Summer of Math Exposition

Have you heard that symmetries can lead to physical laws, but dont really know exactly how it works? In the video, we will derives the laws of collisions in one dimension from first principles using ONLY four symmetries, without assuming any of - Force, Mass, Momentum, Energy, Conservation Laws, or anything else that follows from Newton's Laws of Motion. We will see how the structure of mechanics, and even "mass" can arise from symmetries. Based on my study - https://zenodo.org/records/16966898 (preprint)

A high-level explanation of the EM algorithm, an important tool in statistics.

About a year ago, my friend had a question about triangles that came up at work: it boils down to maximizing the area of a right triangle given a hypotenuse length. Initially I found a pretty basic solution with calculus, but I got interested in solving the problem with more basic methods, which led to a bit of creativity having to make up for the lack of the stronger tools. In this video, I'm presenting proofs that I discovered using tools I would understand to align with high school classes in algebra, basic geometry, and trigonometry. It felt appropriate for me to make that my entry for my first Summer of Math Exposition, because it reminds me a lot of something that Grant says about important facts in math: because important math facts are important because they're related to so many other things, if you take some random thing and dig deep enough you will inevitably find something important. In this case, we end up touching some really fundamental ideas like the Pythagorean theorem, FOIL, Thales' theorem, the isosceles triangle theorem, the triangle angle sum theorem, SOHCAHTOA, and the double-angle identity for sine. Something I'm going for in this video is for the content of these proofs to be a vehicle for starting students down a path of understanding math in a more holistic way, interfacing with it more as a language of truth than a memorized methodology, and seeing where the creativity and problem-solving of math comes into play, as well as how seemingly disparate topics end up getting connected. My hope is that because that overarching idea is more abstract than there being a specific piece of math which is elaborated on throughout the video, it might be able to be adapted into a number of different settings - the proofs are arranged from least knowledge required to most knowledge required partially so that a teacher in a geometry (read: pre-trigonometry) class might be more able to present just the first two proofs, if they feel that would be appropriate. I also think this video could be well-suited to provide a class activity, with the students stopping and trying to solve the problem when the video indicates: I think this problem is actually in a really good spot in terms of difficulty to be attempted by classrooms with different degrees of collaboration across the spectrum of skill going from algebra up to precalculus. I didn't get to say so in the video (ran myself out of production time to add this to the end) but I did just want to say thanks to everyone running the event, Grant in particular, for the opportunity and the inspiration to give this a shot. I think I'm in an interesting spot where the video is more or less what I wanted to make, but I'm aware that there are some spots where it's rough around the edges, and it'll be interesting to see how people react to it, whether people like the core thrust of the idea enough to make up for its shortcomings, and where some areas are that I might be able to improve as a presenter for any future projects like this, or just in my work as a tutor. I'm excited to be getting some real feedback on this after its time in production. Thanks for reading, and I hope you have a nice rest of your day!

This video collects several interesting examples of constructing physical scenarios to prove math propositions, ranging from light refraction, Fermat point, geometry property of ellipse, Cauchy inequality, Pick's theorem, etc. It not only shows different examples, but also explain the common underlying connection: energy minimization. This video can be good material to teach student think in an indisciplinary way.

In my video I present from the ground up how to create a continuum sized chain of sets of natural numbers. A surprising object to exist considering that the whole set of natural numbers is countable. I tried to build everything from zero and skip a lot of details and notations so as it can be approachable to more people. Starting from what we consider finite and infinite, what are bijections, what are the real numbers and why are they uncountable, then going to the density of rationals and reals. Finally I introduce what is the power set and what is a chain of sets and how it all fits together to create this interesting chain. Despite my attempt at explaining everything and starting off easy, I rush through some parts and after all I think the audience of this video should be at least high-school students with some knowledge of math. Although others could enjoy it as well. Created for the Summer of Math Exposition

