Summer of Math Exposition

The Great Fettuccine is a world-renowned mathemagician with an apparently impossible ability. Her assistant selects five cards from a standard 52-card deck, removes one, and rearranges the remaining four. From those four cards alone, Fettuccine can identify the missing card.

The trick raises a fundamental mathematical question: how can we communicate information.

There are 24 possible ways to arrange four distinct cards, which initially seems far too few to identify one card out of 52… However, the assistant is not simply rearranging four fixed cards; he has the freedom to choose which of the five cards to remove. This allows him to choose what information to communicate.

Behind the magic trick is a beautiful idea from combinatorics and information theory. By carefully exploiting the four suits and the 13 possible values, the assistant can encode information into both the choice of the missing card and the order of the remaining cards.

This is an elegant example of a broader mathematical theme: how can we represent and communicate information using limited resources? Problems like this appear throughout mathematics, computer science, coding theory, and information theory, where finding the right way to encode information can turn an apparently impossible task into a surprisingly simple one.

In this video, we uncover the mathematical structure hidden beneath the magic, and see how a little combinatorial ingenuity can make four cards say far more than you might expect.

It’s not magic. It’s maths.

My first Manim-animated project focusing on the geometry underlying King’s Rule, a powerful integration technique exploring the symmetry of integral bounds. While short, the video entirely explores an algebraic and geometric derivation of the rule from fundamentals, making King’s Rule feel like a discovery anyone could have made.

Go is one of the oldest surviving board games in the world (believed to have been created by Yao, an ancient king of China, almost 4000 years ago), possessing more legal positions than there are atoms in the observable universe. Yet, its highly complex endgame led mathematician John Conway to a profound realization: games and numbers are fundamentally the same. This video explores Combinatorial Game Theory by starting from the simple rules of Go and building up to Conway’s Surreal Numbers. We journey from basic territory counting to fractions, infinities, and infinitesimals, exploring concepts like game “temperature” and the “chilling” operator.

Axiom or Theorem is a silly game from mathcamp that allowed us to explore axiom systems. The target audience is anyone who has taken an abstract algebra class before, or is otherwise familiar with structures and first-order logic.

Can some humans really see 100 million colors?

In popular media, human retinal tetrachromacy (i.e. having a fourth cone type) is shrouded in mystery and misinformation. You’ll often hear wildly conflicting claims: some say it doesn’t work at all, others argue it just slightly sharpens standard trichromatic color vision, and many claim these rare individuals can see 100 times more colors than the average person. But what’s the truth?

In this video I debunk these rumors by applying linear algebra to the cone responses of different color visions. To find an objective answer to how functional human tetrachromacy really is, I use Singular Value Decomposition (SVD) as a mathematical framework to evaluate the differing (abstract) color spaces of a diverse set of real and exemplary color visions. I’ll walk you through how to translate biological cone responses into matrices, and use SVD to objectively and approximately calculate:

• The number of distinguishable colors in a visual system.
• The effective dimensionality of a given color space.
• The functionality of the axes that construct a color space.

I run the SVD algorithm on exemplary monochromacies, dichromacies, trichromacies and standard human trichromacy, and then analyze the theoretically “best possible” genetic mutation for human retinal tetrachromacy that we know of so far: SMQL tetrachromacy (where the 4th cone type, labelled “Q”, is shifted approximately 15nm from the M cone origin towards longer wavelengths). You’ll mathematically and visually see why excessive cone overlap fundamentally refutes the “100 million colors” claim.

Finally, we’ll analyze a dichoptic tetrachromacy. This is a fascinating 4-dimensional color vision that can be acquired later in life through specialized spectral filters and sufficient training. And as the SVD calculations will prove, it quantitatively is significantly more functional than anything we know of that humans could get in the genetic lottery.

Whether you’re a math enthusiast curious about the real-world applications of matrix operations, interested in what Singular Value Decomposition (SVD) can be used for, or just fascinated by the limits of human color perception, this video will help how you better understand color vision.

In this video, we use the Yin (yang) Algorithm, one of the most important algorithms in computer music, to improve upon the Music Transcription Algorithm we built in the last video. The algorithm was published in 2002 by Alain de Cheveigne and Hideki Kawahara. Through their algorithm, we learn about auto-correlation, Fourier transforms, basis’s for audio, the difference function and log-normal penalty.

