Summer of Math Exposition

A manim video on numbers construction, I start with peano's axioms for naturals, then pairs and equivalence classes for Z, same for Q, then cauchy sequence for R, and pairs for C (I don't talk about the ring of polynomials and the ideal) It's in french, but english subtitles are available, kindly made by my friend 2swap The targeted audience ranges from curious highschoolers to undergraduates, maybe graduates

Pi is the little easter egg of math that somehow ends up in your path most of the time. In this video I decided to understand how pi related to factorials, but more specifically, double factorials. Throughout the video I cover small bits of topics surrounding infinitesimal calculus. However. I myself hadn't much knowledge on the topic before, so it's not a necessity to know this branch of math fully to understand the video, but it's at least recommended to have some insight on it.

In the video, I introduce (homotopy) type theory and develop the necessary knowledge to understand the univalence axiom in an informal/intuition-based fashion. While the content itself does not have much prerequisites other than how functions work, familiarity with mathematical proofs like induction certainly helps. Knowledge of topology/homotopy likely means that the audience will get more out of it, but it is not at all a prerequisite.

A visual demonstration of curve-sketching fundamentals for Calculus 1.

TL;DR: We created a video on the topic of optimisation, targeted at high-school students. Optimisation as a concept interested us because it is an incredibly powerful mathematical tool while also being relatable and accessible, since a range of mathematical techniques, from simple to complex, may be used. For ease of watching and to share multiple different concepts, we structured our video into five levels. [Although not necessary, an understanding of single-variable differentiation is recommended.] ------------ To start with, we knew that our target audience would be high-school students. The fact that both of us are ourselves recent high-school graduates meant that we felt we could fulfill Grant’s mandate for SoME4. Choosing a topic became slightly difficult then - would a high-schooler with a casual but not frenetic love for math be willing to watch a 40-minute video on complex numbers? But equally, we didn’t want to make a video on topics firmly in the high-school curriculum, like circle theorems or differentiation. We needed an interesting topic, preferably one that extended beyond standard school stuff but was still grounded in the lives and knowledge of our audience. Enter optimisation: everyone who’s used Google Maps, or run their dishwasher at night to save on the energy bill, has interacted with optimisation algorithms. This makes the concept both intuitive and widely applicable. Gaining an understanding of it could actually impact how someone sees the world on a daily basis. Furthermore, the tools of continuous optimisation are largely calculus-based, allowing us to connect to school curriculum while also introducing calculus more complicated than single-variable differentiation. By structuring our video into 5 levels, we were able to build up these concepts in a systematic manner. Each covers an increasingly complex type of optimisation, prompting the viewer to either journey through different concepts while gaining mathematical knowledge or to skip to where their current knowledge ends. We feel that we have been able to explore a variety of concepts and practical problems through our video. The topic of optimisation gave us a vehicle to explain many other foundational mathematical tools in a clear and novel manner. We hope that you find the video entertaining and informative!

This originated as a lesson designed to motivate complex numbers for a precalculus class. I show examples of linear recurrences (generalizations of the Fibonacci sequence), starting with simple population equations. Playing with parameter values leads to instances that create periodic behavior and sine waves, and then we see a specific set of recurrences that create sine waves with half periods very close to 10pi, 100pi, 1000pi. In order to explain this mystery, we build up to the theory of how to solve these recurrences. In finding our sine wave examples have complex roots, we then learn about taking powers of a complex number and how that explains the period of our solutions. Finally, we see how to visualize these exact solutions via a machine which adds the outputs of two 3d complex spirals. Most scenes invite you to play with the same visualization I am showing on Desmos or Geogebra. Here are the links: https://www.desmos.com/calculator/4tj7pu6eon (Scene 1) https://www.desmos.com/calculator/f9353a1d60 (Scene 2) https://www.desmos.com/calculator/pldzawaqok (Scene 3) https://www.desmos.com/calculator/fclt20icqx (Scene 5) https://www.geogebra.org/m/qvxyrxqb (Scene 6) https://www.desmos.com/3d/hwdxpabhib (Scene 7)

What does math sound like? Discover the world of music through the eyes of math, simple and accessible for everyone! Covering everything from the nature of sound, to harmony and ratios. This video takes you through the hidden connections between math and music, with a musical surprise at the end!