In this video I'm visualizing the dimensional math of tetrachromacy. As an aid to this visualization I'm presenting a new type of color, called "interlaced colors", that I've invented to allow for the functional experience of a specific type of tetrachromatic colors for normal trichromats. "Interlaced colors" are based on the concept of optical color mixing. I'm also using "dichoptic colors" (i.e. when each of your eyes sees the same object in different colors) to visualize the math and space tetrachromacy. Watch this video to better understand how tetrachromacy works, how you can experience it yourself through several methods, and how you can use the presented novel color experiences to intuitively understand dimensional math above three dimensions. This video shows how simple math makes tetrachromacy so incredibly more colorful than normal trichromacy. It demonstrates that although trichromacy is a subset of tetrachromatic colors, thanks to the context of a 4th independent axis (and a novel primary color) to the space of color tetrachromatic colors become almost incomparable to normal trichromatic colors—both in quantity and quality. As such, both "interlaced" and "dichoptic" tetrachromatic colors can be used to render and intuitively understand higher-dimensional math visualizations for which normal trichromatic colors aren't enough.

In this video I explore some ideas about computer simulations and quantum mechanics.

We give a visual introduction to the Lighthill-Whitham-Richards model of traffic flow and the method of characteristics. Through this model, we study examples of light and heavy motorway traffic, traffic lights, and the use of variable speed limits.

I designed this video for high-school and undergraduate students who have come across abstract vectors and matrices, but are still developing an intuition for how they can be used to solve real-life problems. I use the example of Pokémon battles. The goal is to find our chances of winning a given battle and use this to develop the optimal strategy, i.e. the sequence of actions that maximizes our chance of winning. To this end, we show how we can represent the game as a Markov chain, from which we construct a linear system of equations that gives us the win probability for each state. Next, we show how the equations change when we choose between different available game actions and how we can use this to make the best choices. Finally, we take a step back from the abstract math and check how useful the strategy is when applied to the real game.

This video presents highly accessible, conceptual approaches to limits, derivatives, and integrals — all without relying on formulas. We include Quick Quizzes designed to foster student-to-student dialogue, while also providing teachers with guidance to develop strategies for anticipating common student misconceptions. Throughout, we pair deep conceptual insights with practical teaching approaches, offering unique ways to understand the core ideas of calculus and support meaningful learning in the classroom for students of any age.

Our entry is a miniature documentary exploring the ideas of Greek geometry, all the way from their roots in Ancient Egypt to their influence on scientific and philosophical thinking to this day. Specifically, it covers how Greek geometry got its start in Ancient Egyptian land surveying, how the first Greek mathematician, Thales, brought these ideas to the Greek world, and how later Greek mathematicians invented the idea of the mathematical proof. The aim of the video is to get middle school to high school kids interested in Greek geometry by introducing it through the intriguing real-world examples that may have inspired the Greeks themselves. We also take care to highlight several of the ingenious, elegant solutions the Greeks employed that make mathematicians like me excited about math to this day. In the end, the hope is that viewers will walk away with a better appreciation for what mathematical proofs are for, and why they’re so important and mind-boggling. Along the way, students will hopefully also gain an appreciation for what excites mathematicians about math—not the calculations and formulas, but the endless sequence of joyfully clever ideas that open the mind to how much can be accomplished with so little if some genius is applied.

This video is about circular sectors (https://en.wikipedia.org/wiki/Circular_sector), specifically their packing density. I determined a lower bound for the packing density of this shape that I haven't seen discussed anywhere. This video serves to document the method and talk about packing mathematics in general. It starts with the very basics of packing and works up to a derivation for this new packing density method. The math is all at a high school level; the most complex equations used are the Law of Sines and Law of Cosines, and a single instance of the floor function.

You have heard of expected value - the math that is supposed to tell us which choices to make, mathematically. But this example gives a very unintuitive result: get $1M in cash, or a 1-in-500 chance to win $1B. Expected value tells us we should gamble, but in reality we would all pick the million! We can do better. In this video, I mathematically model utility to generalize the concept of expected value and show how modeling the world mathematically can lead to real-life insights. We explore the intersection of philosophy and mathematics, unifying something that might feel fuzzy with something that can feel boringly rigid sometimes. My goal with this video is to show the beauty of simple mathematics, show a real-life application of logarithms and a simple mathematical model of the world. It should also teach the viewer about expected value and what it actually means, and hopefully make them curious to explore more. Because math doesn't have to be dry and boring. It can explain our lives.