Github link: https://github.com/Waz-ly/SheetMusicGenerator Original paper: https://www.ee.columbia.edu/~dpwe/papers/deChevK02-yin.pdf

This video is about combining implicit functions using boolean logic. I show why min(), max(), and negation are equivalent to union, intersection, and difference (OR, AND, NOT) and how to use those to create a complex shape using a single equation or inequality. I do this in the context a simple example of building a snowman.

The concepts described are one part of the Theory of R-Functions by Rvachev, developed in the 1960s. Further reading on that by Vadim Shapiro: https://scispace.com/pdf/theory-of-r-functions-and-applications-a-primer-2uvkrb2sp0.pdf

It is also the concept underpinning Constructive Solid Geometry, which is used in CAD for engineering: https://en.wikipedia.org/wiki/Constructive_solid_geometry

I was not able to find much about this concept on YouTube so I thought it would help other people interested in creating art with math.

This video describes a pseudospectral optimal control solution to the brachistochrone problem. It details the continuous optimal control problem, discretization, and demonstrates the optimization.

It’s aimed at students and engineers with a calculus background who want a concrete look at how optimal control problems are set up and solved.

How I discovered a Ramanujan-like pi formula!

An arrow and a wave are the same object — a phasor. Rotating one slides the other. This video follows that idea from Euler’s formula to a fruit fly’s brain, where a rotation is performed by four scaled waves landing in shifted positions and summing.

Wythoff’s Nim is a famous game that is (shockingly) under-explained here on YouTube. There are a few videos, but they all stop after the same point: “Here are the losing positions; good luck!”

This video attempts to go one step further and explain a winning strategy you can perform in your head. It goes through three different, seemingly unrelated, mathematical ideas that are each interesting in their own right, but then weaves them together one by one into a method for determining the best move at any given step. By the end you’ll want to go try out this game with one of your own friends!

What if the average wasn’t just an arithmetic recipe, but a geometric object?

This video starts with a familiar calculation, the sample average, and reinterprets it using linear algebra. If we stack data values into a vector $y$ and project that vector onto the span of the all-ones vector, the projection has the form

The stretch factor $\bar y$ is exactly the sample mean.

From there, the video builds toward ordinary least squares regression. Instead of viewing regression only as “drawing the best line through points,” we reinterpret it as a projection problem: the data vector $y$ is projected onto the column space of the design matrix $X$. The fitted values are the projection, the residual vector is the perpendicular leftover, and the normal equations come from that right angle.

The goal is to show that the sample mean, intercept-only regression, residual orthogonality, and least squares are all part of the same geometric story.

This video is intended for both a general audience, and one familiar with statistics at the undergraduate level.

The idea of binning together random outcomes from a corresponding Probability Density Function (PDF) is introduced. A method for generating random numbers distributed according to an arbitrary PDF, through the use of the inverse Cumulative Distribution function, is explained.

The Domesday Book - a record of settlements compiled in England in 1086 - is analyzed to derive a PDF as a model of medieval settlements. Thus, by the method above, it is possible to generate a random set of population figures for medieval settlements. This can be quite fun! This method is used to estimate the number of cities which should appear in Tolkien’s Middle Earth, assuming medieval England as a starting model.

This video details an approach to using probabilistic, non constructive methods used by Erdos to establish dramatically better lower bounds for Ramsey numbers.

This Sunday, June 21st, is a holiday in Chile: National Indigenous Peoples’ Day. Around this time, Indigenous communities in the Southern Hemisphere celebrate the arrival of the winter solstice, which marks the beginning of a new year for them.

But what exactly is the winter solstice? In this video, I explain it.

Most definitions of sin and cos start with a triangle. That definition breaks completely past 90°.

This video builds sin and cos from scratch using $e^{ix}$, derives $cos²+sin²=1$ from a single exponent cancellation, and arrives at the Pythagorean theorem as a consequence of scaling and not the other way around.

This is not the simplest proof of Pythagoras. It is a derivation of the functions that make Pythagoras work.