This video is a follow up to my previous video “How to Describe Circuit Impedance With Continued Fractions”. In this follow-up video, I’ll dive deeper into the behavior of harmonic resonance modes of ladder circuits. There are many interesting patterns to discover in these harmonic resonance modes, and these patterns can help us understand how resistance affects ladder circuits. The target audience is electrical engineers and math fans interested in continued fractions

This video is about the connection between mathematical logic and two-player games! It is aimed at a roughly high school-level audience (except the one example at the end about continuity might be easier for undergrads to appreciate).

This video is about a lunar eclipse, in which I've tried to answer the following questions: 🔹What really causes a lunar eclipse (and why it’s different from a solar one) 🔹 Why some eclipses are partial and why some last longer than others 🔹 Why the Moon turns red instead of disappearing — the science of Earth’s atmosphere scattering light Although this video isn't very mathematically intensive, I have made this to take my first step towards my objective of quality education for free. I aim to bring an international standard of education in my native language, Urdu. Quality education is almost nonexistent in the Urdu language, and I want to fill that gap. One more thing that while I was busy making this video, my state was flooding with monsoon rains and frequent electricity shutdowns. I hope you will enjoy this.

In this video, I give a visual proof of the inequality of means using areas and simple geometric constructions. Step by step, we see why the quadratic mean is greater than the arithmetic mean, which is greater than the geometric mean, which in turn is greater than the harmonic mean. Instead of relying only on algebra, the argument is built on intuition and shapes, making the inequality both clear and memorable.

This video takes you on a mathematical journey through one of computer vision's most fascinating problems, using image colorization as a gateway to explore fundamental concepts in linear algebra, information theory, and machine learning. Designed for undergraduates and beginners, this presentation provides an intuitive introduction to image processing through a compelling real-world application. We begin with the mathematical foundations: grayscale images as 2D arrays versus RGB's three-channel data representing 16 million colors per pixel. The core challenge emerges from the grayscale conversion formula (Gray = 0.3R + 0.59G + 0.11B)—one equation with three unknowns, making perfect inversion algebraically impossible. Using 3D RGB cube visualizations, we reveal how entire triangular planes of valid color combinations collapse onto single grayscale values. This lossy compression fundamentally destroys information, making colorization a uniquely challenging inverse problem compared to simpler techniques like bit-quantization. The solution lies in the manifold hypothesis: natural images don't randomly fill high-dimensional space but occupy lower-dimensional manifolds characterized by smoothness and semantic consistency. We visualize how real image data clusters on curved surfaces within this vast space, while random noise scatters throughout. The mathematical resolution involves finding the intersection of two geometric objects: the infinite plane of algebraically valid solutions and the curved manifold of natural images. Modern U-Net architectures learn to navigate this intersection through convolutional kernels that exploit spatial locality and skip connections that preserve fine details while building semantic world models. This video transforms a simple computer vision task into an exploration of fundamental mathematical concepts, making linear algebra, manifold theory, and neural networks accessible to students beginning their journey in mathematics and computer science.

An explanation of the Monty Hall Problem, illustrated using paper cut-outs.

In this video, we discuss how many dimensions are needed to embed or immerse a manifold into a higher-dimensional Euclidean space. We cover topics such as vector bundles and their role in obstructions to immersion. The video is aimed at viewers with some familiarity with mathematics, such as upper-level math majors, but not necessarily experts.

A brief introduction to the theory of quantum gravity known as holographic duality or the AdS/CFT correspondence. We cover the basic terminology, the underlying geometry, and some of the entries of the "holographic dictionary". The goal is to give a flavor of the way geometry emerges from entanglement in this theory, thus showing that there is at least a conceivable world in which gravity and quantum go hand-in-hand. I think the median audience is undergraduates in math, physics, and related fields, but there is hopefully still something for high-school students and graduate students.

"When you have a line segment of length 10, and break that into 3 segments, what is the probability of those 3 line segments forming a triangle?" Beautiful and intuitive proof to a simple question, using Desoms.

This video is a journey from the silliest math question ever posed (does every number contain a 0?) to very serious math like Mersenne primes or Zermelo Fraenkel set theory. I hope you'll enjoy it as much as I enjoyed creating it. Even if it's the most unserious piece of math you'll see today. Target audience: high schoolers, undergraduates, math enthusiast.

