Convolutions | Why X+Y in probability is a beautiful mess 3Blue1Brown 2023-06-27 | Adding random variables, with connections to the central limit theorem.Help fund future projects: patreon.com/3blue1brownAn equally valuable form of support is to simply share the videos.0:00 - Intro quiz2:24 - Discrete case, diagonal slices6:49 - Discrete case, flip-and-slide8:41 - The discrete formula10:58 - Continuous case, flip-and-slide15:53 - Example with uniform distributions18:42 - Central limit theorem20:50 - Continuous case, diagonal slices25:26 - Returning to the intro quizThanks to these viewers for their contributions to translationsHebrew: @DavidBar-On, David Bar-On, Omer TuchfeldSpanish: Derek Lacayo------------------These animations are largely made using a custom python library, manim. See the FAQ comments here:3blue1brown.com/faq#manimgithub.com/3b1b/manimgithub.com/ManimCommunity/manimYou can find code for specific videos and projects here:github.com/3b1b/videosMusic by Vincent Rubinetti.vincentrubinetti.comDownload the music on Bandcamp:vincerubinetti.bandcamp.com/album/the-music-of-3blue1brownStream the music on Spotify:open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u------------------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/subscribeVarious social media stuffs:Website: 3blue1brown.comTwitter: twitter.com/3blue1brownReddit: reddit.com/r/3blue1brownInstagram: instagram.com/3blue1brownPatreon: patreon.com/3blue1brownFacebook: facebook.com/3blue1brown
How are holograms possible? | Optics puzzles 5 3Blue1Brown 2024-10-05 | 3d scenes on 2d film, and a diffraction lesson along the way.Instead of sponsored ad reads, these lessons are funded directly by viewers: 3b1b.co/supportAn equally valuable form of support is to share the videos.Gabor's Nobel Prize lecture:nobelprize.org/uploads/2018/06/gabor-lecture.pdfA few resources we found helpful for this videoSeeing the Light, by Falk, Brill, and Storkamzn.to/3NgdiqhPractical Holography, by Saxby and Zarcharovasamzn.to/3ZR2MNNPrinciples of Holography by Howard Smithamzn.to/3ZOihFZTimestamps0:00 - What is a Hologram?3:28 - The recording process11:45 - The simplest hologram17:12 - Diffraction gratings25:15 - Reconstructing the simplest hologram28:24 - Conjugate image31:11 - More complex scenes35:58 - The bigger picture of holography38:27 - The formal explanation------------------These animations are largely made using a custom Python library, manim. See the FAQ comments here:3b1b.co/faq#manimgithub.com/3b1b/manimgithub.com/ManimCommunity/manimAll code for specific videos is visible here:github.com/3b1b/videosThe music is by Vincent Rubinetti.vincentrubinetti.comvincerubinetti.bandcamp.com/album/the-music-of-3blue1brownopen.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u------------------3blue1brown is a channel about animating math, in all senses of the word animate. If you're reading the bottom of a video description, I'm guessing you're 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: 3blue1brown.substack.comTwitter: twitter.com/3blue1brownInstagram: instagram.com/3blue1brownReddit: reddit.com/r/3blue1brownFacebook: facebook.com/3blue1brownPatreon: patreon.com/3blue1brownWebsite: 3blue1brown.com
A cute probability fact (part 3) 3Blue1Brown 2024-09-10 | Why max(rand(), rand()) is the same as sqrt(rand())See Matt Parker's video for more:youtu.be/ga9Qk38FaHM
A cute probability fact (part 2) 3Blue1Brown 2024-09-10 | How do think about max(rand(), rand())The next short finishes the explanation: youtube.com/shorts/lpzUZDefha0
A cute probability fact (part 1) 3Blue1Brown 2024-09-10 | max(rand(), rand()) has the same effect as sqrt(rand())The next short explains why: youtube.com/shorts/sNWDjbaT208See Matt Parker's video for more:youtu.be/ga9Qk38FaHM