A short video that gives some intuition on how the chaos game works, it is a computationally efficient method of forming fractal shapes. Suitable for highschoolers, to undergraduates. The mechanism behind the chaos game is rather simple and visual but the result is fascinating. The video is short as it is meant to pique curiousity rather than be overly detailed.

I explore various rabbit holes with the Fibonacci sequence and it’s friends: tying some historical facts, recursive sequence proofs, generating functions, and other curious ideas together in a string!

This video is a two and a half hour long playdate between you and me---we will eat cookies together and go for a walk. This video is also a multidisciplinary and “autotheoretic” approach towards responding to the following question: why should I care about mathematics, or for that matter, higher education in general?

In this video I respond to this question with a question of my own: but WHO is math to you? To ask this question is to want to shift your perspective of math from something that can benefit you in some way or other, to someone who could be your friend. I demonstrate this perspective shift in three chapters:

Chapter 1, ~33 minutes long (starting at 3:26), is an exposition of the famous proof that √2 is irrational, but it is entirely disguised as a puzzle about tiles. At the end of the chapter, I also talk about why this proof reminds me of the grief and joy of letting go of a very important relationship to me.

Chapter 2, ~22 minutes long (starting at 36:16), is a memoir about difficult decisions I made concerning school, a relationship, future plans, and a job. This is also where we get to go on a walk together.

Chapter 3 part 1, ~44 minutes long (starting at 58:15), is an analysis of how G.H. Hardy, Francis Su, and Simone Weil engage with the question of why we care about mathematics, and in what ways asking “who is math?” builds upon their considerations on this topic.

Chapter 3 part 2, ~38 minutes long (starting at 1:42:25), is a collection of tentative proposals for what we can do in family upbringing, education, and academia contexts to support a vision of mathematics for human flourishing and mathematics as a friend.

Because the full video is far too long, I ask that you choose one of these four sections to judge.

Thank you for your time, and if you wish to reach out to me for any reason, please do not hesitate!

Let’s say a whale magically dropped into a city out of nowhere. What logistical challenges would we have to overcome to move it out of there? This video explains how we can solve that problem using the Rapidly-exploring Random Tree algorithm, which can be used to calculate a path in the case where an object has a non-infinitesimal shape - which means, its rotation is very important, and determines which paths are viable, and which ones aren’t.

Euler’s identity

is often called the most beautiful equation in mathematics because of the way it links five fundamental and seemingly unrelated constants.

However, for the same reason that it is beautiful, the identity poses a barrier to those who first try to approach the problem. How do you raise $e$ to an imaginary power? And what does $e^{ix}$ have to do with circles?

To answer these questions and more, the video builds Euler’s formula from the complex plane and the differential equation $z'=iz$, using maps, animation, and LEGO to make the motion intuitive. Then we put the formula to work through Fourier transformation by using the 2026 movie, Project Hail Mary.

The ultimate goal is not just to prove Euler’s identity, but to make each part of it feel inevitable and intuitive, and to hopefully help you appreciate the beauty of it, too.

Basic trigonometry and introductory calculus are assumed.

While I have created art, comics, and videos that explain mathematical concepts before, this video is more of a “math essay video”, reflecting on how the culture and education system I come from teaches mathematics and, perhaps more importantly, how that can shape the way people come to view math itself.

I hope this reflection can offer a glimpse into how mathematics is taught in my home country, while also prompting a broader question: how much can our culture and education shape the way we understand and feel about mathematics? And if they have that much influence, perhaps we should be more careful about what kind of relationship with math our systems are encouraging people to develop.

My take on the Kalman Filter, introducing it via a simple example that needs only 1D equations, avoiding matrix / vector algebra, and focusing on variance minimization.

See how “number palindromes” relate to the convergence of an infinite series, Pascal’s Triangle, and the Triangular Numbers!

In this video, I discuss the answers to questions like:

• Does changing the base of a set of numbers (like the Triangular Numbers) reveal more palindromes?

• Is there a predictable pattern for finding (some) palindromes?

• Does the sum of the reciprocals of every palindromic number in every base converge?

• Is there a way to make palindromic numbers without needing … uh … numbers?

A number palindrome is a number that is read the same forwards and backwards. The palindromic numbers in this video are in integer bases $B \geq 2$. This video is the product of around 6 months of research and 40 hours of animation and programming, so I really hope you enjoy!