Ferroelectrics are a type of material that are everywhere in our lives---and as we rely more and more on AI, ferroelectrics will become all the more prominent as a backbone for low-energy electronics and other devices. In this video, we introduce ferroelectrics to a general audience. After going through examples and general properties of ferroelectrics, we derive Landau-Devonshire theory by Taylor expansions. We then show how this mathematical theory explains many of the most promising properties of ferroelectrics. We then explain how these properties lead to a vast array of past, present, and future applications, and we end our video by describing some current research in ferroelectrics. This video assumes some high-school level understanding of mathematics, and is intended to be understood by this audience. However, undergraduate or graduate students in physics, materials science, and chemistry may find the topic in question relevant and of interest to their studies.

This video is designed for undergraduate students taking a class on mathematical methods for scientists and engineers. Specifically, this lesson is designed for sophomore physics majors.

The relationship between proofs and programs (Curry-Howard Correspondence), and how a mathematical proof can be type-checked by a proof assistant, along with a few simple demonstrations in the Lean Theorem Prover. Chess is used as an analogy for the formal structure of mathematics. A small demonstration of Lean is also provided. This video is suitable for mathematically inclined high-school students and above.

A nice little video showcasing how the Ising Model (which was originally used for studying magnets) can be used to model melt ponds.

This video follows the creation of the natural numbers, integers, and rational numbers, and a great deal of the operations we use on them. I made it with the intent of being a introduction to real analysis, but since I didn't have the time to get that far, I decided to angle it towards how rigour and proof are so important to mathematics. I'd hope that it's of some use to someone who's in high school who's interested in learning maths at a more advanced level, as I hope this video acts as a light introduction into what that looks like.

This is a step-by-step guide into Heisenberg's famous "Umdeutung paper" in which he created quantum mechanics in 1925. I include the experimental reason for the need of matrices, a deep dive into the four key ideas of Heisenberg's paper, and a detailed worked-out example showing how zero-point energy naturally appears in Heisenberg's theory thanks to an early draft of what years later would become Heisenberg's uncertainty principle. A student with no knowledge of textbook quantum mechanics should be able to follow.

A guide to the birthday problem (aka birthday paradox), the two child paradox and the rare disease testing paradox explained from first principles (without using Bayes' rule). This video introduces conditional probability and probability trees to explain the solution to these 3 classic probability paradoxes. *How the video fits the useful-for-teachers theme:* The video is intended to be used as an aid for advanced high school probability classes (e.g. AP statistics) or early undergraduate classes (e.g. intro stats or intro probability) to help teach conditional probability, which is often a hard to grasp topic for students. The problems here should be useful when just starting out, to introduce some interesting counterintuitive problems you can solve with these methods as motivation for the topic. Each of the 3 problems in the topic can be used as a standalone example problem, for example at the beginning of a class. I've also tried not to overload the video with too many new ideas: I've intentionally kept the notation lightweight so that formalized notation can be built on the main ideas afterwards. After this video, the groundwork for Bayes theorem should be nicely set . The treatment of the birthday paradox as asking "how many people?" also sets up the idea of random variables as a future topic too. See description in YouTube video for some more links to related content.

This video explores arguably the most important discovery in mathematical finance in the last 100 years: the Nobel Prize-winning Black-Scholes-Merton option pricing formula. It is structured as a self-contained lesson that walks through a full derivation of the formula in a way that is accessible to anyone with knowledge of calculus-based probability and statistics. No financial background is needed. We include 6 exercises with hints and sample solutions to encourage working through the derivation on your own.

In this video, we explore the mathematical beauty of spherical harmonics: special functions that form the natural basis for patterns on a sphere. From quantum mechanics to cosmology, spherical harmonics help us describe complex systems in a compact and elegant way. We'll start by understanding how functions on a sphere can be decomposed into simple components and then build toward the multipole expansion - a method for expressing fields like those of gravity or electrostatics in terms of their angular structure. Prerequisites for this video are a basic understanding of multivariable calculus, (partial) differential equations and the Fourier series. Whether you're a physics student or just curious about the mathematics of nature, this is your crash course into one of the most elegant tools in theoretical physics.

