Summer of Math Exposition

This video reports my journey trying to figure out a way of drawing (in a computer) Venn diagrams for large quantities of sets, as well as the things I learned along the way.

An exploration of many algebra topics through vector graphics as motivation

This video is about information theory which explains the concepts of “information” and “entropy” from first-principles. It dives into questions like “where does the equation for information come from?”, “what does entropy really mean?”. It aims to provide a strong intuition which can help you deeply understand these concepts from the world of information theory. The video also goes into conceptual details to answer questions such as why does entropy come up in the world of artificial intelligence (AI), and why do AI models such as neural networks rely on it so often? The ideal audience is undergraduate and graduates in maths, physics, computer science or a related field.

This is a group theory explainer. We start with Galois’s idea of “groups” of permutations. Then we introduce Cayley’s modern definition of a group. We prove Cayley’s Theorem in detail to emphasize how a group can be thought of as a set of functions, and further explore this idea through the definition of a group action. The intended audience is undergraduates, but advanced high schoolers could follow, or beginning graduate students might like as a review or overview.

Gentle Introduction to Logistic Regression. We explore this powerful yet simple machine learning model for binary classification tasks like predicting whether a student will pass or fail an exam! We’ll explain the difference between logistic and linear regression, dive into the intuition behind the sigmoid function, and uncover why cross-entropy loss makes sense for probability-based models. No complex math needed—just clear explanations, intuitive visuals, and a practical demo in Python with scikit-learn.

We give a simple geometric explanation of why the Fundamental Theorem of Calculus is true, specifically the version of the Theorem that says that an antiderivative can be used to find an area (typically called Part 2 of the FTC).

This is a fairly beginner-level video about a technique commonly used in computer graphics effects: signed distance functions (or SDF's). I go over what SDF's are, how to create them, and then go through a bunch of different uses for them in computer graphics, specifically with fragment shaders.

In the video I explain the basic of special relativity, including time dilation, length contraction, and simultaneity. The three phenomena comes together and gives the Lorentz transformation, which I then used to show how to derive velocity addition. I used Minecraft mainly to stay on-brand for my youtube channel, and also because of the moddability of the game, which allowed me to make a shader to demonstrate length contraction. I also wanted to promote physics to more people; I figured that because the Minecraft community has a lot of people, they might enjoy the video as well. In turn, I hope they would enjoy the physics that comes with it.

This video is about the combinatorics of the card game SET. How many cards can you have without any "Sets"? In this video, we answer that question. The solution involves a fascinating blend of concepts that could appear in a course on combinatorics or discrete mathematics.

A cautionary tale for precocious students and a justification for studying the real numbers, in that order.

Use statistical hypothesis testing to decide on which cookie to eat next! The aim of this video is to explain in simple terms the concept of safe anytime hypothesis testing that has been popularised in the last few years through the concept of e-values (https://en.wikipedia.org/wiki/E-values). Hypothesis tests are a way to use the data at hand to make a decision on the average behavior of the data. Here we take the example of preference data (like or dislike and decide on which we prefer on average). Hypothesis tests are a core tool for scientists as it allows one to assess if some observed phenomenon is due to chance or if there is really something to understand from the data. The main idea of anytime hypothesis tests compared to usual hypothesis tests is that we can collect data sequentially and decide to stop whenever we want and still have a controlled probability of error, whatever the reason for stopping. This contrasts with the usual tests which are in general very strict on the methodology one has to use. This video's main target are undergraduate with some basic understanding of what is a random variable, and expectation... However, we tried to make it so that the main concepts are accessible to a broader audience.

What are groups, and why do we study them? In this video, we explore group theory from the ground up with the goal of proving two major theorems: Cayley’s theorem and Lagrange’s theorem, both of which provide fundamental insights into the structure of all groups. I hope to convince you that group theory is interesting and beautiful while also giving you a concrete sense of the techniques that group theorists use to reason about groups.

Polynomial root-finding method that is guaranteed to find a root, even if that root is complex. The algorithm uses a circle test to determine if a given polynomial has a root somewhere in a circle on the complex plane. Then performs a series of circle tests around an annulus to shrink the search space and repeats until it finds a solution.