This video was entirely animated and edited in Blender.

Footnotes and other extra information:

• (Footnote 5) 11:14 Taking the limit of a function $f(x)$ as $x \to n$ means finding the number that $f(x)$ approaches as $x$ gets infinitely close to $n$. When $n=\infty$, this means finding the value that $f(x)$ approaches as $x$ gets infinitely big. For a more in-depth explanation on limits and other Calculus topics throughout this video, see @3blue1brown’s Essence of Calculus series.

• The reciprocal of an integer n is equal to $\frac{1}{n}$

• For time reasons, I never got to explain in the video that there are an infinite amount of palindromes in every base. This can be seen in the set $\{1_B, 11_B, 111_B, 1111_B, \dotsc\}$ that contains every number that is only made of 1’s in a given base, of which there are infinitely many.

Organizers of the Heidelberg Integration Bee 2026 invited me to submit an integral for their competition. In this video I explain the solution and some of the history behind the Feynman trick.

In this video I introduce pure math and proofs to an audience interested in math as a hobby. It’s simultaneously designed for motivated middle schoolers as well as adults relearning math.

I introduce commutativity and associativity of addition, partitions of 5, a solved exercise from Lang’s Basic Mathematics, and Gauss’s proof of the sum of n integers. My presentation includes a split-screen iPad to simulate a live classroom experience (e.g., no video editing), and my hungry feline assistant is featured throughout.

(Note: This is a standalone video that later expanded into a series; no other videos are required for context)

I used a Python script to simulate people’s behavior in a museum, specifically a single-file queueing model where people move in a line but can pass if the person in front of them stops, and to study what a good design is that allows audiences to spend less time waiting, such as changing the exhibits’ placement order or deciding whether to add a U-turn.

I studied the following questions:

• Does waiting time depend on how many waiting spots there are?
• Will the randomness of people’s patience increase waiting time?
• Will placing the most attractive exhibit at the end increase waiting time or not?
• How does peeking at exhibits reduce waiting time?

This video acts as a repository of my knowledge on quaternions for teaching it to other people.

A light-hearted retelling of Pythagoras life, with an introduction of non euclidean geometry and cosine rule.

Any Math or Physics enthusiast can watch and understand the video, including Undergraduates like myself. It focuses on solving the differential equations to produce different sorts of shapes(circular, elliptical, etc.) that an orbit can take in a two body system. It showcases Gravity simulations between planets, and discusses the relation between energy and shapes of the orbits taken by the heavenly bodies.

simple as they are, Tangrams proved to be a challenge to make using mathematical equations, as proved in my Desmos link, https://www.desmos.com/calculator/wgmx38ccqt , and in the video I explain the process of doing so.

Professor Walter Lewin’s famous experiment seems to show Kirchhoff’s Voltage Law failing in a changing magnetic field. Years later, ElectroBOOM challenged that conclusion, arguing that the measurements are correct—but the interpretation is not. In this video, we examine the experiment and what Faraday’s Law really tells us. We’ll start from first principles—beginning with vector calculus, moving through Maxwell’s equations, and finally arriving at Kirchhoff’s Voltage Law.

Overview

This video presents an elementary geometric construction and proof of the determinant formula, developed to fill a gap often left open in introductory and advanced linear algebra literature.

The project originated from a two-year study journey that began in high school after watching 3Blue1Brown’s Essence of Linear Algebra. While the series offers great visual intuition, it relies on the determinant without proving its algebraic formula geometrically. Standard textbooks like Boldrini’s or Axler’s Linear Algebra Done Right either omit the geometric origin or treat the formula as an axiomatic given (e.g., Axler proves $\det(T)$ equals the product of eigenvalues using polar decomposition $T = S \sqrt{T^* T}$, assuming the determinant’s existence and properties beforehand).

Instead of attempting to construct the determinant from eigenvalues, this proof builds the formula ground-up using elementary geometry accessible to high school and early undergraduate students.

