Summer of Math Exposition

An animated explainer on homotopy groups. It presents some geometric intuition for π₁/π₂/π₃ using loopspaces of metric spaces, in way that I don't see talked about enough. This includes an original, nonstandard visualization of the Hopf fibration.

As someone who teaches mathematics to college students (or sometimes high school students I know), I've always been wishing to share something students cannot easily find in the textbook so that they can use it for problem-solving and learn mathematics in more interesting way. I believe this is one of such things that (I am pretty sure) many high school students, college students, high school math teachers, or even college math professors will find interesting to learn. I haven't seen many people who know about this property - I exchanged an email with Grant, and hopefully he remembers this very interesting ratio property of a cubic function I shared with him via email. The goal for this video is to share a ratio property of a cubic function that is very easy to remember and use for a lot of math problems involving a cubic function but still not many people know of. I wish many people get to know this ratio property of a cubic function. According to my experience, it helped me a lot when solving a polynomial equation. So, why don't you check this out? This is a very cool technique I am pretty sure a lot of people would love to learn. Thanks a lot!

The P vs NP problem is widely agreed upon as the biggest unsolved question in computer science, asking whether discovery is harder than recognition -- if the solution to a problem is easily verifiable (like in sudoku, for example), does it also mean there’s an efficient way to find solutions in the first place? Our intuition says this should not be the case -- that solving a sudoku puzzle should be a lot harder than checking the solution once everything’s filled in. In 1956, despite the fact that computer science was a new discipline and hadn’t developed the theory and terminology we’d use today, Kurt Gödel was already pondering what the ultimate limits of computation might be, and he essentially foretold the P vs NP question 15 years before Stephen Cook would formalize it in 1971. In this video, we explore the P vs NP problem through that historical lens, thinking about the problem originally as Gödel did, in terms of a computer program trying to automatically find mathematical proofs, and eventually building up to the actual definitions of P and NP through a series of examples such as graph coloring.

In which we explore the concept of limits using flashlights and laser pointers. We even touch on the formal epsilon-delta definition at the end! I've imagining this would be helpful for a beginning calculus students, as well as people wanting to review the concept of limits in a bit more detail.

An undergrad / very-advanced-highschooler level explanation of the concept of 4D curved spacetime, grounded in the concrete problem of rendering blackholes and other spacetime metrics. Uses extensive animations I created using code, Keynote, and Blender to facilitate explanations. Almost anyone should be able to enjoy about the first 15-20 mins of the video, but some linear algebra knowledge is required for the latter part. Thanks a lot for watching and judging!!

A fictional tale inspired by Desargues’ Theorem — dedicated to secondary and high school students, in the hope of sparking a love for mathematics and encouraging exploration from the simplest ideas. This video is also a tribute to mathematicians — the curious, skeptical, and persistent minds who bring us such beautiful and meaningful theorems. Here, the inspiration is Desargues’ Theorem. My heartfelt thanks to An Minh Phan, Duy Nguyen, Long Do, and Ha Phuong Uyen for their energy, advice, encouragement, praise, motivation, care, spiritual support, and for giving me the best possible conditions to complete this video.

Polytoot, in excruciating detail, introduces the most well-known polyhedra of all time to anyone who, uh, may not be familiar with 3D space yet. Also, yes, this video did take me way too long to make. I'll be making some smaller videos after this one that hopefully shouldn't take as long. I'm aware that this has some graphical glitches in a few places. If I strived for absolute perfection, though, I'd be working on this until the end of time, because I'm just one person. So, I had to make a few compromises. I don't know when the next polytope video is releasing. I don't even know if anyone's going to want to watch this in the first place! But that's okay.

An informal introduction to the negative rows of Pascal's triangle, discussing the motivation and intuition behind some of its basic applications, before diving into discrete calculus, the forward and backward Newton series, umbral calculus, and as a consequence, a quick overview of Stirling numbers, for the purpose of highlighting some deeper, underappreciated treatments of discrete math. The target audience is meant to be a wider net, showcasing fun recreational ideas and basic combinatorics in the early chapters, slowly introducing connections to calculus and linear algebra as the video progressively gets more advanced, but hopefully the ideas throughout the entire video are novel enough that almost everyone will find something interesting within it. The main message conveyed here is that discrete math and combinatorics especially is often underappreciated, in my opinion, and this is just one particular direction that combinatorics can be pushed into, an outline of how its ideas can help shape intuition in another field. The value of pedagogy here for teachers is admittedly not explicitly high, more about the high-level message than about the particular ideas in this video.