How might LLMs store facts | Chapter 7, Deep Learning 3Blue1Brown 2024-08-31 | Unpacking the multilayer perceptrons in a transformer, and how they may store factsInstead of sponsored ad reads, these lessons are funded directly by viewers: 3b1b.co/supportAn equally valuable form of support is to share the videos.AI Alignment forum post from the Deepmind researchers referenced at the video's start:alignmentforum.org/posts/iGuwZTHWb6DFY3sKB/fact-finding-attempting-to-reverse-engineer-factual-recallAnthropic posts about superposition referenced near the end:https://transformer-circuits.pub/2022/toy_model/index.htmlhttps://transformer-circuits.pub/2023/monosemantic-featuresSome added resources for those interested in learning more about mechanistic interpretability, offered by Neel NandaMechanistic interpretability paper reading listalignmentforum.org/posts/NfFST5Mio7BCAQHPA/an-extremely-opinionated-annotated-list-of-my-favouriteGetting started in mechanistic interpretabilityneelnanda.io/mechanistic-interpretability/getting-startedAn interactive demo of sparse autoencoders (made by Neuronpedia)neuronpedia.org/gemma-scope#mainCoding tutorials for mechanistic interpretability (made by ARENA)https://arena3-chapter1-transformer-interp.streamlit.app/Sections:0:00 - Where facts in LLMs live2:15 - Quick refresher on transformers4:39 - Assumptions for our toy example6:07 - Inside a multilayer perceptron15:38 - Counting parameters17:04 - Superposition21:37 - Up next------------------These animations are largely made using a custom Python library, manim. See the FAQ comments here:3b1b.co/faq#manimgithub.com/3b1b/manimgithub.com/ManimCommunity/manimAll code for specific videos is visible here:github.com/3b1b/videosThe music is by Vincent Rubinetti.vincentrubinetti.comvincerubinetti.bandcamp.com/album/the-music-of-3blue1brownopen.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u------------------3blue1brown is a channel about animating math, in all senses of the word animate. If you're reading the bottom of a video description, I'm guessing you're 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: 3blue1brown.substack.comTwitter: twitter.com/3blue1brownInstagram: instagram.com/3blue1brownReddit: reddit.com/r/3blue1brownFacebook: facebook.com/3blue1brownPatreon: patreon.com/3blue1brownWebsite: 3blue1brown.com
In the vector space of all advice... 3Blue1Brown 2024-05-22 | A link to the full video is on the screen, or here for reference: youtu.be/W3I3kAg2J7w
What Follow Your Dreams Misses | Harvey Mudd Commencement Speech 2024 3Blue1Brown 2024-05-18 | I had the pleasure of being invited to give Harvey Mudd's commencement speech this year.Reposted here with permission from the UniversityTimestamps:0:00 - End of Harriet Nembhard's introduction0:45 - The cliché2:28 - The shifting goal5:57 - Action precedes motivation7:02 - Timing10:47 - Know your influence12:05 - Anticipate change------------------3blue1brown is a channel about animating math, in all senses of the word animate. If you're reading the bottom of a video description, I'm guessing you're 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: 3blue1brown.substack.comTwitter: twitter.com/3blue1brownInstagram: instagram.com/3blue1brownReddit: reddit.com/r/3blue1brownFacebook: facebook.com/3blue1brownPatreon: patreon.com/3blue1brownWebsite: 3blue1brown.com
Temperature in LLMs 3Blue1Brown 2024-04-28 | This comes from a full video breaking down how LLMs work. The link is on the bottom of the screen (in the shorts feed at least), or here for reference: youtu.be/wjZofJX0v4M
How word vectors encode meaning 3Blue1Brown 2024-04-11 | This comes from a full video dissecting how LLMs work. In the shorts player, you can click the link at the bottom of the screen, or for reference: youtu.be/wjZofJX0v4M
Attention in transformers, visually explained | Chapter 6, Deep Learning 3Blue1Brown 2024-04-07 | Demystifying attention, the key mechanism inside transformers and LLMs.Instead of sponsored ad reads, these lessons are funded directly by viewers: 3b1b.co/supportSpecial thanks to these supporters: 3blue1brown.com/lessons/attention#thanksAn equally valuable form of support is to simply share the videos.Demystifying self-attention, multiple heads, and cross-attention.Instead of sponsored ad reads, these lessons are funded directly by viewers: 3b1b.co/supportThe first pass for the translated subtitles here is machine-generated, and therefore notably imperfect. To contribute edits or fixes, visit translate.3blue1brown.comAnd yes, at 22:00 (and elsewhere), "breaks" is a typo.