Core Mathematical Insights

1. Oriented Height ($\text{Height}_{\text{oriented}}$):
   Standard $n$-dimensional volume of a parallelepiped is defined as: $\text{Volume}_n = \text{Volume}_{n-1}(\text{Base}) \times \text{Height}$ This proof constructs the determinant inductively by replacing standard height with oriented height relative to a chosen hyperplane: $\det = \text{Volume}_{n-1}(\text{Base}) \times \text{Height}_{\text{oriented}}$

2. Additivity and Opposite Half-Spaces:
   Popular geometric explanations often fail to prove why the additive property (linearity) holds geometrically. This proof demonstrates that for signed volume to be additive across hyperplanes, the measure must yield opposite signs in opposite half-spaces (or half-planoids in $\mathbb{R}^n$).

Planned 4-Part Series Roadmap

This video is Part 1 of a planned four-part series on $n$-dimensional geometry in $\mathbb{R}^n$:

AI Disclosure Statement

WORKFLOW DISCLOSURE:

• Concept, Proof & Logic: 100% original human work created by the author.
• Storyboard & Visual Design: Designed frame-by-frame in Excalidraw by the author.
• Code Implementation: Claude was used to translate the Excalidraw storyboards and timing specifications into Manim (Python) code.
• Final Rendering: Executed via Manim. I am submitting this full disclosure to the SoME organizers to determine formal competition eligibility regarding AI-assisted coding versus AI-generated visual media.

The video describes one way to try and solve the 3-by-3 magic square of squares problem, through extended parametrization using polynomials. Though no solution is found (yet), the purpose is to initiate a community effort, in terms of computing power and hints/ideas.

In my video I discuss the asymptotic equipartition property, a genius and intuitive idea behind data compression. It represents arguably the most important idea to understand data compression.

In this video, I explain ElGamal encryption from the mathematical side rather than treating it as just another cryptographic algorithm.

The video starts with the idea that some mathematical operations are easy to perform but extremely difficult to reverse. From there, I build toward the discrete logarithm problem and show how that asymmetry can be used to create a public-key encryption system.

To make the idea more intuitive, the explanation is framed around a battlefield communication problem involving a general, a subordinate, and an interceptor who can observe what is being transmitted.

The main question is not whether the interceptor can see the communication, but whether seeing it is enough to recover the original message.

We then go through a complete ElGamal example step by step, including how the keys are formed, how a message is encrypted, what information is publicly visible, and how the intended receiver is still able to recover the original message.

The final section explains why the decryption process actually works mathematically, rather than simply accepting the algorithm as a formula.

This video is part of my attempt to explain mathematics visually and rigorously, with the goal of making the underlying ideas understandable instead of reducing them to procedures to memorize.

The main mathematical reference used while developing this video was David M. Burton’s Elementary Number Theory.

Production note

This video is human-authored and AI-assisted.

The mathematics, explanation, script, examples, lesson structure, visual ideas, animation design, and overall direction were created by me.

AI tools were used during production to help implement the animation designs in code and to assist with certain supporting assets such as the thumbnail.

The AI tools did not determine the mathematical explanation or structure of the lesson. They were used primarily as production and implementation tools, while the educational content and creative direction remained mine.

Ancient Greek mathematicians liked to construct geometric shapes using a straightedge and compass: lines and circles. But it turns out the straightedge is actually redundant, in that all the points that you can construct with a straightedge and compass, you can construct with just the compass. Just the circles, and not the lines.

This video explores what pitch is mathematically and physically from the ground up. I’d tried to make this video as accessible as possible without sacrificing any of the conceptual ideas. The video is relatively short (3 minutes) but I’ve included some pretty cool animations of the Fourier Series and other ideas related to pitch.

The video starts with period and frequency, and I went into how the Pythagoreans discovered that consonant intervals correspond with simple integer ratios. I then derived equal temperament from the requirement that twelve equal steps multiply to an octave. I then ask why particular ratios sound good in the first place. I then explain the answer using Fourier Analysis. This video is accessible to most higher level high-school students and should be fine for most undergrads. I hope everyone enjoys it! I made all of the animations with Manim.

This video aims to answer the title question of why the area under a curve is the “opposite” of the slop of the tangent line. In calculus, we learnt that the derivative can be thought of as the slope, and the integral can be thought of as area, and we consider these operations as inverse operations. However, there is no understanding of why the slope should be the opposite of area. This video aims to make that bridge by using the definition of the derivative.

