Summer of Math Exposition

In school, we’re often made to just memorise Doppler effect formulas for light, accept the fact that this is how nature behaves, and move on. But the real story is far more fascinating. In this video, we’ll start simple and break down the complete Doppler effect for light step by step. But things get really interesting at very high speeds. Thanks to Einstein’s theory of special relativity, time itself begins to run differently for the moving source. By the end, you’ll see that these shifts in light aren’t just mathematical tricks or equations to memorize. They’re a direct window into how space, time, and motion really work in our universe.

A brief video summarizing the true meaning behind sin and cos, primarily with respect to polar coordinates

It focus is differentiation. Applicable to a variety of different levels of education.

the video looks at the final problem from the 2024 beijing middle school exam. it's a nice geometry problem, and also gives a sense of how challenging these exam problems are.

The binary addition and subtraction rules are not apparently clear to everyone. Students often resort to rote learning without understanding the true mathematical interpretation of the binary rules. This video explains and breaks down the rules mathematically and how the zeroes and ones interchange and move to different positions depending on whether the operation is addition or subtraction, resulting into a perfect binary output. As a practical example, this video uses two binary numbers, each comprising of six digits and subjecting them to addition and subtraction, in a two-step process. In the first step, I simply operate them under binary rules and thereafter in the second step, through a algebraic operations, breaking each position step by step. Understanding the algebraic operations provides a full clarity to students why the rules behave they way they do, thereby providing full clarity to students.

How to use construction using GeoGebra to solve a puzzle about the area of an annulus.

A summary of my attempt at finding or disproving the existence of magic square of squares. It is made for a general audience, all the prerequisite knowledge required is algebra, but it is quite fast paced.

What do these questions have in common? 1. What number's square equals its sum with itself? 2. What is the product of the slopes of two perpendicular lines? 3. At what temperature are Celsius and Fahrenheit the same? 4. How many squares (of any size) are on a chessboard? 5. At what angle should a projectile be launched in order for its range to equal is maximum height? They all have numerical solutions, even though the questions themselves contain no numbers. In this video, I consider this particular type of question with respect to 2D shapes.

(Note: this video is narrated in Catalan, my mother language, though subtitled in English. In my channel, there is a version dubbed in English, however my pronunciation there is not as firm) The aim of my video is to present a real-life problem where the heat equation can be applied, and go through the steps to solve it, in a space with spherical symmetry. Through this process, I present the steps needed (Laplacians, spherical coordinates, Legendre polynomials, and more), go through the logic that guides them, and reach an end result, but without extensive technicality to make it more of an overview. The reason I chose this topic is that I am an undergraduate currently studying Physics, and I was amazed at learning of these methods to solve problems that, before, seemed too much to tackle. My fellows did not feel this way, however, as they did not comprehend the real reasons behind some steps, and only memorized them. That is why I tried to input what helped me to follow the topic into the video, so that it may help whoever views it. Also note that this is my first time trying to do a video like this, and have no previous experience dubbing, so please excuse my inexperience. Thank you so much!!!

A visual guide that uses the cross product as a stepping stone to make sense of octonion geometry and why 7D is so special.

I go over the idea of a logarithm as the inverse of an exponent. I find most teachers and textbooks go "backwards". I like the version portrayed in this video better. I tried to make a video that front-loads the important concept, So that you can watch the first 10-20 seconds and get the main idea. I think this will be most useful for high school students looking to understand, and maybe even teachers looking for other ideas. I have worked as a math tutor for 20 years, full time for the last 10.

Ever wondered how mathematicians and engineers model the flow of water around obstacles, air over airplane wings, or currents in the ocean? The answer lies in the elegant world of complex analysis, where imaginary numbers become real tools for understanding fluid motion. Whether you're a maths student curious about applications, or simply fascinated by the hidden mathematics that govern our physical world, this video aims to give you a new appreciation for the power and beauty of complex analysis.

