Summer of Math Exposition

This is just a simple video I made to introduce people to higher level mathematics, since it's quite different from the math you know in high school.

In this video, we'll explore how to solve third-degree polynomial equations using Cardano's method. We'll start with depressed cubic equations and learn how to transform any cubic equation into its depressed form. Then, we'll use the cubic formula to find a solution and finally, translate that solution back to the original equation, allowing us to solve any cubic equation. ------------------- There are plenty of videos and educational content visualizing the quadratic formula, making it easier to understand. However, the cubic formula, despite existing for centuries, often gets neglected. To truly grasp it, I had to visualize the problem myself, spending a lot of time sketching and computing the steps. This process inspired me to create an animation that makes understanding Cardano's method much more accessible, allowing viewers to see how we arrive at the solution! 😄

The sum of the first n natural numbers has a nice formula that is a rational expression in n. The sum of the first n reciprocals, another simple pattern, surprisingly does not have such a formula. The remainder of the video justifies that no such formula exists by using asymptotic analysis.

In this video, I introduce the concept of a line integral in multivariable calculus by building it up intuitively from integration in single-variable calculus. Math is all about expanding previous knowledge into new domains, after all! I also try to present the intuition behind line integrals, and finally demonstrate a practical calculation of a line integral for an ordinary function: z = x^2 + y^2, over line segment y = x extending from (1,1) to (3,3).

This video can be seen as a follow up of the famous "The hardest problem on the hardest test" by 3b1b. We compared solutions to three different probelms, one from a macro perspective and the other one from a micro. The trick that is used is also know as the Principle of Deferred Decision.

In this video, we explain what periodic trajectories are and discuss their occurence on billiard tables. We illustrate this by using geometric and tesselation arguments and proofs. The video is intended for freshmen, and high schoolers in their junior or senior year. This video was made by a group of undergraduates under supervision of a doctor of mathematics.

Some hats of different colors are placed in a box and a few people pull a hat at random at the same time. The video highlights how logic could be more useful than probability when trying to figure out the color of your own hat using proof by contradiction.

What is sound? How can we represent it mathematically & in a computer? And how can we manipulate sound by changing its frequency content using Fourier analysis? This videos gives an introduction into the linear algebra behind the Fourier theorem and motivates the underlying math by building a simple low-pass filter in Python (SageMath) to cut off high frequencies of a square wave signal. We will discuss it in a continuous setting first (including the orthogonal decomposition theorem), then motivate the discrete setting (DFT) by analogy. This video is explicitly *not* about the Fast Fourier Transform (FFT).

This video describes the 30 year journey I went on after asking myself what I thought was a fairly simple question. It wasn't. The journey started in a high school Geometry class and ultimately ended in Galois Theory at the conclusion of my Master's classes.

We will be applying techniques of mathematical discovery to a video game, Bloons Tower Defense 6. Our goal is to find metrics to evaluate how good a banana farm is. We’ll find that the banana efficiency is a good metric for most banana farms, but if you want to reinvest some income towards buying more farms, we’ll need to use the banana compound efficiency instead. On the way, we’ll discover the Fundamental Theorem of Bananas and the Monkey Bank Theorem. Audiences with backgrounds from middle school to college and beyond may find value in this video.

This is a video about classical gravity, electricity and complex numbers. A very basic understanding of those subjects is assumed.

How to convert Miles to Kilometers using the Fibonacci numbers, and *why* it works because of the golden ratio. A picture proof using "cobweb" diagrams in Desmos is given.

In this video I show how a projection function can be used to map all points on the xy plane to the unit circle, thus allowing us to see the entire graph of any parametric function

What are the natural numbers? The integers? The rationals? The reals? While we may have an intuitive understanding of these numbers and sets, it is not so easy to actually construct these sets formally. To do so, we must use some axioms of set theory, and using only these assumptions, formally describe what these infinite sets should look like. We will develop various tools in set theory, like ordered pairs, relations, ordering, and equivalence classes, to begin with only zero, and from nothing, build all of the real numbers.

In this video I present the idea that vectors can represent hyperplanes, and how this relates to the concept of motion.

This video gives an intro to FMCW (frequency modulated continuous wave) radar and gives a little insight into why you might want to use it over a traditional pulsed radar.

Basic math with big ramifications for your financial future. The Rule of 70 (or 72) makes it clear why you should invest early and often. Come for the finance, stay for the logarithms and Taylor series. Get in while the doublings are plentiful!

In this video I show how to use only probability to find the volume of hyperspheres with an even number of dimensions. I show how the reasoning can be extended to get an Ansatz for the volume of Lp-balls in any dimensions. The ansatz is finally proven by induction by showing a connection with the beta function.

My approach to understand and attempt to solve Problem 3 from the 2024 International Mathematical Olympiad.