What if you had a Matrix widget that you could program with? The video shows a Python library that let's you add matrices to a notebook that you can interact with. Every number becomes a draggable slider and ... it can also control the rest of the notebook! So it can update charts, re-run cells and interact with numpy/algorithms. This opens up many doors and invites one to do some deliberate play as they explore linear algebra. You're not just watching a video, you can be an active participant as you explore different matrices! The goal of the video is to inspire folks to play along as we demonstrate a few fun things that you can do with these tools.

The Art Gallery Problem involves determining the smallest number of guards (or 3D cameras) needed to ensure surveillance of an entire art gallery, which is generally represented by a simple polygon. This video will guide you through the basics of simple polygons, their triangulation, important theorems with beautiful proofs, and optimization methods to solve this complex problem.

This video gently explores the geometry behind perspective vision. We learn about a puzzle inspired by a work of art from the Renaissance, where the goal is to paint a grid but from a perspective view. We end up rediscovering one of the main themes of projective geometry, a type of geometry that includes points at infinity. The video is made to be accessible, intuitive, and satisfying!

What do file systems, websites, binary search algorithms, and infinite mathematical sequences all have in common? Trees. In this video, we explore the many lives of trees across computer science and mathematics; starting with familiar examples like your file explorer and the DOM, and branching out into data structures like tries and binary search trees. Then, we leave the real world behind and dive into the world of set-theoretic trees: structures that help mathematicians explore infinity, build fractals, and make sense of the strange shapes hidden in logic and topology. Whether you’re new to data structures or deep into descriptive set theory, this video shows how simple rules can create complex objects. Created as part of the 2025 Summer of Math Exposition, hosted by 3Blue1Brown.

My first math/manim video! A friendly introduction to hypothesis testing, with minimal math background required. Most p-value explanations that I've come across focus only on the mechanical process of calculation, without telling students why they're doing it or how to interpret the results. So this video is me attempting to motivate the concept of hypothesis testing from first principles. I had to cut things like error rates, test statistics, two-sided tests, and multiple testing correction for the next video, but Part 1 here should stand on its own. Thank you to all peer-reviewers for your time and feedback!

We introduce and explore deterministic finite automata (DFA) and regular expressions (regex), two tools that are usually taught early in a curriculum of theoretical computer science. We then prove that for every DFA there is a regex that describes exactly the same language. The main goals of the video are: - Build intuition for DFAs and regexes, by looking at many examples and exercises. - Understand the proof why every DFA can be translated into a regex. - Learn about two common proof strategies, namely generalization and induction, that are used commonly throughout mathematics. - Understand why the regular languages form a robust mathematical concept. No prior knowledge is required. The video should be suitable for undergraduates and high-school students.

A lighthearted yet in depth examination of pseudoprimes, and the modular arithmetic underlying primality. The target audience is anyone who loves numbers!

An introduction to some interesting applied mathematics used in Computer Vision. An algorithm based on a Hough Transform allows features like straight lines or circles to be detected in images or video. This topic is typically taught at the end of a Bachelors or Masters degree.

This is a video about torus knots, the Golden ratio, the Fibonacci sequence, musical intervals and cymatics. By combining these we are able to create a mesmerizing way of visualizing fundamental frequencies. We can apply this model to musical theory and perhaps even more.

This video explores the origins of algebra through two remarkable ancient methods, offering motivation for what it really means to “solve for X” or find the unknown in math class. This practice stretches back thousands of years. In Egypt, scribes solved practical problems using the method of false position, cleverly adjusting guesses to reach the answer. In Babylon, mathematicians recorded quadratic equations on clay tablets and solved them through geometry, an early form of completing the square. The video visualizes these problem-solving techniques with engaging animation and adapts the ancient methods into the language of modern mathematics, helping students connect their study of linear and quadratic equations to their deep historical roots. These approaches reveal the ingenuity of early problem solvers and how their ideas shaped the mathematics we still use today. Audience: High school and undergraduate students studying algebra, educators looking for historical context of the subject they teach, and anyone curious about the history of mathematics and human problem-solving.

Math words are weird! The deeper you go, the more you start sounding like a maniac to your friends and family. But if you look at things from the right angle, even the weirdest phrases start making sense and mean something real! Follow the experience of two undergrad math students who, upon finding an unusual phrase in a math textbook, get ambushed by a guy who won't shut up about the basics of probability and how it relates to this weird thing called "measure theory" that they've never heard of.