A video about trigonometry directed towards early high school students that discusses the trigonometric functions on the unit circle, applications of trig functions, and why trig functions are important, along with more info like inverse trig, radians, and graphs

Explore the fascinating world of cryptography and integer factorization with the Quadratic Sieve Algorithm! This video covers the basics of symmetric and asymmetric cryptography, the importance of large prime numbers, and the complexity of factoring. We walk through classic methods like trial division and Fermat's factorization, then dive deep into the Quadratic Sieve—one of the fastest algorithms for factoring large numbers.

Is it possible to find the volume of a shape that doesn't have a formula? To find out, I'm going to use calculus to measure how much water this vase can hold—without pouring a single drop.

This video was created to raise awareness and concern for people with dementia, and to inspire support for mathematical research helping to solve this tragic condition. A new variant of dementia has been reported. The Department of Mathematics for Neurodegenerative Disorders has been assigned to investigate this emerging problem. You are a student researcher who has just joined the lab, tasked with designing a new Graph Laplacian model that can predict and target this anomaly. But there is a mystery that lies in the path towards its solution… one that holds a profound and nearly forgotten memory. Learning is like a story, with its own mysteries, tragedies, and victories. Therefore, this explainer is told as a story. Thematically, it is told in a minimalistic style that captures the feeling of messily working out a problem on a blackboard. As per SoME rules, no artificial generations are in this video.

Beginning of this year I had a little intrusive geometric thought about the Hadwiger-Nelson problem that has sent me into a chase for a solution yet again. I'm cleaning up a lot of my old intuition and try my best to make sense of this unsolved problem that has nearly endured for a century while trying to keep it accessible and engaging even for those without a background in graph theory. It's a journey through optimization and into a rather strange way of measuring sets.

An exploration of integral calculus through geometry.

Have you ever stumbled upon something so simple, yet so astonishing — that it changes how you see the world? In this video, we invite you on a playful journey through the realm of π/2 and the enchanting Wallis Product. Starting from a curious “what if,” we uncover how a seemingly modest pattern blossoms into a profound formula that captures the essence of π—right before your eyes. Prepare for a blend of mathematical elegance and lighthearted discovery. We’ll: - Notice π/2 in a familiar yet unexpected way—setting the stage for deeper insight. - Introduce John Wallis and his remarkable infinite product for π, crafted in the 17th century, with roots in nothing but simple patterns. - Unpack how this infinite product emerges naturally—no heavy machinery, just observation, intuition, and a bit of algebraic magic. - Share the beauty of “aha!” moments—where algebra, geometry, and a spark of curiosity align. Along the way, you'll experience the timeless allure of the Pythagorean spirit: that universal truth can emerge from the most everyday questions — if only we look closely. We’ll also explore: Why Wallis’s Product matters in the broader history of π. The elegance of infinite processes: how endless repetition can converge on one finite, perfect constant. The joy of mathematical discovery—how a simple shift in perspective can illuminate entire worlds. Who is this for? Whether you're a student curious about π, a lifelong learner in love with patterns, or someone who just appreciates the thrill of "aha!", you’ll find something to spark your wonder here. ⏱ Timestamps: 0:00 — Introduction: A Work in Progress 0:22 — Part 1: The Infinite Product 17:18 — Part 2: Hannah's Shift 20:53 — Part 3: Reflections on Infinity Why this video matters Mathematics isn’t just about formulas. It’s about the stories we uncover when we let curiosity lead the way. Here, π isn’t just a constant—it’s an invitation to notice patterns, to ask “why?”, and to follow that question through to something timeless. Join us in embracing the sheer beauty of discovery.

In this video, I explore how to solve a simple algebraic problem with some logical thinking. I show how performing algebraic steps forward and then reversing those steps is crucial for determining all the possible solutions of an equation.

We provide a video that shows several examples of finding an iterative square root, as well as a general explanation of the method used. Our target audience mainly consists of people interested in maths who know what a function is.

Setting up an equation of motion for a guitar string, solving it and applying boundary conditions. Aimed at high-achieving high school students who want to go beyond what they learn at school.

