Summer of Math Exposition
Archive
| Entry | Title | Rank |
|---|---|---|
What P vs NP is actually aboutWhat if we could run algorithms backwards? We discuss how we could (possibly?) do this by turning them into circuits and turning those into satisfiability problems. If we could do that efficiently, it turns out crazy things would happen! | #1 details | |
What School Didn't Teach You About MazesThis video provides a painless introduction to the ideas of Maze Generation and the relating Graph Theory. It also highlights an uncommon algorithm that deserves more attention, particularly in the world of ever-changing mazes. | #2 details | |
How To Make a Computer Create Something Beautiful: String ArtThis is the second video I've made about string art; a technique to make an artwork from a single thread wrapped around nails along a circular canvas. The first video was received very well. I even had many people throw a bunch of really creative ideas at me. I would have never thought about the radon transform if it wasn't for such an amazing community! Thanks so much! | #3 details | |
Programming with Math: The Lambda CalculusThe Lambda Calculus is a tiny mathematical programming language that has the same computational power as any language you can dream of. In this video, we'll first explore this calculus before seeing how we can flesh it out into a functional programming language. After a brief tour of a simple type system, we'll see why the Lambda Calculus has some surprising applications in the field of mathematical logic, and how the implications of this relationship could alter the way that we study mathematics forever. | #4 details | |
Why is this "Fundamental" to arithmetic?An animated video about the fundamental theorem of arithmetic and uniqueness of prime factorisation (UPF). The video starts off by giving examples of why UPF is non-trivial and then proceeds to show why it is truly fundamental again using examples. Afterwards a proof of UPF is provided from first principles. | #5 details | |
Latent Space Visualisation: PCA, t-SNE, UMAP | Deep Learning AnimatedWe explore 3 common dimensionality reduction techniques while keeping in mind the application to Deep Learning. | #6 details | |
How to Calculate 1^3 + 2^3 + ... + 100^3 By HandThis video shows how power sums are hiding in Pascal's Triangle. With this knowledge, we work towards a generalized technique to form a simplified expression for any power sum. | #6 details | |
Penrose Tilings from Five DimensionsThis contribution highlights that aperiodic Penrose tilings can be viewed as projections of a periodic five dimensional cubic lattice. We focus on the geometric properties of this projection and try to motivate the choices that are required for a Penrose tiling. To cope with the five-dimensional geometry, lots of toy models and analogies are exploited. | #7 details | |
Can You Beat The Demon At His Table Game?This is a video about a riddle. We'll explain it, build up the solution and continue to develop upon its basic ideas. | #8 details | |
How Many Symmetries Are ThereExplains how to calculate the number of possible symmetries spherical, wallpaper, and frieze symmetries. Uses a clever argument based on orbifolds and the Euler Map Theorem. The argument is based on a book called the Symmetry of Things. The argument is very visual. This animation illustrates concepts that are hard to present in book format. The illustrations and animations are all original. | #8 details | |
An entire physics class in 76 minutesAn in-depth explanation of nearly everything I learned in an undergrad electricity and magnetism class. Discord: https://discord.gg/XW4tSaxa6w Patreon: https://www.patreon.com/tmodx Source Code: https://github.com/tmodx/an-entire-physics-class-in-76-minutes References, Music, and Further Material: https://pastebin.com/871qFj2s Chapters: 0:00:00 - Intro 0:00:32 - Chapter 1: Electricity 00:21:50 - Chapter 2: Circuits 00:40:44 - Chapter 3: Magnetism 1:03:25 - Chapter 4: Electromagnetism 1:15:46 - Outro Concepts: Coulomb's Law - 0:02:25 Electric field - 00:03:45 Flux - 00:05:57 Gauss's Law - 00:07:05 Voltage - 00:13:47 Four Quantities Review - 00:20:23 Resistance and Current - 00:22:22 Resistors in Series and Parallel - 00:24:54 Power - 00:27:00 Voltage Drops - 00:27:38 Kirchhoff Equations - 00:28:12 Capacitors - 00:29:00 Capacitors in Series and Parallel - 00:32:00 Capacitor Energy - 00:33:37 AC Circuits - 00:35:18 Biot-Savart Law - 00:41:08 Lorenz Force Law - 00:43:42 Gauss's Law for Magnetism - 00:51:47 Ampere's Law - 00:52:53 Faraday's Law - 00:53:50 Lenz's Law - 00:55:20 Eddy Currents - 00:56:05 Transformers - 00:56:59 Types of Magnetism (Ferromagnetism, Paramagnetism, and Diamagnetism) - 00:58:25 Inductors - 1:00:14 Ampere's Law Correction - 1:03:58 Electromagnetic Waves - 1:04:48 Intensity - 1:06:56 Radiation Pressure - 1:09:30 Law of Reflection - 1:11:34 Snell's Law - 1:12:17 Physical Optics - 1:14:20 X-ray Diffraction (Bragg Diffraction) - 1:15:18 | #9 details | |
