Summer of Math Exposition

This video is about how Turing patterns work: how repeated blurring and sharpening produce a Turing Pattern by making a band-pass. I show an alternative method of making a band-pass using two blurs. I look at the reaction-diffusion / Gray-Scott model and where its hidden band-pass is. I cover multi-scale patterns, where several band-passes are made using a cascade of bigger and bigger blurs, and how to efficiently run a large blur using FFT. Bonus: a Turing pattern wavetable, and Turing patterns in 3D.

This video describes a way to hear light, as in, shine a light into your ear and hear music. When I first started this video I actually didn’t realize that this was possible, but as I experimented with various items I discovered it was. There are two key ideas I try to explain in this video: radio transmission and photoacoustic transduction. These two ideas are what lead to the phenomenon I discuss in the video.

Although the video is geared towards a general audience, and there’s little math in terms of equations and such, it is still somewhat technical, so I would say the target audience is undergraduate/graduate students, along with high-school students and teachers/professors. But my hope, of course, is that anyone with a genuine curiosity about science will appreciate the video and understand most of what’s discussed.

In this video, we investigate mathematical knots, and our guiding question is: How can you tell when two knots are the same or different? In pursuit of an answer, we argue that two knots are the same only when one can be transformed to look exactly like the other using a sequence of three basic moves, called Reidemeister moves. Then, we prove that a quantity called the linking number is an oriented link invariant using the fact that it doesn’t change under any of the Reidemeister moves. Finally, we stumble upon the most famous knot invariant – the Jones polynomial – and show how one could’ve devised it from scratch, along with several applications.

Most of the exposition and practice problems come from The Knot Book by Colin C. Adams. However, we have a more elaborate discussion about why the three Reidemeister moves are all you need, and we made modifications to the definition of the X polynomial where we felt was appropriate.

References: The Knot Book by Colin C. Adams https://katlas.org/wiki/K11n34 https://katlas.org/wiki/K11n42

The video unpack the famous “Two-Child Paradox” and how it managed to stump some of the most respected minds in mathematics.

Here’s the problem statement: I have two children. One of them is a boy born on a Tuesday. What is the probability that both children are boys?

This deceptively simple question hides one of the most controversial puzzles in all of probability. The answer seems clear—until you start asking how the information was obtained.

This video is intended to be a “Chapter 0” for the theory of probability, which motivates the definition of a “probability space”, the most fundamental model used in this theory. It is intended for high-school or undergraduate students with a solid grasp on algebra, but no other knowledge of discrete or undergrad-level math. It would best help a student who is just starting their first course in probability or statistics.

In 1900, Henri Poincaré initially conjectured that any 3-manifold that has the homology groups of a sphere must be the 3-sphere, but later in 1904 discovered a counterexample, named the Poincaré homology 3-sphere. In the process, Poincaré also invented homology, the fundamental group and conjectured problems that have continued to interest mathematicians up to the current day. In this video, we study the Poincaré homology sphere by applying foundational results of algebraic topology such as the Seifert-van Kampen theorem and Poincaré duality.

Note: This video is converted from the slides of a talk given to fellow graduate students in a student seminar. I assumed knowledge of definitions and key properties of the fundamental group, homology and cohomology groups that are encountered in a first course on algebraic topology. Even if you have not taken such a course, I hope that the video may communicate some of the underlying concepts and ideas behind algebraic topology :)

Synthetic Aperture Radar (SAR) satellites send out microwaves to earth and use the echo to form an image. They can see through clouds, fog, and smoke. But SAR images are often difficult to interpret, since SAR works fundamentally different from optical imaging.

In this video we’ll explore at how the Golden Gate Bridge seemingly multiplies in a SAR image. It produces three distinct reflections, which look confusing at first glance. But they can actually be used to measure the water level beneath the bridge. We’ll start with the basics of SAR imaging before diving into the physics behind these three reflections.

The goal of my channel is to excite more people about SAR. In my opinion, it’s one of the coolest technologies of our time, yet not many people know about it. I want to explain the signal processing and math behind SAR imagery in a simple way, so that we can “see what radar sees”.

I’ve put together the specific explanations that have helped me understand the foundations of special relativity in the past, and attempted to do it in a way that would connect seamlessly to the deeper concepts of differential geometry to clear the path to general relativity, as best as I could!

We start with the relativity of simultaneity, get to this important effect through the light rays-thought-experiment, and start drawing spacetime diagrams to have them lead the way from the start.