In 2023, our team resolved a 1985 conjecture posed by Paul Erdős. The work was the outcome of collaboration that combined geometry, graph theory, linear programming, harmonic analysis, and artificial intelligence. This video presents the main ideas of the proof visually, using animations, without requiring prior expertise in any of these areas. This video was entirely generated from script using manim, no video editors used. We created a framework for this, that we are happy to share and open source if anyone is interested.

How is a controller designed? This video is the first in a video series that will cover frequency domain controller design. Convolution is introduced and then replaced with the Laplace Transform. P and PD controllers and their mechanical analogues are introduced. High gain feedback and instability is also briefly covered.

In a famous series of letters, Blaise Pascal and Pierre de Fermat invented — or discovered — one of the most important concepts in probability theory: the expected value. But they weren’t even thinking about randomness or chance — they were trying to find a fair solution to a gambling problem. This is the story of how the quest for fairness turned into the foundations of probability theory.

Diffusion models, which are the best class of AI image generators known today, are typically portrayed as models that learn to denoise a corrupted image. This way, they can generate new images by gradually removing noise from a sample of pure noise. This video explains diffusion models through an alternative perspective that is more intuitive and practical; in particular, I show how this perspective results in some interesting realizations, that most other resources seem to miss: 1. Image generation is the same thing as rolling a dice; they can be reduced to the same basic computational recipe 2. That diffusion models elegantly extend the success of gradient descent, the technique used to train all neural networks today, from train time to test time 3. How diffusion models separate the "creative" and "logical" capabilities of image generation into two different actors/players 4. And how this might lead us to a general recipe of solving (almost) any hard problem Given their broad application also to video generation, robotics, drug discovery, music generation, and more, a better understanding of diffusion models is needed, which this video aims to address. The intended audience is broad: 1. AI researchers and engineers looking to get a better understanding of a key technological development in generative modeling 2. People interested in computer graphics, image generation, AI art, and artificial intelligence more broadly 3. General math enthusiasts

An urn contains 3 red balls and 3 blue ones. Imagine selecting a random ball from the urn and permanently removing it without revealing its color. The video discusses a neat little probability puzzle going over how we can use the 5 remaining balls in the urn to determine the color of the removed ball. On the way, we'll discuss Bayes' Theorem, the usefulness of sample size in tandem with signal strength, and how misleading relative probabilities can be.

Historically, the discovery of quantum mechanics was a messy evolution with many doubts and technological challenges. In this video, I wanted to explore a scenario where a physicist unfamiliar with quantum mechanics but with access to contemporary equipment re-discovers the main principle of quantum physics for themselves. It is a mix of math and experiment that eventually leads our imaginary physicist to conclude that they found a kind of physics which does not fit into the classical framework anymore. In the last part, we'll also explore potential applications of this new theory leading to the quantum bomb tester experiment and Grover's algorithm. This video requires only basic understanding of algebra and is hopefully comprehensible for high school students with an interest in physics.

This video is a showcase of an original puzzle I designed, the objective of which is to solve a rather devious system of equations. At first blush it appears impossible, but restricting the variable values to unique sequential integers allows far more unknowns than equations to be resolved (26 variables, 11 equations). It solves similarly to Sudoku or other constraint-style puzzles, but much harder (in my opinion). In addition to the original puzzles, the software used to render all graphics and animations is my own bespoke Java engine Ive been developing this summer. I also composed and recorded all of the included music! Hope you enjoy.

How does one simulate a ray of light traveling through a substance with infinitely changing indices of refraction? How can Snell's Law be pushed to its limits? To analyze this complex topic, we start by discussing gradient index fiber optic cables, their uses, and use ideas from that discussion to build the mathematical tools needed to model a ray of light in these strange new mediums.

With the recent drastic improvements in AI systems, computers seem to be far more capable and intelligent than ever before. However, there are certain problems even the most powerful of computers will never be able to solve. In this video, we explore how the ability of computers to self-replicate and reference their own source code leads to proofs that certain interesting computational tasks are actually impossible! This video is designed for students who are familiar with the basics of coding, but are still coming to terms with what is and isn't possible.

I discuss the importance of considering small components of a system (i.e, infinitesimal objects) when solving physics problems. In particular, the video focuses on solving 4 different problems a few of which most people will have encountered at school. The ideas used in previous problems should hopefully build up onto the next ones. The video also provides a practical application for a lot of the integration techniques taught in school, and whilst the actual computation of the integrals isn't the video's focus, all details are still provided. The 4 problems are about finding the speed of a wave on a string, finding the maximum tension that can be applied to a rough rope wrapped around a pole, modelling atmospheric pressure, and finding the electric field strength around a large charged metal plate. The video closes off with a few follow-ups to the questions that were discussed.