The algorithm that (eventually) revolutionized statisticsAn explainer on the Metropolis algorithm | #9 details | |
The Basics of Circular Motion (and Beyond)!In this video we take an in depth look at the basics of circular motion, and extend it to motion along an arbitrary path with a changing speed. We also investigate a conical pendulum in detail and show that there are no centripetal forces acting on mass! | #9 details | |
The 4D Julia SetA longer from video exploring how one might would expand the Julia set from the Complex numbers into the quaternions. Along the way it is shown, why there isn't a 3D version of the fractal, as well as giving a bit of a historical context for the development of the tools needed to create a image of such a fractal. Quite a large part of the video tries to explain how the signed distance function can be (partially) derived. Along the way there are of course lots of nice fractal images presented :) | #9 details | |
This star almost broke Bohr's atomic modelStory of the mysterious spectral lines of Zeta Puppis, a star that seemed to challenge the atomic model proposed by Niels Bohr. Bohr cleverly solved the issue and turned the observation that nearly dismantled his revolutionary model into a successful prediction. | #10 details | |
How good is Advantage in D&D?In Dungeons and Dragons, "rolling with advantage" means rolling two dice and taking the highest. Can we equate this to a flat modifier? It turns out that this raises some interesting questions about probability. | #11 details | |
Why Quaternions (4d numbers) are usefulIn this video, I show how quaternions are used to describe orientation as well as rotation and the elegant method they provide for computing the transformed position of points. I also visualize what’s happening in the fourth dimension to answer some questions that can arise when using quaternions. | #12 details | |
Eisenstein Primes VisuallyThis video is a discussion of the Eisenstein Integers, applying the ideas of primes to get unique factorization of the integers, and the link between these Eisenstein primes and the natural primes. | #13 details | |
How many times do you ACTUALLY need to shuffle?This video is a deep dive under the hood of the commonly-held belief that you need to shuffle a deck 7 times before it's random. We discuss what this statement actually means, and how to mathematically prove something like this. | #14 details | |
The Moebius functionIn number theory, the Moebius function allows us to decompose complicated functions into simpler parts. The definition of this function can be difficult to understand, so we flesh it out one step at a time. We start with the Dirichlet convolution, we look at its properties, and finally we look for inverses of number sequences. One of those inverses is the Moebius function. We understand how it works by looking at a Hasse diagram. | #15 details | |
Hypergraphs and Acute TrianglesWe solve a problem about acute triangles from the 1970 International Math Olympiad (Problem 6). Moreover, we improve the limiting bound from 70% to 56.15%. The solution uses recent results from the theory of hypergraphs. | #16 details | |
Everything You Need to Know About the Double PendulumI give a detailed explanation of what it means for a Double Pendulum to be unpredictable yet deterministic. Along the way, I derive the equations of motion using the Lagrangian. I also show how sensitive the motion is to the initial conditions with several animations. | #17 details | |
Bresenham's Line Algorithm - Demystified Step by StepBresenham's Line Algorithm is simple, but how exactly does it work? In this video we go through the steps necessary to draw a line inside a grid of pixels using only integers. To do this, we're going to write the algorithm from scratch and derive the decision parameter used in Bresenham's algorithm. | #18 details | |
Fibonacci Nim - Play Zeckendorf First?Fibonacci Nim is a two-player strategy game where players take turns removing ducks from a pile. But how can we win? What does Fibonacci have to do with it? And what other beautiful math can we uncover along the way? | #19 details | |
What is Diffusivity? (Why does it keep showing up? Why do they have the same units?)Diffusivity is a parameter that shows up in molecular diffusion, heat conduction, and fluid dynamics. In each of those cases, it has the same units: m^2/s. To understand why, let's go on an... inebriated stroll... through the mathematical model. [A familiarity with derivatives (basic calculus) is recommended.] | #20 details | |
these compression algorithms could halve our image file sizes (but we don't use them)an explanation of the source coding theorem, arithmetic coding, and asymmetric numeral systems | #21 details | |
