Summer of Math Exposition

A video aimed at high-schoolers and undergrads motivating the study of Cyclotomic Polynomials with a simply stated problem in number theory.

Why does the number e keep showing up where it shouldn’t? In this episode of Deranged Math, I share a personal teaching moment that led me down a rabbit hole of probability and logic. During my first year teaching high school, I gave my students a matching exercise with 10 terms, 10 definitions. I was shocked how many struggled, and a few even got zero correct. Zero! That seemed impossible if they were guessing randomly. As a stats teacher, I had to dig deeper: What’s the chance of getting zero matches on a random matching of n items? The answer? A wild, beautiful surprise. As n grows, the probability that no one matches correctly approaches 1/e (~36.8%). But why does this mysterious constant show up here? We break down the classic “derangement” problem, permutations, factorials, and how pure chaos can converge into elegant order. Just when you think you’ve got it, we reveal how this connects to broader ideas in probability and randomness. This video will twist your brain in a good way. No fancy math needed. Just curiosity, basic math skills, and a willingness to get a little deranged.

Why do so many things in nature—like charges on sphere, snowflakes, ripples, and molecules—turn out perfectly symmetric? Is it just beautiful, or is there a deeper reason? In this video, we dive into why nature leans toward symmetry, and how that leads to real balance, taking Thomson's problem as a primary base. We explore this idea from three perspectives: - Physical intuition: simple experiments and real-world examples - Mathematics: powerful tools like Euler’s formula and group theory - Programming: code that helps us check and visualize our results Whether someone learns best visually, mathematically, or through programming, the video offers something for everyone. We use the Thomson problem as a base, showing how abstract math like Euler's Formula and Group Theory helps explain the real world—proving that symmetry really brings balance. To Explore More 📂 GitHub Repository: https://github.com/intuimation-314/so... 🎮 Interactive UI (Vector Symmetry Explorer): https://intuimation-314.github.io/som...

This entry describes how one can create images by defining a function from R X R to R. The target audience is anyone who knows basic vector geometry (add, scale, planes, dot product...)

In this video, I show you why the normal distribution is in so many cases the proper model to describe statistical data, even though it doesn't exist in reality. We talk about the cental limit theorem and try to simulate different kinds of data in order to see why the central limit theorem is applicable.

A treasure hunt with two metal detectors involving Bayes’ Theorem. We compare true positives and false positives, explore the importance of context, and connect it to medical testing and real-world decision making.

The Combinatorics Roadmap is a collection of mini-classes covering the most basic of to the more complex combinatorics. Learning the basics of the exciting field of combinatorics. I'll walk you through the basics you need to understand in order to master combinatorics in this video. I'll give you a clear and concise road map to help you navigate this challenging and interesting field, from the fundamentals of combinatorial reasoning to diving into more complex subjects. This video is intended to give you a good foundation in combinatorics and encourage you to explore its many wonders further, regardless of whether you're a student or just an avid maths enthusiast. Contents: 0:32 Level 0 - Sets 3:08 Level 1 - Sum & Product Rules 4:40 Level 2 - Permutations 7:14 Level 3 - Variations 9:50 Level 4 - Ring Permutations 12:35 Level 5 - Generalized Permutations 14:37 Level 6 - Combinations 20:05 Level 7 - Pascal's Triangle 22:30 Level 8 - The Binomial Theorem 27:42 Level 9 - Generalized Combinations 31:05 Level 10 - The Multinomial Theorem. 32:52 Level 11 - Repeatable Permutation and Variation 34:14 Level 12 - Repeatable Combinations (Balls and Urns) 37:22 Level 13 - The Pigeonhole Principle 40:37 Level 14 - The Principle of Inclusion-Exclusion (PIE) 44:29 Level 15 - Derangements Big thanks to my dear friend @Awyyn for providing the awesome music in this video! Please check out her other works. Her tunes made the Combinatorics Roadmap even more enjoyable. 🙌