Real analysis is the foundation of many important areas of maths, with practical applications from the Fourier series to complex and functional analysis. Unfortunately, the typical first reaction to this approach is that it overcomplicates simple ideas when intuitions alone convey them well enough. In this video, with the help of the Cantor set, we will disprove two seemingly obvious examples from analysis: relating the size and measure of a set, and the amount of times a function can cross the axes.

In our video, we explore posits, an alternative exposition to floats. We first explain the basic concept of floats and posits and then we proceed with comparing both of them. Do posits beat floats at their own game? Or are they just one more in the long list of unsuccessful efforts to improve floating-point representation?

Why is the area under this hyperbola a logarithm? Why is the base of this logarithm the number e? Why is that so important this is called "natural logarithm"? You'll find all the answers in this video. This content is mainly aimed towards undergraduate students. It touches a bit on integrals towards the end of this video, but the explanation is mainly visual, using the concept of "hyperbolic rotation" to explain why area(a), i.e. the area between 1 and a, has the property that area(a) + area(b) = area(ab): the logarithm property, as well as using the same concept to find the base of that logarithm.

An introduction to using the area model from arithmetic as a way to model algebraic identities visually. The main idea is to let the pictures inform you about what is going on so that you can recreate algebraic identities if/when needed instead of memorization.

One of the hardest puzzles in both Mathematics and Physics is to solve the Navier-Stokes Equations. These equations describe the behavior of fluid and form the foundation of fluid dynamics. They are not only essential for realistic fluid simulations in games and movies, but also play a crucial role in predicting turbulence in air, cyclones, and even the weather patterns. Even proving that whether a correct solution exists or it blows up to infinity would be enough to win the million dollar prize. But, why are the Navier-Stokes equations so hard and difficult to solve? why does this happen? In this video, you and I are gonna explore one of the three strategies proposed by Terence Tao as a possible path to tackle such a problem.

This video explores the true meaning of force through the lens of calculus and offers a visual intuition to differentiation.

This is the story of the Ising model, a simple-looking problem from physics whose complexity explodes exponentially, making it fundamentally unsolvable for large systems. It’s a problem so hard that it secretly represents thousands of other famously difficult puzzles in computer science and mathematics. So, how do you solve an impossible problem? Sometimes, you have to take a leap of faith. This video explores a journey that begins with a simple question: what if the tensions in the system weren't random, but followed a strict, hierarchical rule? We'll follow the thread from a curious observation on a small computer to a bold assumption that reduces an impossibly vast search space to a simple, solvable question. Finally, we'll see how that assumption can be rigorously proven, revealing a surprising and elegant structure hiding in the chaos—a "Great Schism" that a complex society uses to find peace.

Using the well-known bluffing dice game Maier (German; English: Mia), different perspectives on probabilities are explained. The focus is on understanding conditional probabilities and how Bayes’ formula can be applied in this context when using additional information from other players to estimate whether they are lying. Additionally, various research-based visualizations are presented that have already been shown empirically to be helpful strategies for dealing with conditional probabilities. Based on these considerations, decision rules are derived, which are then tested in a simulation to evaluate their practical applicability.

A derivation walkthrough of an alternative version of the quadratic formula I came up with. Meant for enthusiastic high schoolers, and even undergraduate students.

Transformers are now everywhere—from generating game frames and AI avatars to powering YouTube's new LLM-based recommendation algorithm—and at the heart of it all is the attention mechanism. But as models scale, the original attention operation becomes a costly bottleneck. Enter Flash Attention, a breakthrough algorithm from Tri Dao that drastically speeds up attention by optimizing how data flows through GPU memory. In this video, we explore the core ideas behind scaled dot-product attention, the limitations imposed by memory bandwidth, and how Flash Attention uses online softmax and clever memory-aware buffering to reduce quadratic memory complexity to linear. You'll learn how the GPU memory hierarchy (L1, L2, HBM) affects performance, how Flash Attention keeps everything in cache-friendly form, and why even consumer GPUs like the RTX 3090 see over 4x speed-ups with no compromises in accuracy. We also break down how this works mathematically, what makes it so elegant, and how to enable it in PyTorch with just one line of code. If you care about scaling ML, maximizing your hardware, or just want a deeper appreciation of the fusion between algorithm design and GPU architecture, this one’s for you.