The mathematics behind how objects deform is beautiful! Because deformation can be visualized, it allows abstract mathematical concepts to be seen in action. This video dives into the math of deformation and shows how the subject can be used to intuitively understand concepts from linear algebra, using the determinant as an example.

A video about a fun fact I discovered about Wikipedia - the k-core of the Wikipedia graph is made up of tennis players and pro wrestlers! I go into the very basics of graph theory, explain three different centrality measures (including k-coreness), and hypothesize why measuring k-coreness leads to tennis and wrestling. The explanation is kept on a beginner level, but the focus is less "people who are looking to learning graph theory" and more "people who are interested in fun facts about Wikipedia and don't mind walking away having learned some math".

This video walks through various grid counting scenarios including the basic paths on a grid, Catalan Numbers and culminates to an explanation of the Lindstrom-Gessel-Viennot (LGV) Lemma, a fundamental result in algebraic combinatorics for counting non-intersecting paths on a directed acyclic graph (DAG). The video is self-contained and requires no prior knowledge of advanced combinatorics.

The most confusing concept in statistics must be degrees of freedom. Students everywhere leave their introductory stats courses totally bewildered about what degrees of freedom means, and why it seems to show up all over the place, such as in the t, chi-square, and F distributions, and also dividing by n minus 1 instead of n in the sample variance. The answer turns out to be complex but delightful, related to the dimensions of random vectors. Along the way we'll learn a powerful visual way to understand data analysis: the geometry of statistics. If you follow along closely, not only will degrees of freedom make way more sense to you, but lots of other statistical concepts will click too.

The core question is could we intuitively rediscover all the necessary math of general relativity (differential geometry and tensor calculus) from first principles? In this video, we will rediscover the idea of metric tensor from scratch (pythagorus theorem) and see why it lies at the heart of general relativity. And how it helps us mathematically think about curvature of N-dimensional space.

In this video I explain how induction itself works, why induction exercises are the way they are, and how to solve a few typical induction exercises without(!) induction. The video is mostly aimed at students in late high-school and early uni who are currently learning about induction.

Waves are everything around us! I mean even when you think about something "dumb" like the 4 classical elements (earth, fire, water, air): those are all wave phenomenon too. Earth--think gravitational waves, seismic waves, waves through anything that physically moves, i.e. rope. Fire--is an electromagnetic <wave> phenomenon. Water--think ocean waves. Air--think sound waves. Yet so often when this core principle is taught in physics, many details are overlooked. The standard derivation includes assumptions about the amplitude of the wave, assumes that the rope doesn't move at all in the horizontal direction and has constant tension. Doesn't it feel unsatisfying to leave out all these details? In this video, I build an intuition for why these premises are used and the effect they have on the model of the wave. Then I take those ideas and apply them to discover why waves appear everywhere from ocean waves, to music, to the very light that lets you see this very piece of text.

Why can a natural number only be factored in one way as a product of primes? It comes down to an algorithm -- the Euclidean algorithm -- which computes the gcd of two numbers. You can find this in every textbook on elementary number theory. But the way it's always presented, the Euclidean algorithm comes out of nowhere. In this video, I motivate the appearance of the Euclidean algorithm in the proof of unique factorization.

Aliasing is one of those concepts that shows up everywhere - from audio and imaging to radar and communications - but it’s often misunderstood or oversimplified. In this video, we break down exactly what aliasing is, why it happens when sampling signals, and how it leads to distorted or misleading results if you’re not careful. We’ll start with the core idea of how sampling works, why the Nyquist limit exists, and what it actually means to sample “too slowly.” Then we’ll walk through intuitive visualizations and concrete examples to show how high-frequency signals can appear to be completely different - sometimes even disappearing altogether. Whether you're working in DSP, RF, or just curious about signal processing, this video gives you a foundation to understand aliasing and how to avoid it. There’s also a link in the description to a companion Python notebook where you can explore the concepts interactively.

This video is an introductory summary of combinatorics. It addresses the differences between various types of problems, accompanied by accessible visual examples. The main purpose is to instill in the learner the impetus for a deep understanding of the infamous formulas that are often memorized in high school and generate the misleading impression that these are unattainable divine entities separate from oneself: revealed epiphanies from which mathematical work springs. Contrariwise, the inverse approach is proposed here: the formula as a consequence —as necessary as it is beautiful— of every act of reasoning, not as a cause.