In which I try to explain the concept of ratios!! Target audience: 5th graders

This video describes data science as being concerned with using a sample of data to infer something of interest in one of the 2 unseen and theoretical universes out there - the causal universe and the dependence universe. It walks towards making the point that insufficient data is at some level the only thing blocking our view of the dependence universe, and it builds the sharp contrast with the causal universe in which there are far more formidable things blocking our view of the causal universe. In so doing, I try to intuitively introduce the concept of an equivalence class of causal models, without ever introducing the term. The overarching goal of the video is to give students who are beginning their journey into data science a big picture description of what data science is about. One particular thing I am trying to contribute to is informing the latest crops of machine learning students about the fact that there is another entire universe out there besides the prediction/dependence universe. The typical machine learning graduate of today tends to be uninformed of the causal universe and of how very different causal inference and prediction tasks are. When a freshly minted ML student critiques a linear regression (with the relevant variables appropriately transformed) intended to estimate (an otherwise properly) identified causal effect for having an R^2 that is too low, chances are that he/she does not understand the difference between prediction and causal inference. Even ML teachers/pioneers are at times confused about causal inference. Take the classic ISL with R or ISL with Python books as an example. The fact that the following paragraph has survived several rewrites suggests that it is a not due to a typo but is a reflection of a misunderstanding on the part of the authors of what causal inference is and how it is different than prediction "... Therefore, if we determine that there is an association between advertising and sales, then we can instruct our client to adjust advertising budgets, thereby indirectly increasing sales. In other words, our goal is to develop an accurate model that can be used to predict sales on the basis of the three media budgets" (page 15 in ISL with Python and ISL with R). Every machine learning student needs to spend some time to understand causal inference and how it differs from prediction, because causal questions will not cease to be of interest to us as a society; ever. And the machine learning community should not use "linear regression" as a punching bag example only to dismiss it as the ever-useless model, in spite of it being often too simplistic for prediction, because social scientists still use some form of linear regression in most top journal published articles exploring a causal (as opposed to a "prediction") question.

The video offers an intuitive introduction to Lagrangian Mechanics. It starts with a problem from the calculus of variations which, when generalized, leads to the Euler-Lagrange equation—the core of Lagrangian Mechanics. While mathematical details are included, they can be skipped without losing the main ideas.

A video about artistic usages of the sin(x) function in animation, audio engineering, etc. with a couple of cool examples. Tried to make it as basic and simple as possible in order to maybe inspire younger people that might not be that into math and dont see why they should care about it.

This is my submission for the 4th Summer of Math Exposition (#SoME4), where I explore the concept of correlation — a deceptively simple idea that lies at the heart of statistical modeling and machine learning. In this video, I focus on what correlation actually measures, how it can be derived visually through least squares minimization, and why it's more than just a number — it’s a compact summary of linear association. We walk through intuitive visuals, algebraic derivations, and conceptual motivations to bridge the gap between abstraction and understanding. I basically visualize the concept of principle of least error (squares)

Math is everywhere; name anything, and mathematics finds a way to it. Heck, name nothing, math might just get there anyway. It’s an addictive rabbit hole, but hey, so is linguistics. What happens if we merge the two together…? In this video, we will dive into the realm of formal linguistics, going over basic concepts and types of languages, which are compared to Cursed Conlang Circus submissions.

In this video, we learn to create the Dihedral Group using Symmetric operations on basic shapes. For n>2, the nth Dihedral Group Dn is the group of symmetries of n sided regular polygon. This concept is depicted visually in the video which helps the viewer practically understand the concept of Dihedral Group and appreciate the beauty of mathematics. The video starts with introduction of symmetric operation on an equilateral triangle. Then, we first talk about properties of a group in simple terms for those who don't have a mathematics background. (Skip-able if you have studied about groups) Then we create the simplest Dihedral Group D1 using a straight line and D2 using a double sided arc polygon. We then proceed create D3 using an Equilateral triangle. We also show visually that D3 is Not Abelian. Then we create the elements of D4 using a square and write the generalized form of Dn, the nth Dihedral Group. The video ends with the presentation of the 12 elements of D6 using a regular Hexagon and 16 elements of D8 using a regular Octagon. Hope this video helps make this concept more fun and understandable.