Then we connect this to length and time contractions, and find our way to the Minkowski metric, since the contracted lengths on the diagrams don’t look right at first, and demands a change in the way we look at their geometry!

A video about enumerating all of the solutions to a puzzle game, using the language of group theory.

The cellular automaton rule known as Rule 30 generates a universe whose overall behavior is complex. But it has highly structured regions hidden among the chaos. In this video, we’ll discover why these regions behave like they do. They’ll tell us quite a bit about Rule 30 as a fundamental law of physics, specifically about whether we can run it backward in time.

What does it mean to “factor” the sine function? We explore Euler’s brilliant infinite product for sine and show how he used it to solve the Basel Problem. Along the way, we see an infinite product expansion for pi/2 (the Wallis Product) as well as some genuinely beautiful math involving complex analysis, infinite products (and infinite series), Weierstrass/Hadamard factorization, and more.

Why do big exams like the SAT, GRE, ACT, LSAT, or GMAT often feel UNFAIR? Trick questions, ambiguity, and senseless difficulty make these tests frustrating. But to an extent, these tests HAVE to be this way.

This video looks at 4 example SAT or ACT questions that might be considered unfair. Through these questions, we’ll discover a simple formula that’s behind every exam question you’ve ever seen. The formula explains why its necessary to write hard questions that feel tricky or unfair to students. These questions often must have tricky but convincing answer choices (called distractors) or be brutally difficult.

Specifically, the video presents the 3-item characteristic curve for item analysis. This formula, and others similar to it, is used at large companies like College Board to design exams. The video is mostly targeted to high-school students who may be mathematically inclined (or angry about their exam scores), but educators should find value in understanding how to analyze test items in more detail.

Thanks for watching! -John-

The Hodge Conjecture has a daunting reputation as the Millennium Prize Problem so abstract that people have called it “impossible to explain” to a general audience.

My project is an attempt to unravel this math mystery through storytelling. In the video I build intuition from foundational geometry, through projective spaces and topology, all the way to the formal problem statement. This video is for math enthusiasts and lifelong learners with a baseline understanding of undergraduate math.

I used dynamic programming to find the optimal strategy for maximizing expected points in a game of Yahtzee. I then analyzed the resulting strategy to show what this optimal play looked like in various phases of the game, eventually getting to adjustments that would have to be made in multiplayer games.

My goal in this video is not to explain the math as much as it is to show a cool way that math can be used to solve a problem that people might care about. I’m trying to figure out the right balance of how much to talk about math versus just applications in my videos, so I’d love to hear people’s thoughts on this.

The Slope Problem asks a deceptively simple question: What’s the smallest number of different slopes of line that n points can define? Answering that question requires a surprising amount of ingenuity.

What’s prettier, rainbows or math? Good news, you don’t have to choose!

In this video, following many cues from Vladimir Arnold’s book Mathematical Understanding of Nature, I attempt to explain the rainbow in a self-contained way, without assuming any calculus or trigonometry. Arnold’s approach is remarkably simple. Accessible to anyone with a bit of patience, really. Yet it can explain many of the complexities of the rainbow. In this video, we see why the rainbow always appears at the same angle, why that particular angle, and why it appears to be lit from below but dark from above.

If you persevere, there’s also an Easter Egg at the end.

Supplee’s submarine paradox asks a deceptively simple question: what happens to a neutrally buoyant submarine when it starts moving relativistically?

In this video, we explain why the submarine sinks—and show how the apparent contradiction between different reference frames is resolved. We solve the paradox in two different ways, highlighting how relativistic density, force, and gravity fit together consistently.

A compact but surprisingly deep application of special relativity and the equivalence principle.

Double cube hides every square root from √1 to √6, but √7 is nowhere to be found. Why? The answer isn’t in the geometry of this solid, but in the algebra. It all boils down to a number theory question that took over 2000 years and the work of many great mathematicians like Euler, Lagrange, Legendre and Gauss to answer. This is the origin story of additive number theory, starting from the Pythagoreans’ polygonal numbers, through 18th century world-class brainstorm, to the final piece of Fermat’s [redacted].

It’s a topic close to my heart since I’be been working on it for roughly 3 years now. So… enjoy!

This video explains how probability problems can be solved by representing them as state diagrams, also known as Markov chains. Using this concept, we solve five probability problems, starting off easy and building up to hard.