Analyzing the sums of the rows in Pascal's triangle when treating them in numbers in different bases including atypical bases like a Fibonacci base system, factorial base system, and a base x and factorial combined base system.

The Math behind how computers generate RANDOMNESS

A Playful stick-figure animation where Ploophi confronts a mysterious bouncing balloon - revealing a surprising take on Set Theory. This is the entry for the Summer of Math Exposition 4 #some4 which is created by the 3Blue1Brown #3blue1brown Inspired by Animation vs. Math By Alan Becker, this is a visual math story told entirely through expressive animation

I motivate the formula for the expected maximum among the components of a random vector, each following their own distribution. To this end, I provide Python code animating the empirical order statistics given by the Beta distribution. And this itself is motivated by a dice roll scenario.

This video provides an overview to the potential of graph theory in a short and (hopefully) engaging way. This video does not explore concepts in depth and should be thought of as the "trailer" to a more complete introduction to graph theory. For example, similar videos could be created for various university-level courses, allowing the students to quickly check what the material entails before choosing which topics to follow.

Pretty much a summary of all of the calculus one can look forward to doing in the "Honors Precalculus" class which I did last year. Not very rigorous, but the justifications are good enough to be used on a high-school tier free-response. It is very fast-paced, so pausing is encouraged if you feel like I moved on from something too quickly. Not very serious, (and even contains swearing that I am surprised is not against the rules), but I assume that most people who will watch this are in their mid-late teens. Enjoy!

Tensors are an incredibly confusing concept when you first encounter them. What doesn't help very much is the topics that they tend to be buried behind. In this video, I hope to give you the intuition behind this important mathematical concept and show why you should care about it.

This video shows how one can use geometry in place of differentiation with an example.

I came up with a pedagogical/visual tool for singular value decomposition during the spring semester of 2024, while teaching an introductory course in linear algebra. I call this tool the "map of SVD". During a review before the finals I had helped my students make their own maps of SVD, but there was no public and recorded description of the idea. In the video I go over the map, make one from scratch that the audience can follow along, and finally mention some of the algebra details underneath. While from an arts-and-crafts point of view the video is suitable for a broad audience, in my estimation to really understand and get the most out of the video and the map one would need to be at least somewhat familiar with the more abstract ideas typically developed in an introductory linear algebra class at the undergraduate level. These ideas are approximately (a few reincarnations of) the Fundamental Theorem of Linear Algebra (aka rank-nullity theorem), eigenvalues, eigenvectors, orthogonal transformations, diagonalization and spectral theorem for symmetric matrices. Initially I had also thought of including more advanced reincarnations of the same idea; for instance consider trying to applying the spectral theorem to a sequence of matrices. Although this is very natural from a dynamical point of view, and leads to Lyapunov exponents and works of Furstenberg-Kesten, Oseledets, Karlsson-Margulis etc., ultimately I decided to keep things simple. Perhaps in the future I'll expand on the map of SVD in these directions.

This video shows how changing coordinate systems transforms the way we see functions. Starting with UVs in shaders, we move from cartesian to polar coordinates, and visualize how the same sine wave shifts from a straight line into a circular pattern. The focus is on visualizing the math, building intuition around it, rather than going deep into derivations, making it approachable for anyone curious about math and graphics, and hopefully making them look more into it.

In mathematics a function is a rule that connects an input value with an output value. The rule that connects the input and output values is in the form of an expression. In the video a few techniques for sketching functions in a 2D coordinate system given their expression is demonstrated.

I wanted to share a visualizer for Cayley graphs that I am quite pleased with. Although I did not write it myself, it is fair use to explain concepts using it and teach people how to use it. I also wanted to share some "philosophy" about groups: as both a bunch of Nouns and a bunch of Verbs. Although I do not go into depth, hopefully I do explain enough that people understand (1) **how to use the visualizer** and (2) **how the nodes and edges are generated**, and take away some sense of how to (3) **see both Nouns and Verbs in the Cayley graphs**. I hope that even say middle schoolers can receive these 3 takeaways I intended for this video, even if some words are too advanced.