The Matrix Transpose: Visual IntuitionLet's look at what the transpose of a matrix means intuitively. We'll understand how the transpose of a matrix is needed for trying to find pairs of vectors that have the same dot product before and after some linear transformation. We'll also use the Singular Value Decomposition to get a better geometric intuition for how these transformations appear geometrically. #matrix #linearalgebra #transpose #svd | #22 details | |
AI Powered Next-Gen Graphics Rendering || Watch a Cybertruck Come to Life!This video is about the technical details and implementation of Neural Radiance Fields from scratch. We cover all the fundamental math and coding to build your own NeRF! | #23 details | |
Writing Software to Solve This One Taskmaster TaskA showcase of the making of the real-time anagram solving software I wrote to aid in a speech-writing task in an episode of Taskmaster, and a look into the optimization I found to make it run super fast using bitwise operations. If you are interested in optimization algorithms or creative writing--or you're just a fan of Taskmaster--this video is for you. | #24 details | |
Visualizing 4D Pt. 1The first video in a multi-part series on understanding and visualizing four dimensions, from a mathematical point-of-view. | #25 details | |
when you don't know the answerLink to interactive Web.VPython simulation: https://trinket.io/library/trinkets/1fe7d9cdd6 Music Ravel: Gaspard de la Nuit - Ondine (Performer: Bernd Kreuger) https://commons.wikimedia.org/wiki/File:Maurice_Ravel_Ondine.ogg “Impact Prelude” - Kevin Macleod https://www.youtube.com/watch?v=on7qf4jUbgw “Late Night Radio” - Kevin Macleod https://www.youtube.com/watch?v=QhtL-6Cf5Xk Credits MIT Runge Kutta Derivation https://web.mit.edu/10.001/Web/Course_Notes/Differential_Equations_Notes/node5.html Atomic Vibration Clip (electrons technically) https://www.youtube.com/watch?v=ofp-OHIq6Wo NASA Hurricane Video https://youtu.be/p-3aB9hJ8Hc?si=qha6kCicKlDLHOxo Solar System Clip https://youtu.be/z8aBZZnv6y8?si=3VX2OkpMlKOix76k | #26 details | |
A beautiful example of double countingDouble counting is an interesting problem solving technique in discrete mathematics. This video introduces this technique through examples. | #27 details | |
Differential equations & the Lotka-Volterra rulesWith differential equations, we can model many phenomena in nature because describing the rate of change of something is often simpler than describing the quantity itself. Let me show you a few examples. | #28 details | |
Zorn's Lemma DemystifiedThis video explains Zorn's lemma: exactly what the statement means, and when and how it is used. The audience is people that know some math and have heard of Zorn's lemma but have no idea what it's actually about. | #29 details | |
Graph Traversal AlgorithmsThe Graph Traversal Algorithms BFS and DFS are some of the most fundamental algorithms in graph theory. In this video, we will investigate the large application range of these algorithms as well as their implementations. | #30 details | |
Finite Element Method Explained in 3 Levels of DifficultyThe finite element method is difficult to understand when studying all of its concepts at once. Therefore, I explain the finite element method in three levels with increasing complexity. | #31 details | |
The 1/3-2/3 ConjectureThe 1/3-2/3 Conjecture is one of the biggest unsolved problems in Order Theory. In this video we introduce posets, the 1/3-2/3 conjecture, what the current state of the art is on the problem, and why it matters. | #32 details | |
Carving the Column [Mapping a Grid onto a Curved Surface]Ever wonder how 2D designs get applied to crazy, curvy 3D surfaces? In this video we look at one specific case, the tapered column, and how a very useful mathematical tool (rhymes with Molar Subordinates) transforms this problem from daunting to totally do-able. | #33 details | |
The Math for Folding OrigamiThis video goes over three theorems about crease patterns. Crease patterns are the resulting lines made from folds after unfolding an origami piece. Target audience: Middle schoolers and above The topic choice was picked to be a novelty to most viewers | #34 details | |
So you're tired of drawing free-body diagramsThis video explores how an intuitive approach to path optimization problems leads from Newtonian Mechanics to Lagrangian mechanics. Aimed at people who have studied calculus and a little bit of physics, but not necessarily studying physics as a career path, this video hopes to interest people in the subject of physics beyond first-year physics courses. | #35 details | |