------------------Here are a few other relevant resourcesBuild a GPT from scratch, by Andrej Karpathyyoutu.be/kCc8FmEb1nYIf you want a conceptual understanding of language models from the ground up, @vcubingx just started a short series of videos on the topic:youtu.be/1il-s4mgNdI?si=XaVxj6bsdy3VkgEXIf you're interested in the herculean task of interpreting what these large networks might actually be doing, the Transformer Circuits posts by Anthropic are great. In particular, it was only after reading one of these that I started thinking of the combination of the value and output matrices as being a combined low-rank map from the embedding space to itself, which, at least in my mind, made things much clearer than other sources.https://transformer-circuits.pub/2021/framework/index.htmlSite with exercises related to ML programming and GPTsgptandchill.ai/codingproblemsHistory of language models by Brit Cruise, @ArtOfTheProblem youtu.be/OFS90-FX6pgAn early paper on how directions in embedding spaces have meaning:arxiv.org/pdf/1301.3781.pdf------------------Timestamps:0:00 - Recap on embeddings1:39 - Motivating examples4:29 - The attention pattern11:08 - Masking12:42 - Context size13:10 - Values15:44 - Counting parameters18:21 - Cross-attention19:19 - Multiple heads22:16 - The output matrix23:19 - Going deeper24:54 - Ending------------------These animations are largely made using a custom Python library, manim. See the FAQ comments here:3b1b.co/faq#manimgithub.com/3b1b/manimgithub.com/ManimCommunity/manimAll code for specific videos is visible here:github.com/3b1b/videosThe music is by Vincent Rubinetti.vincentrubinetti.comvincerubinetti.bandcamp.com/album/the-music-of-3blue1brownopen.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u------------------3blue1brown is a channel about animating math, in all senses of the word animate. If you're reading the bottom of a video description, I'm guessing you're 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: 3blue1brown.substack.comTwitter: twitter.com/3blue1brownInstagram: instagram.com/3blue1brownReddit: reddit.com/r/3blue1brownFacebook: facebook.com/3blue1brownPatreon: patreon.com/3blue1brownWebsite: 3blue1brown.com
How large language models work, a visual intro to transformers | Chapter 5, Deep Learning 3Blue1Brown 2024-04-01 | Breaking down how Large Language Models workInstead of sponsored ad reads, these lessons are funded directly by viewers: 3b1b.co/support---Here are a few other relevant resourcesBuild a GPT from scratch, by Andrej Karpathyyoutu.be/kCc8FmEb1nYIf you want a conceptual understanding of language models from the ground up, @vcubingx just started a short series of videos on the topic:youtu.be/1il-s4mgNdI?si=XaVxj6bsdy3VkgEXIf you're interested in the herculean task of interpreting what these large networks might actually be doing, the Transformer Circuits posts by Anthropic are great. In particular, it was only after reading one of these that I started thinking of the combination of the value and output matrices as being a combined low-rank map from the embedding space to itself, which, at least in my mind, made things much clearer than other sources.https://transformer-circuits.pub/2021/framework/index.htmlSite with exercises related to ML programming and GPTsgptandchill.ai/codingproblemsHistory of language models by Brit Cruise, @ArtOfTheProblem youtu.be/OFS90-FX6pgAn early paper on how directions in embedding spaces have meaning:arxiv.org/pdf/1301.3781.pdf---Timestamps0:00 - Predict, sample, repeat3:03 - Inside a transformer6:36 - Chapter layout7:20 - The premise of Deep Learning12:27 - Word embeddings18:25 - Embeddings beyond words20:22 - Unembedding22:22 - Softmax with temperature26:03 - Up next
Simulating the electric field and a moving charge 3Blue1Brown 2024-01-26 | A link to the full video is at the bottom of the screen.Or, for reference: youtu.be/aXRTczANuIs
How the Mandelbrot set is defined 3Blue1Brown 2024-01-24 | A link to the full video answering this is at the bottom of the screen. Or, for reference: youtu.be/LqbZpur38nwThanks to these viewers for their contributions to translationsBulgarian: Martin GrozdanovFrench: GiveMeChocolate, YoyodotpyGerman: Josh, dlatikayHebrew: Omer TuchfeldHindi: rajeshwar-pandeySpanish: Marcelo Lynch