My first ever animation, on an in depth explanation of what entropy is. It's a long format with some prerequisites (Lagrange multiplier technique, integrals, log properties). Any advice is welcome, I am well aware that the video has some problems. Entropy is often poorly understood, even among physicists. The goal of this video is to bridge de gap between information theory and statistical physics. It also tries to introduce Entropy very naturally by trying to solve a simple problem. At the end of this video you should be able to understand : * What information theory is * What entropy means in information theory * What the maximum entropy principle is * Why we need to use statistics to solve the dynamics of many particle systems * What entropy becomes in statistical physics * How to compute the microstates of a system in quantum and classical mechanics * Why the second law of thermodyanics is basically (almost) always true * What temperature really is, and why heat flows from high to low temperature

Suppose there are n colleagues who all wish to have access to a secret. Giving the full secret to each person is too risky - one untrustworthy colleague could leak its entire contents. Splitting the secret into n parts and giving each colleague a piece is too impractical, as it would require all n participants to combine their pieces to recover the secret. Can we divide the secret into n parts, such that only a certain number of colleagues, k, need to combine their parts to recover the secret? This video explains how this is possible with Shamir's Secret Sharing, devised by Adi Shamir in 1979.

This video shows a simple way of generating a random variable with a given density probability density function f, which I think is a good starting point for introducing this type of random variables and their properties. It is aimed at high-school or undergraduate students that have followed a probability course.

In this video, I motivate the study of interval arithmetic for reasoning about real-world measurements, and use that to motivate a precise definition of continuity and derivatives. This video is designed to be interesting and accessible to both students of Calc 1 who have and have not seen continuity and derivatives before. For educators who prefer a more active classroom, I have also added a visualizer for plugging in intervals of values to functions to my piecewise function grapher: https://trkern.github.io/pw_limit

This video is in french, but it is translated into english thanks to youtube. For people that don't like AI voiceover, they can use the subtitles. We focus on the issue of interchanging limit and integral : but fundamentally, what prevents us from interchanging limit and integral ? This approach, which involves examining what does not work, will be rich in lessons : it will allow us to discover the phenomena of mass loss, but also the 'forbidden law' that governs the theory of integration. By the end of this video, your intuition will be strengthened, and the theorems learned in preparatory classes or undergraduate studies will seem much more natural as we will see how to recreate them.

Newton's method is a powerful technique for approximating the roots of functions. But with a clever substitution we can construct functions where every value gets stuck in an endless cycle. Better yet, if you know integration you can find this function for yourself! This video is an entry to 3Blue1Brown's "Summer of Math Exposition #4". It is aimed at an early undergraduate audience, and was recorded in one take in the style of a live tutorial. This is also my first time making a maths video ... lots to learn :)

A look at complex numbers from the perspective of physics: what are they, what are they even good for, and why on earth are they all over quantum mechanics?

What connects DNA and graph theory? Interval graphs can be represented by intervals where edges are encoded by their intersections. This video begins with the historical motivation from Seymour Benzer’s work on gene structures in DNA. It then explores how to decide whether a given graph has an interval representation. You’ll learn about two key characterizations: minimal forbidden induced subgraphs (Lekkerkerker & Boland) and consecutive orderings of maximal cliques (Fulkerson & Gross). The video also introduces PQ-trees, a data structure by Booth and Lueker used to test these orderings efficiently. It concludes with my 2010 research on when partial interval representations can be extended to full ones.

How would you cross a field full of cows at minimal risk 🐄? And what is the quickest way to calculate the best path? This video explores three mathematical approaches and their runtime performance: From expressing danger with a cost function, to using a Voronoi diagram, to a combo of the Depth-first search (DFS) and Union-Find algorithms for building the safest path.

Audience: Real analysis students and calculus students who can't wait to learn real analysis! Description: Ever wished you could use mathematical induction on the real numbers instead of just the natural numbers? In this video, we dive into real induction — a powerful but little-known proof technique in real analysis that can simplify some of the subject’s most important theorems. Not only is it super-interesting by itself, but it allows you to prove some classic results in real analysis in just a few lines. Created by: Nikunj Goyal <[email protected]> and David Jelinek <[email protected]> for the channel InterContinentalBallisticMath

In this video, we will explore different types of reasoning used to make and verify conclusions in mathematics. This video is targeted especially towards those making the transition into high school level geometry because making proofs is a skill students typically first learn in this class.