Some of the most celebrated results in geometry and topology are often perceived as elegant, yet intimidating. Euler’s Formula or the Gauss-Bonnet Theorem are sometimes presented through abstract arguments that can feel inaccessible to many learners.

The goal of this video is to change that perspective.

Using a set of interlocking polygonal tiles, each proof becomes a hands-on process. My intent is to use these tiles to build polyhedra, allowing viewers to follow the mathematics. Quantities such as faces, edges, vertices, and angular defects become tangible objects that can be manipulated and observed throughout the proof.

I hope this approach offers a useful example of how physical models can help communicate deep mathematical ideas while preserving both the beauty and the rigor of the original proofs.

Let me know what you think! DPM

Starting with the natural numbers, we construct the integers, then the rationals, and then the real numbers. The construction of the reals using Dedekind cuts is often taught in a confusing way, and we want to make the idea much clearer, more visual, and easier to understand. Dedekind cuts will go from being mysterious to being almost obvious.

And as a bonus, we show how to construct the complex numbers, just in case you don’t believe in “i”.

Featuring original topological transformations and insights into Hilbert curves.

Part 4 of an ongoing series exploring 4D visualization. In this video we use 3D motion and spacetime to derive the basic rules of 4D geometry - how lines, planes, and hyperplanes interact in 4D

Reflectivity is an extremely important concept for understanding weather radar data, because it’s often the first product you look at.

In this video, I’ll explain how reflectivity is computed, why it’s useful, and give you resources to further learn on your own!

This is an attempt to build a picture of why waves are a universal feature of nature, focusing on intuition and physical arguments and trying to distill the ideas to as simple of math as possible.

It is aimed at high school students and early college students, and at a general lay audience. I believe the approach and intuition provided would also be of interest to more advanced students and practitioners of STEM fields. Most of it can be understood with knowledge from a basic physics course, although some parts may require pre-calculus.

Computers aren’t just for entertainment or work. They’re our modern-day cave paintings. They record the world around us: how we lived, what we did, what we saw, what we heard. But there’s a catch. We live in a continuous world, and computers live in a discrete one.

So here’s the question: once a sound is recorded digitally, can a computer know the exact continuous sound that created it? Not an approximation. Not a good guess. A mathematical guarantee?

Surprisingly, yes!

This video demonstrates a visual proof of the Nyquist-Shannon Sampling Theorem. The idea that a continuous signal can be perfectly reconstructed from its samples, as long as those samples were captured at least twice as fast as the signal’s fastest-changing component.

No prior background in signal processing, engineering, or computer science is required. By the end, you’ll understand not just that perfect reconstruction is possible, but why, and exactly what sampling rate guarantees it.

In wanting to make a gravity simulation with particles, I discovered an algorithm that has never been covered in full on YouTube before, which immediately caught my attention due to how fast it claimed to be. Normally with $N$ particles, calculating all pairwise gravity forces takes $O(N^2)$, however this algorithm achieves an approximation in $O(N)$ and also has a bounded error which can be tuned.

The title may be somewhat clickbait since I am only in high school, but as my teachers had never heard of it, and judging from the lack of material on YouTube, I think it’s pretty unknown, even though it’s much faster than the Barnes-Hut algorithm ($O(N\log N)$), which a lot of videos do cover.

In this video, I go over the mathematics behind the algorithm in how it computes forces and manipulates certain objects to achieve the speedup. I plan to discuss my implementation in a future video, where I also learnt a lot when trying to optimize it. The final result is a simulation of 120,000 particles updated at 60 fps.

Finite fields, or Galois fields are mathematical structures that behave in familiar ways under addition, subtraction, multiplication, and division, but have only a finite number of elements in them. In this video, I introduce how to construct Galois fields both for simple modular arithmetic and the more complex prime power fields.

Target audience: undergraduate math students and math enthusiasts who are familiar with modular arithmetic.

Present a geometric/physical definition of e and figure out a way to compute the constant based on it.

This is a 9 hour introduction to the mathematical structure behind the proof of Fermat’s Last Theorem, beginning with roughly high school level mathematics.

Fermat’s Last Theorem states that the equation $x^n+y^n=z^n$ has no positive integer solutions when $n>2$.

Many explanations of Fermat’s Last Theorem focus primarily on its historical story, while the mathematics behind the proof is replaced by a few broad metaphors. Appropriate metaphors can certainly be useful, but some people may also want to understand the argument in the actual language of mathematics. This video was made for those people.