In this video, we explore spectral graph theory through the lens of heat diffusion. We start by explaining what community structure in a graph means, and why identifying clusters is important. Then, we introduce the heat equation and its discrete version on graphs, showing how temperature spreads over time. Finally, we use the slowest mixing temperature distribution (the second eigenvector of the Laplacian) to reveal the graph’s natural clusters.

The Pokemon video games can spawn rare variants of each Pokemon called Shiny Pokemon, which simply have a different color scheme. By understanding how the game calculates the random numbers when spawning a Pokemon, we can influence the game to spawn Shiny Pokemon. This video demonstrates how computer random number generation works, how the game uses this to generate pokemon, and how we can represent the random number generation as a transformation using a GF2 matrix. It then demonstrates how we can invert the matrix and use information from the game to reverse engineer internal data to enable us to find the steps to force a shiny pokemon to spawn. This video contains some high level computer science and linear algebra topics, but is meant to be approachable by presenting these topics through the lens of a video game.

Not many people know that the light from the sky is polarized, and even fewer understand how or why. Yet the polarization of the sky is something we can not only see with our own eyes (almost), but also recreate at home with a simple experiment. I’m a postdoctoral researcher in nanophotonics, and I believe that understanding this effect offers a perfect introduction to wave optics and light scattering. In this video, I aim to strike a careful balance between accessibility and scientific precision, because oversimplification often leads to misconceptions. I also place a strong emphasis on illustrations: they may look completely different at first, yet often depict the same physical idea, just from a different point of view. To follow this video, all you really need is a rough idea of what an electric charge and electric field are. No advanced physics required. Although the hairy ball theorem isn’t essential here, it still quietly shows up, and makes everything just a bit more beautiful.

Geometry class has convinced us that dimension is inherently a macroscopic property; i.e., that you can't understand the dimension of a shape without "zooming out" and seeing how much space it takes up. It has also convinced us that all points are created equal: that is, that all points are zero-dimensional. Yet, surprisingly, there is a meaningful way to describe the (possibly nonzero) "space" taken up by a point from the perspective of algorithms. More precisely, every point has an effective dimension related to how hard it is to produce approximations to that point -- or, equivalently, how hard it is to compress its digits. And the geometric/fractal dimension of any shape is computed as the maximal effective dimension across all the points in that shape. So, the dimension of a set comes from its least compressible elements. In this video, I discuss all these concepts, state the main theorem (The Point-to-Set Principle) which relates effective dimension of points to the fractal dimension of sets, and show how one may use this theorem to calculate the fractal dimension of a standard fractal known as the Middle-1/3 Cantor Set.

In this video, I try to convince you to hang your sequences on a clothesline, count some stuff with them, and keep you awake with stop-motion dinosaurs.

What happens when you connect two harmonic oscillators together? This is known as coupled oscillation, and has many interesting properties. This video covers the introduction to this topic, explaining harmonic motion, the differential equations which describe the phenomena, and how the system changes with the addition of extra bodies. It then explains why this movement is not periodic, despite being a superposition of multiple periodic motions. Due to time constraints, this videos scope was constricted, and so there are certain simplifications in the video, and many interesting subtopics that were not mentioned. If you'd like to learn more about this topic, there are some interesting videos and papers teaching the material in the video description.

The question is simple: How to choose between 3 options, using a coin! We provide two solutions, and compare the expected value of the number of flips needed for each approach.

This video covers the fundamentals of quaternions and demonstrates how to implement a quaternion class in Python to rotate and animate a 3D model, such as Earth.

In this video, we explain how convert from one unit to another, both conceptually and practically. We go through some examples, including converting areas and volumes. There are even some challenge questions for you at the end! The content is aimed at high school to college students taking science classes, who need a mini-lecture or refresher on how to convert from one unit to another. This video was created at Siena University in the summer of 2025 by students Eva Pflomm (Math) and Grace Jones (CS) and Professor of Physics, Matt Bellis. At the time of this work, Eva and Grace had just completed their freshman year. The animation was done with the manim (https://github.com/3b1b/manim) software, specifically the manim community edition (https://www.manim.community/), developed by Grant Sanderson for his 3Blue1Brown YT channel ( / 3blue1brown . We made extensive use of ChatGPT using the paid "Plus" level when we developed the actual python code. ChatGPT provided no input to the actual content of this video, which was derived from a video recording of a whiteboard lecture by Matt. All the python code was written and developed by Grace and Eva. Audio was recorded with OBS (https://obsproject.com/) and the video editing was done with Kdenlive (https://kdenlive.org/). The music was written by Eva Pflomm and Matt Bellis.