Taylor Expansions, √(-1), and something my professor said that really stuck with me.

Do you really know derivatives? This video introduces matrix calculus to single-variable/vector calculus students through visualizations, it utilizes mainly Manim but also PowerPoint and handwritten derivations.

Visualising the Law of Total Expectation and Linearity of Expectation, using examples and animating them.

This is a video where the basics of building an AM radio and reaaciver are explained as well as with how Quadature Amplitude Modulation works

A visual journey into Project Euler’s “Chromatic Conundrum” (#544). We start with the simplest 2×2 case, explore why there are exactly 18 valid colorings, and build up a geometric graph that reveals patterns and symmetries. This isn’t a full solution, just a set of hints and visuals to guide you toward tackling the puzzle yourself.

Does light have shape? It does and the shape of light lets us feel rotation even if we cannot see the rotation. In this video, i start with describing that property of shape of light that tracks rotation, and visualize how shapes of light from a far away star can tell us if it is rotating or not. This far away star is a special source that emits in certain windows which have a period of 16.34 days. Naturally, we would like to know if this period is due to its rotation. Watch the video to find out. Although we have some data, it is not conclusive beyond doubt to realize rotation. This is how science is. It will take more time to be sure and we are working on it. Stay tuned to find out more. This video explains how polarization of light, namely, the Polarization Position Angle can track the rotational motion. It further builds on the idea and tries to understand the periodic active window behavior of repeating Fast Radio Burst 20180916B by studying the Polarization Position Angle variations. This video is an informal description of the results of the following paper https://arxiv.org/abs/2507.07651 :Constraining the origin of the long term periodicity of FRB 20180916B with Polarization Position Angle. Links: https://en.wikipedia.org/wiki/FRB_180... https://en.wikipedia.org/wiki/Effelsb... Galaxy image from Tendulkar et al. (2021). Created using manim and Davinci Resolve.

We explain how to implement the k-opt algorithm for the TSP (Travelling Saleperson Problem). Prior Knowledge Required: - High school combinatorics - Some knowledge of Python

Getting a better idea of prime density by combing exponential growth with exponential decay.

Discusses mathematics of various pursuit curves and possibilities of capture.

The magic square is an ancient puzzle that is not fully understood today. The 3x3 has only one solution but the 4x4 has 880. I was very surprised to discover how difficult the 4x4 magic square is and I conducted an investigation to find all the solutions. This style of Maths investigation can be useful for everyone. I wanted to demonstrate how I handle a problem when I don't know which maths techniques are involved. I tried to make the video as colourful as possible. This actually was how I solved the problem on paper first by thinking about coloured grids and I tried to represent that on screen as a demonstration in problem solving rather than displaying maths that I already understood.. Although the problem is connected to very difficult computer science theory I kept this video completely algebra free so it can be watched by high school and even younger students. However the ideas of Boolean algebra and computational complexity are related to degree courses. I

In this video, we’ll take a look at how to solve indefinite integrals, but instead of using the usual ‘u’ substitution , I decided to use a visual approach so everyone can track what’s going on during integration. Many people merely apply the trial and error method when solving, which can feel frustrating, so my goal is to make the process much easier to follow.

Explaining my process for writing shader code. Meant to demonstrate problem solving methods.

This video is meant to show why the cross product describes motion of rigid bodies, but I didn't really get to finish it in time. This is what I have so far. 2 mins to submit, so that's all I can say

Complex numbers are vectors!!! And yes they are!!! We'll see a lot of different ways to look at complex numbers as vectors!!! And they can also be viewed as linear transformations even though the latter isn't the most rigorous approach of them all!!!

A lot of people have seen nunchucks before, whether that be from Teenage Mutant Ninja Turtles or from Bruce Lee footage. But what if I told you they serve a deeper mathematical purpose? That is the aim of this video. This video dives deep into Nunchucks and their relationship with the wonderful world of Topology. It goes over various topics, like S03 Rotation Groups, Closed Surfaces, the concept of Rubber-Band Topology, etc.