In this video, I show various Desmos files I've made relating to special relativity, including the fundamentals of the Lorentz transformation, the the spacetime interval, the tunnel paradox, and the barn paradox. Further, I use Desmos 3D to visualise the case of two spacial dimensions and one time dimension, explaining effects like the abberation of light and doppler shift.

This video explores a simple question about Elo ratings: if n chess players starting with equal Elo ratings play a total of k games with predetermined outcomes, how high could the group coordinate to get one of their ratings? This video gives the asymptotic (in k) answer when n=2, and provides close asymptotic (in k) upper and lower bounds when n=infinity. It's based on a my paper that appeared in the proceedings of FUN 2024, linked here: https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2024.29 The two mathematical ideas that are highlighted are (1) the conversion between discrete and continuous processes and (2) mathematical invariants and potential functions.

Content: This video shows how Archimedes in Ancient Greece derived the formula for the volume of a sphere. It uses only the knowledge Archimedes himself had, so the audience can appreciate the brilliance of his method. By the end of the video, the limitations of this method will be revealed, along with the significance of calculus. Audience: High school students who know the formula for the volume of a sphere but do not know how to derive it without calculus. College students who are just beginning to learn calculus. This video demonstrates the importance of calculus by contrast. People who are interested in mathematics and the history of mathematics.

Shows why 2^x grows faster than x^2, using bananagrams tiles to show a visual intuition. For high school students learning about exponential functions, or a general audience curious for new perspectives on simple math, or even a more advanced audience interested in new perspectives.

This video is an explanation of cartographic hillshading. It starts by discussing what a directional light is and then works through the required mathematics. In doing so, a basic lighting formula is derived for hillshade. The video closes with a few footnotes about variations, an interesting illusion, and hillshade’s incredible realism. My target audience is an undergraduate student in math or computer science (or an advanced high school student) -- though I did my best to make the video as accessible as possible. While directional lighting is a topic in computer graphics, I make few explicit references to that area and prior knowledge of graphics is not required to understand or enjoy the video. I hope you enjoy the video! I had a lot of fun making it!

We present a method to calculate the similarity between words based on their phonetic transcription (their pronunciation) using the Needleman–Wunsch algorithm. It was originally devised to compare amino acid sequences of two proteins. Here, we explain every detail of the algorithm, including a Rust implementation. The video is accompanied by a paper, see the GitHub repo for more details: https://github.com/Splines/phonetics-graph

The 90s "Magic Eye" pictures were kind of magical — 3D scenes that arose from 2D images with no coloured glasses or special equipment needed. There was a little spate of YouTube videos reminiscing about them this summer, so I thought I'd recreate my old code for building them that I wrote way back when — probably as a literal child on an Atari STe. With the backing of modern hardware and an adult brain, it's obviously a lot nicer now. This is a kind of accompanying video explaining how it all works — there's a slightly-interactive text explainer on the website as well if you prefer to read: https://www.andrewt.net/stereograms/ — ideally I suppose the whole thing is one big package but I've classed it as "video" because, well, half of it is a video and there's no option for "video & ¬ video"

This video goes over the number theory required to solve the problem as an introduction to the world of number theory. This includes modular arithmetic, Fermat's little theorem, and multiplicative order. The goal was to make it as beginner friendly as possible, but there was a lot to go over. Target audience: High school and above, mathematical maturity and/or prior experience strongly suggested

There's a beautiful connection between the quadratic formula and the Gaussian curvature of surfaces. I aimed to explore this connection, starting from defining curvature of curves, all the way until computing and discussing the Gaussian curvature of surfaces in R^3. I tried making the video accessible to a general audience. Most of the video is a discussion of geometric concepts, but someone with a basic understanding of calculus should be able to follow most of the computations. There's just one computation that requires multivariable calculus.

A very basic video giving an introduction to index notation. Many people I've tutored have had essentially zero familiarity with it more than halfway into undergrad, and I think its an essential tool for most of physics. I procrastinated on this one wayyy too much! Lab work is intense...

We use deep understanding of Hilbert space and operator theory to dissect the Fourier Transform and extract its eigenfunctions numerically in MATLAB.