This video explores Lill's method, a forgotten visual method to solve polynomials. Instead of crunching formulas, the method involves geometry, sketching lines and circles, and even using origami to arrive at the solution. The video starts off with a general explanation of the method, followed by a proof, and also shows links to Thales' theorem for circles when solving quadratic equations. We then move onto a beautiful geometric method for deriving even the complex roots of quadratics, using a mixture of lines and circles, before doing a similar method for cubic equations. The method for cubics also involves some origami techniques, folding paper in a particular way to arrive at the solutions of the polynomial.

Quantum computers are FAST. They can solve problems that would take a classical computer millions of years. But is there a limit to how fast they are? The answer is yes, and the limit is the speed of information. The speed of information is closely related to Shannon's entropy and information gain. Learn it all in this friendly video!

To make topology more visual, pure mathers often describe it as analyzing complicated shapes. This is such an oversimplification that it misses nearly every application of the field. As time passed, this oversimplification became the face of topology. Its true meaning was forgotten, along with its applications. This video uncovers what topology ACTUALLY describes, and why it shows up in nearly every field relating to computational mathematics.

TL;DR: The fundamental motivation, concepts, and identities of elementary trigonometry visually from the ground up. ========== If you are familiar with the 3Brown1Blue channel, you also likely aware of Grant's excellent "Essense of Calculus" and "Essense of Linear Algebra" series. That experience is what I attempted to recreate in this film; I wanted to present a perceived abstract and tedious topic from the bottom up in a visuals-prioritized form. I've seen many students struggle with Trigonometry, and I personally have also experienced how frustrating it is trying to understand the intricate topics. I also know many students (including myself) have been asked to "memorize" these concepts, which I personally believe is very ineffective in motivation and learning. Thus, I attempted this summer to create learning material to help students intuitively grasp the topic, with a focus on motivation and fundamental definition and understandings. The general topic outline is as follows: 0. Introduction. 1. The Motivation Behind Trigonometry - Why It's So Important. 2. Very Quick Review of Fundamental Concepts/Definitions - Soh Cah Toa. 3. Radian Angles Measurements - Why Would We Put π in Angles? 4. Basic Trigonometric Angles (30, 45, 60 and how they are derived + 0 and 90), and how they are computed primatively. 5. Intuitively Extending Trig Angle Inputs Beyond 90 degrees. 6. The Connection to Unit Circle. 7. Basic Identities and their Geometric Visualizations. 8. Misc. & Conclusion. Note that I skipped topics like Inverse Trigonometric Functions or the Geometric Proof of the Law of Cos in interest of time and keeping in line with the mission. ========== The video was posted to the Steakmath YouTube channel at https://youtube.com/@steakmath.

The Coin Puzzle That Tricks Everyone

Difference of Squares is a really interesting and wide-ranging topic! In this video, I want to first give an intuition for the concept, then provide many examples of where you'll encounter it in the future. The multiplication trick is one I wish someone had taught me in elementary school! And I believe if we teach it well to young people, they will be better prepared to tackle algebra and geometry problems in the future.

Every gamer knows what damage drop off is, but very few will know the math needed to make this happen. In this video, I will show the math needed to make this happen and try to improve it. And possibly overcomplicate it.

What happens when you rotate spacetime—literally—using complex numbers? In this video, we explore the surprising simplicity behind complex spacetime transformations and how they connect to electricity, quantum mechanics, and special relativity. We introduce four-vectors, geometric algebra, and the power of rotors to simplify Lorentz transformations. You'll see how real physical quantities like mass, energy, and momentum emerge naturally from elegant mathematical structures. This is Part 1 of a multi-video series unpacking the geometry of spacetime through complex analysis, Pauli and Dirac matrices, and geometric products—all in the spirit of Einstein and modern physics. From the elegance of U(1), SU(2), and SU(3) symmetries to the deep structure behind quantum fields, we’ll explore how seemingly abstract math reveals some fascinating patterns. Perfect for students and enthusiasts of math, physics, and anyone curious about the deeper structure of reality.