A challenging puzzle about subset sums 3Blue1Brown 2024-01-22 | A link to the full video answering this is at the bottom of the screen. Or, for reference: youtu.be/bOXCLR3WricThanks to these viewers for their contributions to translationsFrench: GiveMeChocolateHindi: rajeshwar-pandeySpanish: Yago Iglesias
Ellipses have multiple definitions, how are these the same? 3Blue1Brown 2024-01-19 | A link to the full video is at the bottom of the screen.Or, for reference: youtu.be/pQa_tWZmlGsThe full video this comes from proves why slicing a cone gives the same shape as the two-thumbtacks-and-string construction, which is beautiful.Editing from long-form to short by Dawid Kołodziej
Three levels of understanding Bayes theorem 3Blue1Brown 2024-01-17 | A link to the full video is at the bottom of the screen.Or, for reference: youtu.be/HZGCoVF3YvMEditing from long-form to short by Dawid Kołodziej
The medical test paradox (well paradox) 3Blue1Brown 2024-01-15 | A link to the full video about Bayesian thinking is at the bottom of the screen.Or, for reference: youtu.be/lG4VkPoG3koLong-to-short editing by Dawid Kołodziej
Positioned as the hardest question on a Putnam exam (#6, 1992) 3Blue1Brown 2024-01-12 | A link to the full video is at the bottom of the screen.Or, for reference: youtu.be/OkmNXy7er84Editing from the original video into this short by Dawid Kołodziej
Why does light slowing imply a bend? (Beyond the tank/car analogy) 3Blue1Brown 2024-01-11 | A link to the full video is at the bottom of the screen.Or, for reference: youtu.be/Cz4Q4QOuoo8That video answers various viewer questions about the index of refraction.Editing from long-form to short by Dawid Kołodziej
The cube shadow puzzle 3Blue1Brown 2024-01-09 | A link to the full video is at the bottom of the screen. Or, for reference: youtu.be/ltLUadnCyi0Thanks to these viewers for their contributions to translationsChinese: ZstringXFrench: GiveMeChocolate, YoyodotpyGerman: Josh, dlatikayHindi: VaMErYT, rajeshwar-pandeyKorean: tebaioioo
What does it mean that light slows down in glass? 3Blue1Brown 2024-01-08 | A link to the full video is at the bottom of the screen.Or, for reference: youtu.be/KTzGBJPuJwMThat video unpacks the mechanism behind how light slows down in passing through a medium, and why the slow-down rate would depend on color.Editing from long-form to short by Dawid Kołodziej
Why do we call them scalars? 3Blue1Brown 2024-01-06 | A link to the full video is at the bottom of the screen.Or, for reference: youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_abEditing from long-form to short by Dawid Kołodziej
A beautiful international math olympiad problem 3Blue1Brown 2024-01-03 | The link to the full video is at the bottom of the screen. For reference, here it is: youtu.be/M64HUIJFTZM
Definition of a bit, in information theory 3Blue1Brown 2024-01-02 | A link to the full video is at the bottom of the screen.Or, for reference: youtu.be/v68zYyaEmEAThat video describes using information theory to write a bot that plays WordleEditing from long-form to short by Dawid Kołodziej
The Newton art puzzle 3Blue1Brown 2023-12-29 | A link to the full video is at the bottom of the screen. Or, for reference: youtu.be/-RdOwhmqP5sThanks to these viewers for their contributions to translationsFrench: PyStLSpanish: Yago Iglesias
What is a group? 3Blue1Brown 2023-12-27 | A link to the full video is at the bottom of the screen.Or, for reference: youtu.be/mH0oCDa74tEThat video introduces group theory and the monster group.Editing from long-form to short by Dawid Kołodziej
How to derive a formula for π 3Blue1Brown 2023-12-25 | A link to the full video is at the bottom of the screen.Or, for reference: youtu.be/NaL_Cb42WyYThat video explores how this question leads to a quandary on prime numbers, and how a pattern in primes allows for a clean final answer.Editing from long-form to short by Dawid Kołodziej
The limit of limiting arguments 3Blue1Brown 2023-12-23 | A link to the full video is at the bottom of the screen.Or, for reference: youtu.be/VYQVlVoWoPYThat video gives multiple examples of lying with visual proofsEditing from the original video into this short by Dawid Kołodziej
For anyone who might not know how links in shorts work 3Blue1Brown 2023-12-21 | YouTube disabled links in descriptions and comments, but we can add links to videos on the shorts player itself. For reference, the one at the bottom of this screen is youtu.be/KTzGBJPuJwM