If you're watching this, hello! This is my first time ever doing something quite like this, but I've been a big fan of many math education youtubers for a while, such as vihart, numberphile, and 3blue1brown, that I decided I had to give it a shot. I don't have a formal education in math and I'm fairly new to the whole video editing/creating biz, so as a result it's definitely a tad bit rough around the edges (particularly in some of the small animations because my laptop is NOT built for this), but it was still really fun to make! Hope you enjoy! EXPLANATION FOR WHAT THE VIDEO IS: I go over the 12 different non reflected pentominoes and different features and facts about them and how they connect together, and then a fun puzzle to attempt for the last third of the video.

Parallel Matrix Multiplication on a GPU using CUDA C.

A video about matrix multiplication explained with ice cream ratios.

In this video, we explore the relational model developed by E. F. Codd and its history. This video was made in collaboration with my friend Torge: https://www.youtube.com/@TorgeBlunck My website: https://www.emilien.sh/ The majority of this video was made using the awesome Manim Community library: https://www.manim.community/ Relational Completeness of Data Base Sublanguages by E. F. Codd: https://www.inf.unibz.it/~franconi/teaching/2006/kbdb/Codd72a.pdf RelaX (relational algebra calculator): https://dbis-uibk.github.io/relax Relational playground: https://relationalplayground.com Music by Vincent Rubinetti Download the music on Bandcamp: https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown Stream the music on Spotify: https://open.spotify.com/playlist/3zNK20qC96mVSww60lVi1k Color palette (Catppuccin): https://catppuccin.com

A video explaining how to render 3D graphics using a 2D game engine. Explains concepts such as perspective, isometric projection, rotation matrices, and shaders. The viewer needs a basic understanding of linear algebra and geometry to follow through the video.

Despite the title, this is the second video of a series, but it should be possible to follow without having seen the first video. In the last video, we saw intuitively what sets are, but didn’t specify the defining rules for sets. In this video, we establish the language we’ll use to precisely define these rules: the language of logic, which is also the language used to prove mathematical statements.

Dijkstra's algorithm, a fundamental algorithm of finding the shortest path from one vertex to another in a graph. We explain the algorithm with extremely clear animations, then we discuss its complexity, and finally we formally prove its correctness.

Our voting systems often come under fire for their less-than-representative results. Come with me on a whirlwind journey as we explore alternative voting systems and the issues they entail, with particular emphasis on proving the well-known Arrow's Impossibility Theorem.

Iteration means repeated application of a function. Iterating a function known as the "logistic map" gives a simple model to simulate population dynamics. In this video we introduce some key concepts and lay the groundwork for part 2, in which we take a deeper dive into one aspect of chaotic dynamics.

Hedging is an important aspect of managing a portfolio in financial markets. We give an example of a simple contract on the result of a die and showcase how contracts with a zero or even negative expected value can help reduce the overall risk / variance of a portfolio.

Probability often has the feeling of being a completely separate field of math from all others, and something that requires a unique set of skills. But why, and what can we do to make it easier? This video is a broad introduction to basic probability theory ideas with an emphasis on problem-solving strategies.

I will show you an elegant trick for computing the area of a circle of radius R using integrals. In one dimension, the integral is too tricky. The two dimensional integral of a constant function 1 over the circle is simplified by switching to polar coordinates. But if we are not careful, the integral value changes to 2𝛑R. To fix the mistake, we have to add a local weight factor w(r)=r. This video derives and geometrically explains the integral substitution formula. It shows why it contains the derivative of the substitution function.

The Double Pendulum Fractal & It's Chaotic Beauty Dive into the mesmerizing world of chaotic systems with "The Double Pendulum Fractal." This video explores the double pendulum, a fascinating example of chaos theory where small differences in initial conditions lead to dramatically different outcomes. Motions in chaotic behavor is based on nonlinearity of the mechnical systems. However, chaos is not a random motion. As you have seen, the motion can be described with a specific nested structure, which is called fractal. While the chaotic behavior of the double pendulum is well-known, its fractal nature based on initial conditions remains a relatively uncharted territory. Using advanced numerical simulations, we reveal the intricate fractal patterns that emerge over long timescales in the double pendulum's time evolution. You'll discover how energetics shape the gross structure of these fractals, exhibiting quasi-self-similar properties reminiscent of classic fractals like the Mandelbrot and Julia sets. Let's unravel the dynamic pendulum's secrets and the beautiful, chaotic fractals hidden within. Whether you're intrigued by the butterfly effect, dynamic pendulums, or the satisfying gradients of periodic motion, this video is a simple explenation through the double pendulum chaos and the stunning fractal landscapes it creates.

You might know what a prime number is, but have you ever heard the term "long prime"? If not, that's understandable. Long primes aren't that talked about because frankly, they don't seem to be all that useful. But when has that ever stopped mathematicians? Long primes actually have lots of interesting patterns and can lead us to pretty cool claims due to their restrictions. This video is an example of patterns we can decipher from such numbers and is mainly following\ my journey into discovering them and analysing them.