What are complex numbers, really? In this animated video, I break down the geometric meaning behind complex numbers using clear visuals and intuitive animations. Learn how complex numbers work on the complex plane, how multiplication affects rotation and scaling, and why 𝑖 is more than just a symbol — it’s a tool for 2D transformations.

Demonstrating precisely tracking faint moving objects using the accumulation of motion extracted pixel values projected into a voxel grid from multiple low resolution cameras.

A couple of years ago I tried to light up a hallway I built in Minecraft, but I couldn't space the lanterns equally. This led to a problem in number theory that deals with all possible positive integer solutions to the equation L = an + bn + a, and a neat visualization to view all possible solutions.

In this video, I talk about Pythagorean Triples and how they are generated.

I teach general problem-solving techniques by focusing on a problem related to machine vision and positioning systems. I chose this problem because it can be broken down into many smaller ones. We learn how to identify these sub-problems, how to prioritize them and what to do when they’re interconnected: how to isolate them so they can be worked on independently and how to determine if our solution is correct. We build a method from the ground up using just simple reasoning and high school-level math. We first figure out how to solve the problem by hand, then by automating the steps with simple but powerful algorithms. These will be implemented from scratch and explained step-by-step using visuals. The content is aimed at beginners, such as high school students, who want to sharpen their skills and better understand why those things they study in math class are important in practice.

A video on the epsilon delta definition of a limit, motivated by the problem of whether 0.999... = 1 is true.

The Second Isomorphism Theorem in Group Theory is very opaque at first glance. Here's what's actually going on!

This video is intended for undergraduates with a linear algebra background, but is also approachable for high school students. The primary goal is to introduce symplectic geometry by way of a novel way of graphing complex lines, that is, functions of the form y=mz where m is a complex number.

Struggling with logarithms? Here’s the simple strategy I’ve used since day one: Take the base, go to the other side, equals what’s left. I explain log notation, rules, and show step-by-step examples - including a real world finance problem. Let’s make logs easy! #logarithm #SoME4

This is one of the most fundamental statements in set theory. It allows us to imagine incredibly powerful objects with great precision. It molds the infinity known as the continuum, allowing us to play with difficult sets like the real numbers. It is a statement that impacts fields like analysis, measure theory, combinatorics, and topology. Finally, it stems from a technique of set theory, called forcing, which is one of the most powerful and interesting concepts in all of mathematical logic, allowing us to construct incredible mathematical models of a vast set of axioms. We will study the statement known as Martin's Axiom, which is useful not only in its applications to the world of mathematics but also as a great introduction to the more advanced aspects of set theory.

In my entry, I show a bit about what eigenvalues and eigenvectors are, how to build an intuitive understanding of them and also a cool application in automation and control theory. I tried to conciliate the needs and wants of both struggling students as well as people more into STEM by providing a basic refresher and a more advanced application as well. I start by introducing a real world problem (the stability of a pendulum), then proceed laying the theory groundwork until we have enough notions to finally solve the problem. In conclusion, a small digression on discrete-rime systems gives another view on the same topic from a different perspective. Unfortunately I caught a cold and I'm not an native English speaker, so the audio is not too clear. I apologize for that! All diagrams, graphs and animations were made by myself in Octave, Paint.net or other free software. Stock videos and background music used are from Pixabay under a free commercial use policy.

While Maximum Likelihood Estimation is one of the core technique in supervised machine learning, using it naively in interactive systems leads to consistency problems: it does not anymore converge towards the true parameter. But, towards what does it converge?... Pre-requisites - Basic notions of machine learning - Notions of statistics - Algebra in R^2

I made this video with the hindsight I gained after finishing my own physics undergrad. Rather than jumping directly into rigorous derivations of various quantum equations of motion, I thought it might be valuable to get a birds eye view of how they all fit together. This is how I personally envisage them all, and this view helps me to know when each one is useful. See the very last screen, which summarises the whole shebang :)

This video is a (to the best of my knowledge) new proof of the Pythagorean Theorem which I came up with. It involves slicing two circles and rearranging the pieces to make a larger circle. Then, the viewer is challenged to generalize this proof to prove the Law of Cosines and this proof is drawn. I think this video is perfect for a high school geometry class, as a video to be played for the class after they learn about the Pythagorean Theorem (or the Law of Cosines). I hope that it will offer a more visually appealing proof than the ones students would otherwise encounter (and one with infinite rotational symmetries!) I think that the challenge of generalizing the proof to non-right triangles is also a great one for teachers to pose to students in the classroom.