Infinite Lighthouses and π 3Blue1Brown 2023-12-21 | A link to the full video is at the bottom of the screen.Or, for reference: youtu.be/d-o3eB9sfls
Can you even imagine 2^256? 3Blue1Brown 2023-12-16 | Originally written as a supplement to an explanation of the cryptography behind Bitcoin: youtu.be/bBC-nXj3Ng4(An active link is on the bottom of the video player)
Order from chaos 3Blue1Brown 2023-12-15 | A link to the full video on the Central Limit Theorem is at the bottom of the screen.Or, for reference: youtu.be/zeJD6dqJ5loThanks to Dawid Kołodziej from long-to-short editing
The surface area of a sphere 3Blue1Brown 2023-12-14 | A link to the full video is at the bottom of the screen.Or, for reference: youtu.be/GNcFjFmqEc8
A short on shorts 3Blue1Brown 2023-12-13 | Animations taken from this video: youtu.be/-RdOwhmqP5sAnd this one: youtu.be/LqbZpur38nw(Description links are not active in the shorts player, but you can follow the link at the bottom of the video screen itself)
A pretty way to add weighted dice 3Blue1Brown 2023-12-12 | A link to the full video is at the bottom of the screen.Or, for reference: youtu.be/IaSGqQa5O-MIt describes convolutions in probability, extending to the continuous caseEditing from long-form to short by Dawid Kołodziej
A simple image convolution 3Blue1Brown 2023-12-12 | A link to the full video is at the bottom of the screen.Or, for reference: youtu.be/KuXjwB4LzSAThat video introduces convolutions, as used in image processing, probability, and signal processing.Editing from long-form to short by Dawid Kołodziej
These integrals all equal π, until... 3Blue1Brown 2023-12-11 | A link to the full video is at the bottom of the screen.Or, for reference: youtu.be/851U557j6HEThese are known as Borwein integralsEditing from long-form to short by Dawid Kołodziej
The split necklace puzzle (with a surprise topological solution) 3Blue1Brown 2023-12-11 | A link to the full video is at the bottom of the screen.Or, for reference: youtu.be/yuVqxCSsE7cA puzzle about stolen necklaces, from a video about the Borsuk Ulam theorem in topologyEditing from long-form to short by Dawid Kołodziej
Error correction is incredible 3Blue1Brown 2023-12-10 | A link to the full video is at the bottom of the screen.Or, for reference: youtu.be/X8jsijhllIAEditing from long-form to short by Dawid Kołodziej
The chessboard and coins puzzle 3Blue1Brown 2023-12-10 | Video with the solution: youtu.be/as7Gkm7Y7h43b1b on a meta-puzzle: youtu.be/wTJI_WuZSwE(These description links aren't active in the shorts player, but you can follow the link on the bottom of the video screen itself)This comes from a collaboration I did with Stand-up Maths, where on his channel we covered the solution, and here on 3blue1brown we analyze a meta-puzzle.Editing from long-form to short by Dawid KołodziejThis comes from a collaboration I did with Stand-up Maths, where on his channel we covered the solution, and here on 3blue1brown we analyze a meta-puzzle.Editing from long-form to short by Dawid Kołodziej
Prime spirals 3Blue1Brown 2023-12-09 | A link to the full video is at the bottom of the screen.Or, for reference: youtu.be/EK32jo7i5LQThanks to Dawid Kołodziej for editing this short from the original.
The barber pole effect 3Blue1Brown 2023-12-09 | A link to the full video is at the bottom of the screen.Or, for reference: youtu.be/QCX62YJCmGkFilming by Quinn BrodskyEditing from long-form to short by Dawid Kołodziej
Fourier series 3Blue1Brown 2023-12-08 | A link to the full video is at the bottom of the screen.Or, for reference: youtu.be/r6sGWTCMz2kThat video tells the story of how this concept was originally invented to solve the heat equation.Thanks to Dawid Kołodziej for editing together this short
Seeing with sound 3Blue1Brown 2023-12-08 | A link to the full video is at the bottom of the screen.Or, for reference: youtu.be/3s7h2MHQtxcEditing from the original video into this short by Dawid Kołodziej
How prisms work (video linked above shows why light slows down) 3Blue1Brown 2023-12-07 | A link to the full video is at the bottom of the screen.Or, for reference: youtu.be/KTzGBJPuJwMThere, I wanted to dig deeper to understand light slows down, and why this would depend on the color.