In this video about Special Relativity, I share some of the "paradoxes" - problems - that have helped me deeper understand the theory. We'll touch up on all the fundamentals and clarify confusing points for someone either learning or relearning the subject, with the help of spacetime diagrams and visualizations!

I coded a Fourier drawing machine in Minecraft. This video is explanation of how these machines work and an introduction to the fundamental mathematical concepts of Fourier series and the Discrete Fourier Transform. This video is aimed towards a math-interested audience who may or may not play Minecraft. Minecraft is simply used as a fun way to visualise the Fourier drawing machine, allowing the player to fly around as the machine can draw the shape using particles or even blocks.

We use the hidden summation formulas within Yang Hui’s triangle to deduce the formula for the sum of higher integer powers of the numbers from 1 to n. We also demonstrate how the successive difference method can accelerate these calculations. This method first appeared in Yuan Dynasty Zhu Shijie’s book “Precious Mirror of the Four Elements” and was proved by modern Chinese mathematician Hua Luogeng. Lastly, we briefly review an application of this summation formula to compute a definite integral.

In this video I try to explain the core ideas behind Monte Carlo Integration. Understanding of undergraduate probability is assumed.

This video reveals the surprising connection between billiard table geometry and fundamental concepts in number theory — including: the greatest common divisor (GCD) the least common multiple (LCM) and prime factorization By tracing the paths of billiard balls, the video offers a rich geometric perspective on these elementary number theory ideas. It is designed for students who already have a basic understanding of these topics and are looking to deepen their intuition through visual and conceptual exploration.

Digging into a subtle point that tripped me up when I first learned how magnetism emerges as a relativistic effect of the electric force.

In all natural human languages, the order of the sounds/signs (phonemes) in an utterance conveys information. But is it possible for a language to convey no information with phoneme order? And if so, what properties does such a language have?

Aimed at programmers and high school math students, this video is about multisets and how we can extend Boolean operations to the natural numbers. On the way, we prove common misconceptions wrong and link seemingly disjoint fields such as probability, data structures, and number theory together. With over 22 drawings and 2200 lines of code, this is by far my most mathematical and time consuming endeavour, so please consider supporting the channel. I literally made this video while working on multiple school assignments.

If you are handed an expression that proclaims, "Work equals integral of F times dx", it is an understandable gut reaction to conclude that F is the force component while the differential is the displacement component. But as we see in the video, this is not always the case! We go through examples of a pulley and of pumping liquids through a straw to see how important it is to interpret an integral properly. In the latter case, we see that the dx actually finds its way inside the force part! This video is for calculus teachers to better understand the physics in the background and to share with their students. It also includes an outro song as a little treat :)

As a different approach to teaching reinforcement learning, in this video we start from a basic supervised learning setup and derive a state-of-the-art reinforcement learning algorithm (PPO) simply by trying to train the model with a non-differentiable loss function. Aimed at undergrads with experience with basic deep learning principles.

The Normal Distribution is perhaps the most important distribution in probability. Everybody learns about it in high-school, but its exact form is usually dropped down from on-high and it isn't explained how anyone came up with it. I thought that the best way to scratch this itch was to find out where it came from historically, which this entry covers by showing how the normal distribution originated with the work of De Moivre (though he didn't yet know that his discovery would be important in probability).

Yeah... Algebra scared me when I first saw it. So many questions just went unanswered and the chapter was finished within seconds while I was left blank. Tighten your seatbelts coz I'm gonna show you a few of the "scary" questions from my childhood algebra class, turn them into fun puzzles and try to answer questions that my little self never got answers to.

Four circles, sixteen regions… right? Not quite. Try to draw them, and you’ll only ever get fourteen. Why does this happen, and what does it really mean to “do a Venn diagram”? In this video we uncover an idea that makes Venn diagrams work — how circles fail, why blobs succeed, and the deeper principle that makes diagrams work at all. At the end of the video, we’ll see how the same idea powers whole fields of mathematics, from topology’s rubber-sheet geometry to probability’s random variables. We demonstrate the art of discovering mathematics: When faced with a problem trace your thoughts conciously, write the details carefully, and the solution will stare at you. To teachers: I have taught non-routine mathematics to students from 8th-12th grade over the last 15 years. In my experience, students find set theory dry and abstract, while doodling blobs (Venn diagrams) feels concrete and fun. But here's the thing—when students draw a blob and call it a subset, they're actually performing a mathematical function that maps abstract sets in their minds onto concrete shapes on paper. This realization can transform set theory from boring to brilliant. The video explores the benefits of this perspective. Hopefully this approach helps your teaching!