This video aims to intuitively explain about Brachistochrone or cycloid curve in a fun way. While mathematical details are included, they can be skipped without losing the main ideas.

In 1900, the mathematician David Hilbert presented a list of 23 problems to the International Congress of Mathematicians. The first question on the list was the continuum hypothesis : a question about the size of infinite sets proposed by Georg Cantor in 1878. In 1963, Paul Cohen introduced a technique called “forcing” to construct a new mathematical universe where the continuum hypothesis is false. In 1940, Kurt Godel already constructed a mathematical universe where it’s true. Godel and Cohen’s results imply that the continuum hypothesis will never be solved.

In this video, we explore Cohen’s forcing technique using boolean valued models to construct new mathematical universes.

In this video I dive deeper into the hole of measure theory that I had started touching upon last time. I built up a hyperfinite counting measure without ultrafilters from scratch.

This short video answers a simple question: What is the most irrational number? We vaguely define what “most irrational” means and use continued fractions to discover the answer.

This video is about how to use the discrete Fourier transform to construct polygons. The topic is also known as Symmetrical Components in electrical engineering and also as the PDN Theorem.

Using Manim to explain the essence of programming thinking. Visualizing concepts such as for loop, function, data structures for the students.

This video examines why digital platforms and technologies are often designed around a supposedly “average” user, even though people differ substantially in their skills, resources, constraints, and ability to adapt. It develops the idea of a design mismatch: a system may work efficiently for one group while imposing hidden learning costs, access barriers, or exclusion on others. The video explains how firms choose design standards, why market competition does not always eliminate these inequalities, and how network effects, switching costs, and unequal demand can reinforce a dominant but insufficiently inclusive design. It also considers the role of incentives, accessibility standards, regulation, and public investment in encouraging technologies that serve a broader range of users. Combining economic reasoning, formal models, and real-world visual examples, the video shows that inclusive design is not merely a technical challenge—it is also an economic and policy choice about whose needs are prioritized and who is able to participate in a changing technological environment.

What does an expression like (2+3-4+5) actually mean?

At first glance, this seems like a very simple question. But the expression can be represented by several different binary trees, corresponding to different ways of grouping and reading the operations. Our conventional notation hides much of this structure, and the usual rules of BIDMAS tell us how to evaluate an expression without necessarily telling us the most useful way to see it.

This video explores an alternative perspective on algebraic notation.

Starting from binary operations and syntax trees, I look at what happens when subtraction is rewritten as the addition of a negative, and division as multiplication by a reciprocal. Once these inverse operations are normalised, familiar algebraic properties such as associativity and commutativity become much easier to exploit.

Instead of reading

[ 2+3-4+5 ]

as a sequence of binary instructions, we can read it structurally as a sum whose terms are

[ +2,\quad +3,\quad -4,\quad +5. ]

Likewise, products can be understood as collections of factors rather than simply chains of multiplications and divisions.

This leads to a broader question: how much of the difficulty of algebra comes from the mathematics itself, and how much comes from learning to read the notation?

I am a secondary mathematics teacher in the UK, and this question has led me to develop an approach I call SPI Maths — Sums, Products and Indices. Towards the end of the video I briefly show some experimental tools I have been building to make mathematical structure more visible: a syntax-tree builder and an algebraic transformation tool based on command verbs.

These are not presented as a finished solution, but as examples of what mathematics education might look like if structural reading were given a more explicit role. https://spimaths.com/demo3

The video finishes with several pedagogical hypotheses arising from this perspective and an invitation to think differently about something we usually take for granted: https://spimaths.com

How should we teach people to read mathematics?

Wilf’s Method Global Bisection for finding roots of polynomials in the complex plane which is guaranteed to find all roots regardless of complexity and multiplicity. It uses a series of Sturm sequences as on a square in the complex plane to identify how many roots exist inside of the square. It then gets subdivided into four squares which are also tested and the search continues until all of the squares containing roots are sufficiently small.

The International Mathematics Olympiad has the reputation of being among the hardest mathematics competitions in the world (as well as the most prestigious).

But the first IMO problem each year is usually more accessible. The reason is not only to give all contestants an accessible entry point, but also to entice casual mathematics enthusiasts from the general public (that’s us!), allowing us to experience the joy of recreational maths and put our problem solving ability to the test!