This Manim based video aims to draw a connection between Music, Maths, and the Mind. It touches on the fascinating Mathematics of fugues, and their intrinsic (and little known) relationship to strange attractors and chaos theory. To build on this, we introduce and try to understand differential equations and the limits of our predictive abilities, and some conflicting ideas in quantum mechanics which cause this.

This video is designed to help clarify the many formulas involved in the study of basic combinatorics. It is intended for high-school students, but it can also be useful for anyone who struggles with math and is looking for clear methods with minimal prerequisites to understand the fundamentals of combinatorics. The video features short and easy-to-remember explanations that can be used to tackle all the most common combinatorics exercises. I truly hope this video can be helpful to someone, and I’d love to grow this channel to share more content, always aiming for simple and memorable techniques that require few prior skills. Oh, I forgot: that's my first video! Let me know how to improve!!!

This is a visualization of the arguments given by de Broglie in his 1924 Thesis, where he postulated that a mass particle could be a wave. We we see how we can discover de Broglie's famous formulas about the frequency and wave number of the particle using the special theory of relativity.

The video is about my personal experience coming across the Catalan numbers and trying to find my own proof of their closed-form expression. The point of the video is to teach viewers about the Catalan numbers, but also to show what the process of mathematical problem solving can look like.

Often the most famous theorems have hundreds of different proofs. But why do so many mathematicians “waste” their time reproving the same theorem over and over again? To answer this question, we look into three different proofs of Euclid’s Theorem (i.e. there are infinitely many primes) to see that each proof provides us with new insights and is therefore not just a repetition. If you only have time to watch around 10 minutes of the video then I recommend 00:00 - 05:45; 14:37 - 17:03; 24:08 - 26:10

An exploration of the two’s complement number system. This video explores how computers store negative numbers, how it differs from how humans write negative numbers, and the benefits and drawbacks of each system.

Imagine you're the first person to discover a triangle. What's the shortest name you could give it so that someone who has never seen one before could recreate it? Would you describe its angles? Its sides? The way it sits on the page? Does the order in which you name these features matter? These seemingly simple naming decisions lead us directly to the heart of **congruence**: When are two triangles really the same triangle? Each way you choose to describe them reveals something essential about triangles while leaving other details hidden. As we search for the one true way to name triangles, we discover something profound: rigidity emerges from constraints. The symmetries we preserve reveal the hidden identity shared across all triangles. Efficient naming unveils underlying structure and mathematical unity. By the end, we realize that there is no more fundamental insight in mathematics than giving the same name to what could otherwise have been given a thousand different names. P.S Math teachers reach out to us we would love to design an accompanying lesson plan to bring this hands-on visual approach into your mathematics classroom!

Deep dive into Ramanujan's famous pi formula by reviewing his original paper.

This is a video about counting the expected number of intersections when ten random chords are drawn in a circle. It covers many topics in probability theory, and is targeted at those with limited or no experience in probability. Topics covered: Naive probability, multiplication rule, sample spaces, events, outcomes, partitions, intersections of events, conditional probability, the law of total probability, the continuous law of total probability, probability density functions, random variables, expected value, indicator functions, counting using the binomial coefficient.

An overview of special relativity in 11 minutes, covering time dilation, length contraction, spacetime diagrams, and the world's most famous equation! (Notes and time stamps below) Notes: - The speed of light in a vacuum is constant, but the speed of light CAN change in different mediums. Light is just an electromagnetic wave, if you have conditions that drastically alters the electric permittivity, you can slow down light. Some very active research is dedicated to slowing down light until you can actually see individual photons - Don't come at me for my poor drawing of the aether, I don't know how to draw something that's invisible and probably doesn't exist (this may not bode well for when I try to illustrate dark matter) - Sound travels faster in water. Sound is a pressure wave, the more dense a material is the quicker the wave can propagate. Since water is denser than air, it moves faster - See here for more details on the Michelson-Morley experiment: https://en.wikipedia.org/wiki/Michelson%E2%80%93Morley_experiment - For the light clock, you might ask (as I have), why the experiment can't be done in any direction? Why does the movement have to be perpendicular to the light ray bouncing? It's because of length contraction. Try repeating the same thought experiment but with the mirrors moving parallel to the light ray bouncing! Since the mirrors are moving in that direction, the distance between the two mirrors should contract, resulting in the same answer as before - 87% the speed of light is not currently achievable for transporting humans, so I wouldn't get your hopes up trying to travel to the future - I go through it quickly, so see https://en.wikipedia.org/wiki/Lorentz_transformation#Hyperbolic_rotation_of_coordinates for more information on the derivation of Lorentz transformations, and their relation to hyperbolic rotations - Note I use a (- + + +) metric signature, which is for relativists. If you are a particle physicist, (+ - - -) might be more familiar to you. It does not matter Time stamps: 0:00 - Warning 0:08 - Intro, what does it mean to be relative? 0:57 - The Fall of Galilean Relativity 1:50 - Special Relativity Postulates 2:12 - Light Clock and Time Dilation 3:15 - Length Contraction 4:08 - Twin Paradox and Relativistic Doppler Shift 5:34 - Simultaneity 6:22 - Spacetime Diagrams 8:39 - E=mc^2 10:38 - Outro Once again, thanks for watching! Let me know what you think down in the comments and feel free to ask me anything or request certain topics! If you've made it this far down in the description good job, here's a cookie for you: 🍪