Moments of force, or torques, are an essential concept to understand in physics and engineering. This video provides a visual explanation of what moments are, how they work, and demonstrates two approaches to calculating moments created by angled forces.

I created a math puzzle game inspired by New York Times Connections. Using the Desmos Graphing Calculator, I rebuilt Connections, but swapped the connected words for number patterns. I love creating math learning/puzzle games and was inspired by SoME to record and publish my process.

This video introduces the Finite Difference Method, a numerical method for solving partial differential equations, using the Poisson equation to demonstrate. We will briefly review the Poisson equation and then discuss using the FDM to solve Poisson boundary value problems on rectangular, annular, and trapezoidal domains, with Robin or Dirichlet boundary conditions.

Classical mechanics states that if every force involved in a system is conservative, then the system's energy is constant. In this video, a binary gravitational system, such as the Earth-Sun system, is analyzed from two different reference frames. A paradox then arises: in one frame energy is constant, as it must be, while in the other it seems to vary over time.

A visual intuition behind the arc length formula most commonly taught in Calculus 2, along with an example problem. Designed for students who are familiar with the concepts from Calculus 1.

We provide an actual use case for Galois Theory: solving a question of Ramanujan. How do we denest this repeated cube root? We do this the old fashioned way, bashing it out instead of using slick tricks, and along the way, we see the power of the Fundamental Theorem of Galois Theory

This video is meant to show the intricacy and genius that went into creating a line-drawing algorithm made by Jack Bresenham that is still used to this day. The video delves deep into the proof as to how the algorithm functions, trying to create an intuitive understanding behind each choice that the original paper makes. It is aimed at people with high-school knowledge of mathematics.

Strange things start happening when you mix quantum theory with classical probability. Entirely different looking descriptions turn out to be physically indistinguishable, and it is impossible to say whether the randomness of measurement outcomes is due to quantum superposition or classical uncertainty. The two cannot be separated. Are they even different things at all? The standard wavefunction description can be misleading. Density matrices and the algebraic approach, which I discuss here, handle this much better. This is aimed at an audience with some knowledge of quantum mechanics, most likely at an undergraduate or graduate level. However, it should be accessible to anyone with an interest in the subject.

The video is a gathering of results from combinatorics that, albeit very useful, are not usually taught at an entry to intermediate level even in a math course.

Usually when people say "some" of something they also mean "not all" of it, even though it's not explicitly part of the word's meaning. The video attempts to explain the principles behind this linguistics phenomenon and a mathematical theory which encapsulates its behavior.

When I first learned about the binomial theorem, it felt like an algebraic trick, and I dismissed it quite quickly as a speed math hack. But as I thought more about it, I realised that there was a really interesting visual way to understand it, and that made me finally appreciate it. I wanted to share that appreciation in this video. (Sorry about the animations! I'm going to upload another, non-glitchy version later. Unfortunately, I couldn't make it in time for the competition deadline.)

What connects the Parthenon to sunflowers and the Mona Lisa? How is the musical interval that splits the octave responsible for the death of a Pythagorean, and why is it known as the devil in music? These questions are all answered in this video, which asks how beauty and irrationality are linked, and explores continued fractions, a representation that lets us define "how" irrational a number is. What is the most irrational number possible? Can this number be beautiful? In this video we weave a thread through thousands of years of history, from the discovery of irrational numbers to jazz and modern art. Far from being diabolical, these numbers contain an intrinsic beauty that displays the complexity and interconnectedness of the universe we inhabit. Rethinking the real line https://www.youtube.com/watch?v=uFWJuZQLKJs&t=133s @ProofofConceptMath Myth of the tritone https://www.youtube.com/watch?v=3MhwGnq4N9o @AdamNeely ​