What Is UNIFORM Convergence ? Why Do We Care ? | Episode 1 - Sequences Of FunctionsWe know how to define without any ambiguity the convergence of a sequence of reals, and more generally in finite dimension, the convergence of a sequence of vectors, thanks to the equivalence of norms. On the other hand, it is much more difficult to define the convergence of a sequence of functions. Several natural, useful but not equivalent definitions are possible, among which we distinguish simple/pointwise convergence and uniform convergence. Those are the two modes of convergence that we we will discover intuitively and VISUALIZE together. The emphasis is on visualizations, as well as the transmission of the mathematical "feeling" that is hidden behind the sometimes off-putting formalism. For those who would like to support me: https://fr.tipeee.com/kobipy/ A comment, a like or a share are just as appreciated! 😉 The animations were made in Python, with the Manim module : ttps://www.manim.community/ The background music is : Piano Music by: Bensound.com/free-music-for-videos License code: NUU8ZZSW39K0YG4M Some of the images were taken on : https://www.vecteezy.com/ ⏲ Timestamps : 00:00 - Introduction 00:23 - Pointwise convergence 02:35 - Uniform convergence 08:29 - Conclusion Other videos on this channel that might interest you : ◆ What is an area under the curve ? https://youtu.be/E0Ryo199-Gs ◆ Everything you DREAM to know about The LEBESGUE Integral : https://youtu.be/Oigh-j52CqE ◆ The Myth of the Horizontal LEBESGUE Integral ? | Épisode 3 : https://youtu.be/zUg_KVBd7e4 ◆ Where Is The Circle ? | Gauss Integral : https://youtu.be/U2xmox321_k ◆ How Many Squares on the ChessBoard ? | Bijection & Counting : https://youtu.be/2adERtKWCek ◆ The Weierstrass Function is Everywhere Continous, Nowhere Differentiable : https://www.youtube.com/watch?v=37tG_qvBb3M ◆ The BATMAN Equation : https://youtu.be/5SVRLxCBu9E ◆ The Time Table IS Inside your Cup : https://youtu.be/tBoE568elCE ◆ Can you solve the 6 enigma ? https://youtu.be/ikDqP5b9lgM | #35 details | |
The ALMOST Perfect NumbersThe perfect numbers are some of the most famous in all of number theory. But since perfection is so hard to achieve, we're naturally drawn to the mathematical quest for perfection. That manifests itself in the form of many types of almost perfect numbers. | #36 details | |
Probability "Paradoxes" Explained with Common SenseA video about probability. | #37 details | |
Brent’s Method of Finding Roots and Inverse FunctionsA description of Brent's Method and how to use it to compute an inverse function. Its application to the solution of the Saha Equation - a fundamental part of plasma physics - is also discussed. | #38 details | |
What do Lifeguards, Lasers, and Archery Fishing Have in Common?In this video we use math from a standard lifeguard-path optimization problem to understand why light refracts, and then modify the math to measure the speed of light in water using a ruler, a cup of water, and a laser pointer. | #38 details | |
Is My Dad Cheating?My family loves playing Settlers of Catan, and yet I always find myself losing to my dad. I think the dice are rigged, so I decide to do a chi-square goodness of fit test to check my hypothesis. | #39 details | |
Make Better Decisions With Less Data - Bayesian Statistics (Part 2)Summary: This video continues our Bayesian series launched earlier this summer. In Part 2 (Part 1: https://youtu.be/NLKWLBJ-b9E), we will learn how Bayes' theorem can be used in statistics! We will come up with our estimate of a model parameter via the Bayesian approach, and we will use our Bayesian result to answer a realistic question. This video provides a slight twist on the infamous 'sun rise problem', with alterations made to make everything a bit more clear, hopefully (e.g., modeling everything discretely). Note: Thank you for hosting this year's competition. The deadline was an excellent motivator to wrap up this video. I am looking forward to reviewing other submissions. | #39 details | |
Entropy Actually Explained [Machine Learning | Information Theory]Entropy is one of the most important, yet least understood statistical quantities. This video will clear up your questions, and is a deep dive into Entropy as it appears in different areas of mathematics, in particular Artificial Intelligence. We build an ML Decision Tree Classifier from scratch! | #40 details | |
Randomly playing chessIn this video, we explore what happens to chess pieces when they randomly move around the board. In particular, we explain a simple formula for a somehow complex question: if a single knight starts from a corner of the board and randomly moves around, how long will it take to return to its original location? | #41 details | |
The Geometry of AI Minds in Superposition (By Anthropic): Visually ExplainedWhat truly goes on in the inner thoughts of AI language models like ChatGPT? We will explore the strange geometric structures that concepts in their mind are arranged in. This video is based on this research paper: https://transformer-circuits.pub/2022/toy_model/index.html It is also related to my other video on AI mind reading based on recent OpenAI and Anthropic work: https://youtu.be/krINuMZhJmU | #42 details |