The average is almost exclusively talked about by means of its "sum of things divided by numerous of things" formula in formal education. While this formula's use may be justified for abstract mathematics, it is not an intuitive way to mentally approximate an average of some concrete set of objects. In this video, we discuss tracking the average of a discard pile in a card game, and discuss one alternative method to consistently determine if the average of the pile is 6 (or any other fixed guess) and what card needs to be discarded to make the new average be 6.

an example of a commutative ring with unity. made for undergrad math students in abstract algebra — ACKNOWLEDGEMENTS — The mathematics in this video is based on material from "Algebra" by Michael Artin, Chapter 11, Section 1. The specific example is drawn from Exercise 1.7(a). Visuals were created using Manim Community Edition (https://www.manim.community/) —====================—

Has someone ever told you that the fourth dimension is this unimaginable and unthinkable mathematical construct? Have you ever heard someone say that it is not possible to know what a 4D world would look like, and that the only way we would be able to get close to having any sort of intuition for it is to take a 3D slice of it? Have you believed these claims, and have they ever made you feel disappointed, because deep down you felt like there just has to be a way? Whether or not you identify with this description or have no clue what I mean when I say "four dimensions", this video not only debunks some common myths shared about 4D, it also quite literally approaches it from a completely new angle. Understanding 4D means to first understand 3D, and the only way we can truly comprehend the seemingly intimidating nature of higher dimensions is to re-think some of the most basic and commonly accepted rules of life, about things like vision, perspective, movement, and rotation. Come join me on this adventure into a new realm of endless possibilities that is simultaneously so different and yet so familiar to the flattened experience we're used to!

In this video, I introduce the topic of turbulence. I discuss what makes turbulence difficult to study, and I show how the scale invariance of turbulent flows (the fact that they look the same at different length scales) gives us a powerful tool to make progress in this field. I demonstrate the idea of scale invariance using examples, give a heuristic explanation for why it appears in turbulence, and then use it to quickly derive a famous result in the theory of turbulence.

In this video, we take a quick look at the holy grail of cryptography, fully homomorphic encryption. This is a video intended for a general audience, without much expertise.

Musical Constellations is a little experimental instrument I've been working on the for the past few months, which seeks to answer the question: if the constellations in the sky were melodies, what would they sound like? This video is a deep dive into the mathematical algorithms behind Musical Constellations, which takes you on a musical journey while hopefully teaching you something new along the way. Subjects of the video include procedural generation, Poisson disk sampling, music theory and graph theory.

When we think about probability, statistics, and distributions, we often focus on modeling a single variable. Doing so, however, can make us miss how variables interact — and these interactions can completely change our decisions and insights. In this video, we explore covariance and correlation, showing why looking only at individual (marginal) distributions can be misleading. You’ll see how the joint behavior of variables reveals insights you wouldn’t get from marginals alone, with applications in fields like finance. We’ll cover: - Marginal distributions and expected values - Joint distributions, covariance, and correlation - A realistic finance example By the end, you’ll understand how considering relationships between variables can reveal insights and guide better decisions that you would otherwise miss.