Intended audience: Those familiar with basic E fields in AP Physics C: Electricity and Magnetism, or undergrad-level electrcity course. This video explores the derivations for E-fields and cool facts about the infamous ring of charge in electric physics. It examines both a flat ring of charge and a raised point charge. It also delves deeper into the E field equation and help visualize dynamic graphs for varying the length of the ring and height of point-charge with respect to the angle.Lastly, the video makes comparisons to the E-field equation, which is similar to Coulomb's Law with some trigonometric factors.

Discusses how computers use the binary system to represent number and perfrom arithmetic on them. Also covers the two's complement and how it can be used to represent negative numbers and the way adder/subtractor circuit can be built from logic gates. Audience is everyone, but mainly high school and undergraduate students.

In the first video of a five-part series, we look at how vectors describe the world around us.

Love song of Plato, who loves his platonic solids platonically.

Hi, I'm Michael. I had a bit of fun in this video. What's one way we can find out who's the best at tennis? I attempt to answer this question according to a natural mathematical model that's already popularly used among games like chess, go, and many competitive first-person shooters.

Guess Who meets math. Two players race to find a hidden card first. Simple rules. Sharp decisions. You will see: • Why binary search wins in solo play. • When a worse average wins more head to head. • Where binary search stops being best. • Why mixing strategies at random becomes optimal. You will build a question tree, rank questions by entropy, update beliefs with Bayes, and test greedy picks against planned sequences. Quick visuals and small experiments keep focus on choices which cut the board fast. These ideas connect to Rock Paper Scissors, Poker, and Scrabble, where mixed strategies and payoffs shape smart play. Try the approach in your next match. Ask halves, update after each answer, track progress. Fewer turns, stronger wins.

In my entry, I present a seemingly paradoxical device - the Fabry Perot Interferometer (Resonant Cavity), which consists of 2 semi-transparent mirrors. When the resonance condition is fulfilled, 100% of light can be transmitted, which might seem unintuitive at first. Even if you are familiar with interference, the video contains enough nuances to keep you engaged.

Unlock the hidden components of mathematical functions, unveiled in signals and systems! This video dives into Fourier Series and the Fourier Transform, showing you how to decompose signals and view them from a frequency domain perspective. This time, let's utilize the manim visuals! We'll explore: (1) What Fourier Series are and how they approximate functions like triangle and square waves. (2) The intriguing Gibbs Phenomenon at discontinuities. (3) The power of complex Fourier series and frequency spectrum analysis. (4) How low-pass, high-pass, band-pass, band-reject filter work. (5) Understanding the fourier transform visually Transform your understanding of signals today!

Geometry problems in math competitions can feel impossible, but what if there was a cheat code? In this video, I introduce the Method of Moving Points, a powerful problem solving technique that helped me solving every geometry problem on every contest I had in my last year of competing. The basic concepts of the method are internationally known, however this alone is not enough to solve hard problems. That's why I made an extension of the method, with which the hardest problems can be solved too. In this video I explain the basic version of the method, and demonstrate it on an example problem from the IMO. I made a document containing the extended version, which can be found in the description.

This video seeks to give an overview of three common mathematical tools but from an Engineering context. The Laplace, Fourier and Z-Transforms are often taught individually but, when used together, they allow a much clearer picture of a system. Here, the application is about control systems with an example of a car suspension system.

I provide the equation that determines an individuals path to financial freedom. In this video, I present the formula, explain it using an example and then show how to derive the formula. We see how an individuals financial data all combine to inform when you can stop working and let your wealth sustain you. I note this is an idealised financial model — it assumes constant growth rates and inflation-adjusted spending — but it provides a powerful framework for answering key questions like: -How much earlier can you retire if you cut spending? -What impact does investing more have on your financial freedom age? -Can you afford to work less?