Im still astounded this is true 3Blue1Brown 2023-12-07 | A link to the full video is at the bottom of the screen.Or, for reference: youtu.be/jsYwFizhncESliding blocks on a frictionless plane, counter their collisions, and...Thanks to Dawid Kołodziej for editing together this short
Dont let it fool you! 3Blue1Brown 2023-12-07 | A link to the full video is at the bottom of the screen.Or, for reference: youtu.be/YtkIWDE36qUThanks to Dawid Kołodziej for editing together this short
How well do you understand refraction? | Optics puzzles 4 3Blue1Brown 2023-12-03 | Why bending, how can light go "faster" than light, and moreLessons are primarily funded directly by viewers, who get early access to new videos: 3b1b.co/supportAn equally valuable form of support is to simply share the videos.Much of the last video, as well as this one, is based on the following Feynman Lecture:https://www.feynmanlectures.caltech.edu/I_31.htmlLooking Glass Universe videos on the refractive index:youtu.be/uo3ds0FVpXs?si=Q12Rgz9vN1JMo_diTimestamps:0:00 - Why slowing implies bending3:36 - Recap for how slowing happens5:08 - Birefringence6:19 - The barber pole8:20 - When the refractive index is less than 1Thanks to these viewers for their contributions to translationsFrench: LE PRAT Ronan, PyStLHebrew: Omer TuchfeldPortuguese: rose ✨Russian: iverid------------------These animations are largely made using a custom Python library, manim. See the FAQ comments here:3b1b.co/faq#manimgithub.com/3b1b/manimgithub.com/ManimCommunity/manimAll code for specific videos is visible here:github.com/3b1b/videosThe music is by Vincent Rubinetti.vincentrubinetti.comvincerubinetti.bandcamp.com/album/the-music-of-3blue1brownopen.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u------------------3blue1brown is a channel about animating math, in all senses of the word animate. If you're reading the bottom of a video description, I'm guessing you're 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: 3blue1brown.substack.comTwitter: twitter.com/3blue1brownInstagram: instagram.com/3blue1brownReddit: reddit.com/r/3blue1brownFacebook: facebook.com/3blue1brownPatreon: patreon.com/3blue1brownWebsite: 3blue1brown.comtesting an addition
But why would light slow down? | Optics puzzles 3 3Blue1Brown 2023-11-30 | How the index of refraction arises, and why it depends on color.Quotebook Notebooks: 3b1b.co/storeThese lessons are primarily funded directly by viewers: 3b1b.co/supportAn equally valuable form of support is to simply share the videos.Looking Glass Universe videos on the index of refraction:youtu.be/uo3ds0FVpXsMuch of this video is based on the following Feynmann lecturehttps://www.feynmanlectures.caltech.edu/I_31.htmlThe explanation for why the phase of a wave produced by a plane of oscillating charges is a quarter phase behind the wave of a charge in the center of that plane, and hence a quarter phase behind that of a light wave inducing the oscillations, is given in the previous chapter:https://www.feynmanlectures.caltech.edu/I_30.htmlSections:0:00 - The standard explanation3:14 - The plan5:09 - Phase kicks8:25 - What causes light?13:20 - Adding waves16:40 - Modeling the charge oscillation20:59 - The driven harmonic oscillator26:57 - End notesThanks to these viewers for their contributions to translationsGerman: OleHebrew: Alon Altman, Omer TuchfeldHindi: SaurabhPortuguese: @doryhelioairesUrdu: Momini0------------------These animations are largely made using a custom Python library, manim. See the FAQ comments here:3b1b.co/faq#manimgithub.com/3b1b/manimgithub.com/ManimCommunity/manimAll code for specific videos is visible here:github.com/3b1b/videosThe music is by Vincent Rubinetti.vincentrubinetti.comvincerubinetti.bandcamp.com/album/the-music-of-3blue1brownopen.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u------------------3blue1brown is a channel about animating math, in all senses of the word animate. If you're reading the bottom of a video description, I'm guessing you're 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: 3blue1brown.substack.comTwitter: twitter.com/3blue1brownInstagram: instagram.com/3blue1brownReddit: reddit.com/r/3blue1brownFacebook: facebook.com/3blue1brownPatreon: patreon.com/3blue1brownWebsite: 3blue1brown.com