The course begins with mathematical ideas accessible to someone with roughly a high school level background. It then introduces the major concepts needed to understand the skeleton of the modern proof, including linear algebra, abstract algebra, elliptic curves, modular forms, and Galois representations.

By the end of the video, viewers should be able to understand the mathematical skeleton of the proof of Fermat’s Last Theorem. The proofs of Ribet’s theorem and the modularity theorem proved by Wiles and Taylor are not included, but the video explains what these theorems say and why they combine to imply Fermat’s Last Theorem.

The course consists of six chapters:

1. Basics: Elementary number theory, congruences, and proof of n=3/4 of FLT
2. Linear Algebra: Vector spaces, linear transformations, matrices, eigenvalues, and representations
3. Abstract Algebra: Groups, rings, fields, quotient structures, and $p$ adic numbers
4. Elliptic Curves: The group law, reduction modulo primes, and Galois representations
5. Modular Forms: The upper half plane, modular transformations, Fourier expansions, and Hecke operators
6. The Endgame: The Frey curve, Ribet’s theorem, modularity, and the final contradiction proving Fermat’s Last Theorem

The course was designed top down. I began with the final argument and worked backward to determine which ideas had to be introduced for that argument to make sense. My aim was for every chapter to form part of a single mathematical story, rather than for the video to feel like a collection of unrelated lectures.

I wanted the course to feel almost like a film. In Chapter 6, all the concepts introduced throughout the earlier chapters finally come together, and I hoped viewers would feel the excitement of seeing the entire structure suddenly click into place.

The video is deliberately long. Fermat’s Last Theorem is often treated as a result whose proof is simply too advanced to explain. Not every technical argument can be reconstructed from high school mathematics, but the overall structure does not have to remain mysterious.

The purpose of this project is not to reproduce every technical detail of Wiles’s original work. It is to make the mathematical journey toward the proof visible: what the major objects are, why they are introduced, how they are connected, and why the final argument works.

When you flip a fair coin a million times while tracking whether heads or tails is in the lead, most people (myself included before researching this topic) assume the lead will trade back and forth evenly. In reality, it doesn’t. For almost any long, perfectly fair 50/50 process, one side ends up ahead for the overwhelming majority of the time. In fact, the scenario that feels fair (where both sides split the lead 50/50) is actually the least likely outcome. Through Math animations and MS Paint illustrations, this video explores the counterintuitive math behind Feller’s Arcsine Law, breaking down why fair leads are so sticky using the Reflection Principle, generating functions, and calculus.

A presentation done as part of the undergraduate math research course at the university of guelph. Covers some basic neural network knowledge, some compression methods, convolutional neural networks, then gets into the result of the paper being an output aware compression method result.

Imagine you have to colour the regions of a map on a torus in such a way that adjacent regions are not the same colour. How many colours would you need? The notorious four color theorem is well known in mathematical circles, and in this video we talk about its generalization to arbitrary surfaces, and derive an elementary yet tight bound for a wide variety of surfaces.

In doing so, we take a journey through central ideas in graph theory and topology of surfaces such that it is accessible to a wide audience: high school and above.

We fill a soup with 4000 random programs — no design, no selection, no goal — and let them run into each other. For thousands of epochs, nothing happens. Then, almost from one moment to the next, the chaos collapses into order: some of the tapes have started copying themselves, and they take over everything.

Correlation is not causation - except when it is! This video is a light introduction to causal inference, a field of statistics focused on when correlation can tell us about causal effect sizes. It covers causal diagrams (Directed Acyclic Graphs, or DAGs), slope in linear regression models, the problem of confounding, controlling for a confounding variable in multiple regression, bad controls (mediators, colliders, and outcome variables), and very briefly the idea of natural experiments using the method of instrumental variables.

When I turned 40, I built a DIY math museum for my birthday party - and the games section was the hit of the night. That sparked an idea: turn one of my favorite areas of math, permutations and symmetric groups, into a real board game anyone could play. In this video I walk through the math behind it (composition, cycles, transpositions, and conjugation), then show how it all becomes the rules and winning strategies of Swaps Battle, the game I designed around it. Whether you’re a math enthusiast curious to see abstract algebra come alive on a game board, or a strategy-game fan looking for something with real depth under the hood, this video has something for you - and you can try a free solo version of the game online right now.

