwQGAoC-

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The split necklace puzzle (with a surprise topological solution)

$How well do you understand refraction? | Optics puzzles 4$ $Why does light slowing imply a bend? (Beyond the tank/car analogy) A link to the full video is at the bottom of the screen. Or, for reference: https://youtu.be/Cz4Q4QOuoo8 That video answers various viewer questions about the index of refraction. Editing from long-form to short by Dawid Kołodziej$

The DP-3T algorithm for contact tracing (via Nicky Case)

Holograms are wild (link above explains how this works)

Why do prime numbers make these spirals? | Dirichlet’s theorem and pi approximations

$How are holograms possible? 3d scenes on 2d film, and a diffraction lesson along the way. Instead of sponsored ad reads, these lessons are funded directly by viewers: https://3b1b.co/support An equally valuable form of support is to share the videos. Thanks to everyone who helped with this project: Paul Dancstep, for help writing, and for all the 3d modeling Craig Newswanger and Sally Weber, for making the central hologram shown Kurt Bruns, for the artwork of Dennis Gabor Phoebe Tooke, Wayne Grim, and Rick Danielson, for filming at the exploratorium Quinn Brodsky and Mithuna Yoganathan, for footage of lasers through diffraction gratings Vince Rubinetti, for writing the music Cliff Stoll for the Klein Bottle Hologram credits (thanks to the commenter tovedelenius1228): The Microscope is by Walter Spierings, 1984 Lucy in a Tin Hat is by Patrick Keown Boyd, 1988 The Star Wars-themed Direct-Write Digital Holograms were produced by Zebra Imaging. The Shakespeare embossed animated integral hologram was made by Applied Holographics. Mathematical corrections: 1) In the analysis for the distance between zone plate fringes, we should do a Taylor approximation about d=0, not about x=0. If you this right, the result at the bottom will look like x / sqrt(L^2 + x^2), which conveniently cancels out another (much sillier) mistake, which is how x / L in this case is not sin(theta), but tan(theta). Thanks to those who spotted that, I guess I must have been happy enough to see the desired sin(theta) at the end that I didnt properly double-check how we got there. 2) In the end, I referenced treating |R^2| as some real number, so that its only scaling O. This only makes sense to do because the amplitude of R is constant. Or at least, it varies only very slowly around a point. In this way, what I say a few moments later about making no assumptions about R is not quite right, we do need to assume its a wave with relatively constant magnitude across the film. Gabors Nobel Prize lecture: https://www.nobelprize.org/uploads/2018/06/gabor-lecture.pdf A few resources we found helpful for this video Seeing the Light, by Falk, Brill, and Stork https://amzn.to/3Ngdiqh Practical Holography, by Saxby and Zarcharovas https://amzn.to/3ZR2MNN Principles of Holography by Howard Smith https://amzn.to/3ZOihFZ Timestamps 0:00 - What is a Hologram? 3:28 - The recording process 11:45 - The simplest hologram 17:12 - Diffraction gratings 25:15 - Reconstructing the simplest hologram 28:24 - Conjugate image 31:11 - More complex scenes 35:58 - The bigger picture of holography 38:27 - The formal explanation 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$

$e to the pi i, a nontraditional take (old version) Wait! Theres an improved version: https://youtu.be/mvmuCPvRoWQ Also, for the calculus-savvy, youll prefer this one: https://youtu.be/v0YEaeIClKY Home page: https://www.3blue1brown.com/ The enigmatic equation e^{pi i} = -1 is usually explained using Taylors formula during a calculus class. This video offers a different perspective, which involves thinking about numbers as actions, and about e^x as something which turns one action into another. For more information on viewing exponential functions in this new light, check out this article: http://goo.gl/LPyc4R Music: Wyoming 307 by Time For Three, http://www.tf3.com/$

What does area have to do with slope? | Chapter 9, Essence of calculus

The Brachistochrone, with Steven Strogatz @3blue1brown