I go over how I made a fluid simulation from the ground up, and explain everything along the way with the help of some animations. (also my first video, please leave some feedback :) )

Taylor's polynomial expansion is a core part of high-school level calculus. However, I was never satisfied with the way it was taught to me, as the motivation for it seemed to come out of nowhere. In this video, I show how Taylor's polynomial, an explicit formula for the error of the polynomial approximation, and a generalized version of Taylor's polynomial with multiple centres, are all the result of just applying the fundamental theorem of calculus over and over again.

This video explores methods to find eigenvalues and quick facts and shortcuts that make calculating eigenvalues easier. It is intended for anyone who knows the basics of linear algebra and would like to learn more facts about eigentheory. The purpose of the video is to be a one-stop-shop for anyone studying eigenvalues who is wanting to gain intuition for these methods or practice finding eigenvalues. Additionally, it offers a good review on eigenvalues for an upcoming test or just to refresh one's mind.

Risk neutral probabilities is the foundation of most mathematical finance: How much would you pay to enter a game with some payout, despite being risk-averse or risk-seeking. Most videos explain it in a much more challenging context, so my goal of this video is to introduce people to the idea, so when they dig deeper into the topic, it becomes much clearer.

This aims to clear some confusions (that the authors formerly had) about the hyperbolic plane. We explain how to do geometry on curved surfaces, then construct the hyperbolic plane, then discuss other things sorounding its models.

How in the world is THIS POINT in the middle of nowhere part of the Mandelbrot Set? How is the point RIGHT NEXT TO IT not? How is the next point in the same direction back in the set again? I dig into how the Mandelbrot set works, visually, to eventually show why points in the middle of nowhere end up in the Mandelbrot set, and more so, how a tiny movement at that point pulls it out of the set, and doubly more so, how the same movement puts it back in the set.

Our brains are for thinking not remembering, that's why I refuse to remember any elementary or middle school geometry formulas. Instead I try to reason their areas using visual methods. In this video I'm showing such methods.

We prove two combinatorial identities using "trijections"--equivalence classes with size three--rather than standard "bijective" counting arguments.

A video related to this sequence: https://oeis.org/A006075 Making better visuals using manim for these proofs: http://www.contestcen.com/knproof.htm

Most of us know that e is used as a common base for expressing exponential relationships, but why? In particular, why do we need to convert from any other base to base e? What does this conversion allows us to do? And how to understand the mechanics of such conversion? We will explore these questions in this video. Through this journey you will gain a deeper understanding about the special properties of math languages, a topic which we rarely think about. After this video you will see how the difficult concepts of e, log are naturally connected.

A proof that the halting problem is undecidable for Turing machines. However, real computers are not Turing machines, and neither are theoretical Artificial Neural Networks. A thesis related to this is made at the end of the video.

Integration by Parts is one of calculus's most famous integration theorems. However, although the way it is usually proved as a result of the product rule of differentiation is effective, it lacks any visual intuition. This video aims to correct that by providing an interesting way of viewing integration by parts visually. The video is based on a proof without words, and thus does not contain audio to keep the "without words" nature of the proof. This video was based on a "Proof without Words" titled "Integration by Parts" by Richard Courant. This proof comes from Volume 1 of the book Proofs Without Words: Exercises in Visual Thinking by Roger B. Nelson. Works Cited: Nelsen, R. B. "Trigonometry, calculus, & analytic geometry." Proofs without words: Exercises in visual thinking, American Mathematical Soc., 2020, p. 42.

This is a video about the last Thurston geometry we have not previously explained in our videos, "the universal cover of the 2x2 special linear group over reals". Why such a name? An exciting travel through spaces of motion, product, and twisted product geometries! By Zeno and Tehora Rogue This is a continuation of our very irregular series on explained visualizations of non-Euclidean geometry. Watching the earlier episodes might help understand this one better.

In this video, I discuss the etymology of some of the most common terms and symbols in Mathematics.

NOTE BASED ON THE COMMENTS: This video is part of a series of videos. I uploaded all the videos in the series before the deadline. The follow up video is the one in the card at the end of this one. The reason for breaking up the full exposition in several parts is that I wanted to give time for people to think about this problem before revealing an answer. If you keep going down the rabbit hole of videos, you might get to some pretty outstanding mathematical technique! In this video, we explore a classic puzzle with a twist: Alice picks a secret number between 0 and 15, and Bob’s mission is to figure it out by asking yes or no questions. Sounds simple, right? But there's a catch—Alice is allowed to lie once! This video isn’t just about finding the solution—it’s about challenging yourself to think differently. Join us as we delve into this intriguing game of logic, deception, and strategy. Can you come up with a better approach than Bob? Share your thoughts in the comments, and stick around for the next video, where we push the boundaries even further. Keep your mind sharp and stay mathy!

The video teaches about good coding practices when building a fun project. It emphasizes how understanding math concepts can result in simpler, easier to understand code that also usually correlates with fewer bugs.

This video includes: - Matrix notation, - Basis vector notation e_i, - Summation notation - Defining structures In this video I juxtapose several related applications of matrices, in hopes of inspiring the viewer to form connections between them.