I describe a concept that is central to modern algebraic geometry - schemes. It allows us to do geometry over pretty much any number system you can think of, including the integers, p-adics, and so on. It's used in the proof of Fermat's Last Theorem by Andrew Wiles. I first define locally ringed spaces by generalizing properties of continuous functions on a topological space. Then, I reverse this process and construct a locally ringed space called Spec(R) from any commutative ring R. I achieve the goal in the video by defining a scheme as a locally ringed space that is locally isomorphic to Spec(R). Finally, I touch on some advanced topics such as Proj of a graded ring, quasicoherent sheaves, invertible sheaves, sheaf cohomology, relative Spec and Proj, and blowing up. And I make a connection with number theory via the ideal class group. This is a submission to 3Blue1Brown’s #SoME4 contest. Works Cited Foote, Richard, and David Dummit. Abstract Algebra. Danvers, John Wiley & Sons, 1991. Hartshorne, Robin. Algebraic Geometry. New York, Springer, 1977. Jacobs, Konrad . “File:Alexander Grothendieck - Face.jpg - Wikimedia Commons.” Wikimedia.org, 21 Oct. 2024, commons.wikimedia.org/wiki/File:Alexander_Grothendieck_-_face.jpg. Accessed 1 Sept. 2025. Munkres, James R. Topology. New York, Ny, Pearson, 1974. Vakil, Ravi. The Rising Sea: Foundations of Algebraic Geometry. 8 Sept. 2024. Wiles, Andrew. “Modular Elliptic Curves and Fermat’s Last Theorem.” The Annals of Mathematics, vol. 141, no. 3, 1995, p. 443, https://doi.org/10.2307/2118559.

This video introduce the method to construct all rational numbers, square root, cube root. and find the root of liner, quadric, cubic equation use paper folding.

Understand how engineers design earthquake-resistant structures using the Response Spectrum method. How do engineers predict the maximum impact of these forces without running complex simulations every time? In this video, we explain the Response Spectrum Method, a powerful seismic analysis tool that helps engineers assess the maximum structural response based on natural frequency and earthquake time history. We break it down using simple analogies, starting from a ball-on-stick model, introducing equilibrium equations, and leading up to how the design spectrum is generated.

Based on the Fundamental Theorem of Calculus, you might assume that every derivative is integrable. Shockingly, this is not actually true! In this video, I show you the construction of a function which has a derivative which is not integrable, delivering a counterexample to this wrongful assumption. The video consists of three parts: first up is a crucial lemma for proving nonintegrability, second is the construction of a set on which the derivative will be discontinuous, and last is the construction of the function and its nonintegrable derivative. The video is aimed at undergraduates who have completed their first year of mathematics, specifically at people who have taken a course in Real Analysis.

Video Explaining PD controllers -- a very nice but surprisingly useful tool in mathematics used to nearly universally balance systems which honestly changed my view on problem solving

This video will guide you through polynomials and vectors.

In this engaging lesson, viewers explore how additional information can dramatically change the likelihood of an event — a concept brilliantly illustrated through a humorous story involving Henry Kissinger and a Siberian lumberjack. Using intuitive examples with fruit boxes and medical testing, students learn to calculate conditional probabilities and apply Bayes’ Theorem. By the end of the lesson, they’ll not only understand the mathematics behind updated beliefs but also recognize its relevance in real-world decision-making, from diagnostics to debunking misleading statistics.

From Chess and Go to Olympiad-level math, DeepMind’s AI has tackled some of the hardest thinking problems known to humans. But here’s the surprising part: models like AlphaZero and AlphaProof don’t actually “think” like us — they turn these challenges into search problems. In this video, we explore how Go can be represented as a massive search tree with more possibilities than atoms in the universe, and how AlphaZero uses Monte Carlo Tree Search combined with neural networks to master the game. We’ll then look at AlphaProof, which takes a simpler but smarter approach to searching through proof steps, enabling it to solve Olympiad-level math problems and even reach a silver medal standard at the IMO. By the end, you’ll see how DeepMind reframed thinking itself into search — and why that’s such a powerful idea for the future of AI.

It is sometimes said that the complex numbers cannot be ordered. This video explains why that is, and how we fix it.