Take exactly four copies of the digit 2. Using mathematical operations, how many different numbers can you make?

I give an overview of divergent series and about how they arise naturally in Fourier series. This is targeting upper division undergrad or beginning graduate students in Mathematics.

This video is on mechanical degrees of freedom and Maxwell’s counting argument, and how it can be used to explain the rule that you should make frames out of triangles in engineering. It ends with an intuitive explanation for the Maxwell-Calladine index theorem without using any linear algebra.

This video is intended to be as accessible as possible with minimal background knowledge required, and can hopefully be understood by a layperson. However it also should hopefully work for someone more involved with maths.

This video is about the lemniscate constant, a constant that shares astounding similarities with pi. We explore its definition, and move onto its various representations, as well as striking parallels it has with pi, from its continued fraction, to its integral representations, and so on. We finally look at how it directly interacts with pi, from the arithmetic-geometric mean, to the gamma function, and we explore its deeper significance to the rest of mathematics

I will be discussing how temporal grammar can be linked to lambda calculus, particularly focusing on de Bruijn indices and the significant implications of this equivalence.

Also, I will be exploring some aspects of linguistics and the etymology of the pluperfect tense, and how that logic can be extended to name the new temporal tenses, one being the ‘perpluperfect’ tense and explain the motivation behind it and why English as it currently stands is temporally ambiguous.

Sometimes a problem looks tricky just because a hidden symmetry is sitting right in front of us. Once you spot it, everything clicks, and you realize how beautifully simple it actually is. This puzzle is a classic example of that!

In this video, we’ll see how electromagnetism can be derived from geometry, curvature, and symmetry!

Summer of Math Exposition (go check out all the other entries!): https://some.3b1b.co/

Detailed Notes: https://drive.google.com/file/d/153px…

A visualization of the Knight’s Tour problem, exploring its most fascinating patterns and their properties. Based on the books and website of George Jelliss. For chess and math lovers.

We explore group theory through the childhood board game Funny Bunny and discover how it relates to the pattern of the holes that appear during the game. This allows us to understand why exactly two squares are missing (and create some pretty visuals along the way).

We then dive deeper into beautiful graph-theoretic ideas and a surprisingly advanced formula at the end, using only simple visualizations and intuitive reasoning.

Be warned, this video has a “slight” French accent, and I truly hope it is still fully understandable. Also, this is not the original video (which you can find on the same YouTube channel). I created this unlisted English-dubbed version so you wouldn’t have to rely on either the original French audio + english subtitles or YouTube’s AI-generated dubbing (YouTube does not yet allow me to add an additional manual audio track to my videos…), but if you feel like it, the original is definitely much better!

In this video, we explore a geometric proof of the Basel problem:

$1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\cdots=\frac{\pi^2}{6}$

Our story begins with a circle that repeatedly doubles in circumference, gradually flattening into the number line. When lights are placed at the odd integers, a hidden system of right triangles appears between consecutive circles. Together with the inverse-square law and the inverse Pythagorean identity, this geometry preserves the total observed brightness as the circle expands. By the time the circle becomes the number line, every odd integer has arrived at its final position—and the value of the Basel sum is waiting for us. This video was inspired by Grant Sanderson’s beautiful lighthouse proof for 3Blue1Brown. My goal was to explore another story hiding within that proof: the expanding circles that provide its architecture, and the way each integer appears to carry light forward to the next generation. The appendix proves a geometric fact assumed during the main argument: why the integers on consecutive circles align perfectly. CHAPTERS 0:00 Introduction 1:34 Part I — Building the Architecture 5:40 Part II — Conservation 9:07 Part III — The Hidden Geometry 11:05 Part IV — Passing the Light Forward 13:55 Part V — The Reveal 18:27 Appendix — Why Do the Integers Line Up?

In the Leibniz notation, we write differentials as fractions, e.g. $\frac{\text{d}^2 y}{\text{d} x^2}$. But can we really treat them as such? While it works for the first derivative, it breaks down already for the second. In this video, we find out why and rebuild the second derivative with the product rule, so that it behaves like an algebraic object. As a bonus, we arrive at the second-order inversion formula in a few steps.

This video is based on the paper “Extending the Algebraic Manipulability of Differentials” by Bartlett & Khurshudyan (2018).

A 20-minute video documenting my yearlong attempt to find a magic square of square. Little knowledge is required but is faced paced. Has some interesting simple math for those that are curious.