This video introduces the idea of z-scores as the common language of the normal distribution. We’ll see how measuring values in terms of standard deviations allows us to compare across different normal curves, apply the empirical rule, and ultimately compute probabilities. The z-table then becomes the practical tool that connects these ideas, letting us move between raw data, standardized scores, and areas under the curve. Intended for students in an introductory high school or college statistics course, this video emphasizes understanding the big picture: how z-scores unify the many applications of the normal distribution into a single framework.

Discover how a simple soap film can solve complex optimization problems. In this visual demonstration, we use physics to naturally find the shortest time path solution, showcasing the fascinating connection between nature and mathematics. Perfect for math enthusiasts, physics lovers, and the simply curious. The beauty of this topic lies in realizing that nature can be our most powerful “calculator”. By mapping mathematical problems into physical systems, we can let nature work out the solution for us. This video is suitable for middle school, high school, and undergraduate students. There’s something here for everyone!

Pattern Discovery is a model for teaching math based on building intuition through finding patterns.

Proof of Pascal's theorem using Bezout's theorem.

In 1951, the great musical satirist Tom Lehrer, who was then teaching calculus at Harvard, wrote an especially Lehrerian optimization problem - in 72 lines of verse! - for his students. Tom gave me a copy of this problem 25 years ago. I've never seen it published before, so as a tribute to Tom, who passed away in July, I present it here.

Complex, self-similar patterns emerge from simple rules, and fractals are all around in nature. But how can these phenomena be captured on a screen? This video explains how to generate fractals using the chaos game, log-density plotting, gamma correction, and affine transforms, and how to render them in real time on the GPU, so you'll never have to leave your screen

An exploration of what goes on when your browser renders this description. This video ended up less directly mathematical than I'd expected, but I still thought it'd be worth it to submit to SoME as I originally planned.

Today we cover 6 increasingly difficult math problems. They might seem tricky at first — that’s because they are… if you’re not creative. Timestamps: 0:00 Intro 0:25 Problems 0:40 Discrete Math 0:53 Exploding Lights 2:31 The 3 Exams 3:38 Algebra 3:52 Distortion 5:30 Triple Trouble 6:55 Geometry 7:02 Pihedron’s Quad 9:20 Captured (Probability) 14:25 Outro Made with @pihedron. This is our entry to the Summer of Math Exposition 4. Check out his channel for similar content, and vote for our submission!

A unique derivation of the equation for the sun's elevation. Hi everyone, this is a problem I have been working on for a while. I had a lot of fun with it. I hope it is just as interesting for you! This is also my first-ever submission to the Summer of Math Exposition #SoME4! I greatly appreciate 3 Blue 1 Brown for all he does to make math engaging for everyone. All graphs were made using GeoGebra 3D Calculator. I would be very happy to hear your feedback! ~Heart to Universe

This project was inspired by 3Blue1Brown’s interactive quaternions video—the moment where you don’t just watch the math, you touch it. I wanted that same feeling for spherical coordinates. The result is a short, highly visual lesson paired with an interactive Desmos scene that mirrors the video one-to-one. Watch the video from start to finish to see the structure unfold cleanly; at the end there’s a quick walkthrough showing exactly how to use the Desmos file to recreate everything you’ve just seen—right down to the final picture. The focus is conceptual clarity you can feel: how the forward map (r,θ,φ)→(x,y,z) paints the sphere, how the inverse map (x,y,z)→(r,θ,φ) “reads” points back into angles, and where the representation gets delicate near the poles. No prior knowledge is required; the visuals do the heavy lifting while the narration keeps the rigor intact. Then the Desmos model turns the ideas into a sandbox: drag a slider, nudge an angle, and watch the geometry breathe. I built this to help the material sink in through doing—first a beautiful, guided pass, then hands-on reconstruction. It’s meant to be enjoyable to look at, satisfying to tinker with, and faithful to the math. Everything is free by design so students and teachers can use, remix, and share. If it resonates, please pass it on; the more people who interact with the ideas, the better the ideas stick. Here is a Desmos link with the clone with everything seen in the video: https://www.desmos.com/3d/og7qio7wgz It is recommended to watch the video from start to end, to then interact with everything that was in the video and also watch the Desmos walkthrough at the end for the perfect Desmos experience: https://www.desmos.com/3d/og7qio7wgz This could be perfect for any classroom interactive visualization from high school to undergraduates as no priors are required to interact.