This video targets the audience who are highly interested in Geometry and Topology, and whose math level is around high school level. I've work 3 months for this video, completed it using manim coding and video editing, all of those done by myself (I'm a 16-year-old Chinese high school student). This video introduces the Kakeya Problem, Basic dimensional knowledge, and the Katz-Tao Program to solve the Kakeya Conjecture, and the basic approaches Wang Hong and Joshua Zahl published in their paper claiming that they proved the 3-dimensional Kakeya Conjecture. I want to start from a really fundamental and anti-intuitive problem to lead the audience learn deeper about geometry and algebra, and understand how mathematicians worked their way to the success of proving a math problem. I believe this kind of video is educational. Although I learned all the manim and python techniques on my own, I managed to finish this 23-minute-long video. This is my first video, and for the standards of this, I'm quite satisfied, and I sincerely hope that you can recognize my video. Thank you.

Many aspects of our society are shaped by the repeated occurrence of small, random events. This idea that the persistent repetition of random events can profoundly impact individuals was insightfully captured by Sir Francis Galton. In this video, we revisit the mathematical principles underlying Galton's board and leverage them to go further the Central Limit Theorem (CLT) and see what can happen when the hypothesis of the CLT are not satisfied. Given a different perspective of the same object in math can lead to a better understanding of it, and sometimes lead to new results or a broader understanding and this is what we are going to do in this video.

A Discussion on the history and mathematics of the The Kalman Filter. A recursive state estimator used in the Apollo 11 trip to the moon and which has an enormous impact on life in the modern day

This video is aimed at anyone interested in seeking an intuitive resolution to the paradox that it should make sense to consider the expressions '0.999...' and '1' as being representations of exactly the same numerical value.

“Psychotic Millennium” is a cross between a one-man play, a philosophical TED talk, and a classroom math lesson. It is a highly theatrical romp through probably my favorite math story of all time: the story of the Millennium Problems, the tantalizing million dollar prizes for solving them, and the man who (spoilers!) heroically solved one and subsequently disappeared (more or less). This video is an edited recording of this play’s 3 public performances from just a few weeks ago. One of the inspirations for writing this play was actually Grant Sanderson’s TED talk about the importance of “story” to math engagement — as someone who has always loved theatre, I was excited by the possibility of creating something that would tease out the “story” behind some of my favorite math to weave together my passions for both math and theatre. As I stated in the playbill for the show, I have always had friends who hated math, which always made me a bit sad, because I felt like the “math” they hated was different from the math that I had always loved. I felt like if I could just show them a little bit of why I loved math, then maybe they wouldn’t hate it so much. Because of this, I made every effort to make this show as accessible as possible, even to a more “math-skeptical” audience, with the goal that anyone with a basic understanding of the main math operations, and maybe some introductory algebra, could follow everything I was saying. (Although I made sure to include plenty of fun tidbits for the math lovers as well.) I also endeavored to include enough entertaining anecdotes that even those who didn’t quite follow everything could still enjoy the performance. As may become clear from watching the show, I don’t really go too deep into the Millennium Problems themselves (except, perhaps, for one), but I more so use the story behind them to engage the audience and get them to buy into an exploration of some other fun math. Although this is somewhat of an unconventional submission to this contest, I have a strong belief in the power of the story I tell in this show, and I hope that it can inspire just a little bit more confidence and enthusiasm in even the most math-hating among us. Thank you for your consideration!

Welcome to the Academy of Useless Ideas! In this video, we begin our journey to solve "the hardest logic puzzle ever." But first, we’ll make a detour to a classic riddle—the two doors and two guards problem from the movie Labyrinth. We’ll build a tiny logical toolbox to construct a mathematical model of the puzzle. This isn't just about finding a clever solution; it's about learning a powerful method to engineer questions using propositional logic and its five key connectives: NOT, AND, OR, IMPLIES, and IF AND ONLY IF. By the end, you'll see how this formal language helps us design a question that works every time and how this same approach can be used to tackle even more complex problems.

This video addresses motivation and the attitude of kids towards mathematics. It presents 3 levels of maths, similar to levels in a game and illustrates them with an example.

I decided it was about time to start learning to code AIs this summer, and then the competition notice for SoME came out so we did a little sidetracking.. I'm not sure the level of math needed for this is, but curious high-schoolers should probably be able to get it.