Summer of Math Exposition Archive

This video is about how Turing patterns work: how repeated blurring and sharpening produce a Turing Pattern by making a band-pass. I show an alternative method of making a band-pass using two blurs. I look at the reaction-diffusion / Gray-Scott model and where its hidden band-pass is. I cover multi-scale patterns, where several band-passes are made using a cascade of bigger and bigger blurs, and how to efficiently run a large blur using FFT. Bonus: a Turing pattern wavetable, and Turing patterns in 3D.

This video describes a way to hear light, as in, shine a light into your ear and hear music. When I first started this video I actually didn’t realize that this was possible, but as I experimented with various items I discovered it was. There are two key ideas I try to explain in this video: radio transmission and photoacoustic transduction. These two ideas are what lead to the phenomenon I discuss in the video.

Although the video is geared towards a general audience, and there’s little math in terms of equations and such, it is still somewhat technical, so I would say the target audience is undergraduate/graduate students, along with high-school students and teachers/professors. But my hope, of course, is that anyone with a genuine curiosity about science will appreciate the video and understand most of what’s discussed.

In this video, we investigate mathematical knots, and our guiding question is: How can you tell when two knots are the same or different? In pursuit of an answer, we argue that two knots are the same only when one can be transformed to look exactly like the other using a sequence of three basic moves, called Reidemeister moves. Then, we prove that a quantity called the linking number is an oriented link invariant using the fact that it doesn’t change under any of the Reidemeister moves. Finally, we stumble upon the most famous knot invariant – the Jones polynomial – and show how one could’ve devised it from scratch, along with several applications.

Most of the exposition and practice problems come from The Knot Book by Colin C. Adams. However, we have a more elaborate discussion about why the three Reidemeister moves are all you need, and we made modifications to the definition of the X polynomial where we felt was appropriate.

References: The Knot Book by Colin C. Adams https://katlas.org/wiki/K11n34 https://katlas.org/wiki/K11n42

The video unpack the famous “Two-Child Paradox” and how it managed to stump some of the most respected minds in mathematics.

Here’s the problem statement: I have two children. One of them is a boy born on a Tuesday. What is the probability that both children are boys?

This deceptively simple question hides one of the most controversial puzzles in all of probability. The answer seems clear—until you start asking how the information was obtained.

This video is intended to be a “Chapter 0” for the theory of probability, which motivates the definition of a “probability space”, the most fundamental model used in this theory. It is intended for high-school or undergraduate students with a solid grasp on algebra, but no other knowledge of discrete or undergrad-level math. It would best help a student who is just starting their first course in probability or statistics.

In 1900, Henri Poincaré initially conjectured that any 3-manifold that has the homology groups of a sphere must be the 3-sphere, but later in 1904 discovered a counterexample, named the Poincaré homology 3-sphere. In the process, Poincaré also invented homology, the fundamental group and conjectured problems that have continued to interest mathematicians up to the current day. In this video, we study the Poincaré homology sphere by applying foundational results of algebraic topology such as the Seifert-van Kampen theorem and Poincaré duality.

Note: This video is converted from the slides of a talk given to fellow graduate students in a student seminar. I assumed knowledge of definitions and key properties of the fundamental group, homology and cohomology groups that are encountered in a first course on algebraic topology. Even if you have not taken such a course, I hope that the video may communicate some of the underlying concepts and ideas behind algebraic topology :)

Synthetic Aperture Radar (SAR) satellites send out microwaves to earth and use the echo to form an image. They can see through clouds, fog, and smoke. But SAR images are often difficult to interpret, since SAR works fundamentally different from optical imaging.

In this video we’ll explore at how the Golden Gate Bridge seemingly multiplies in a SAR image. It produces three distinct reflections, which look confusing at first glance. But they can actually be used to measure the water level beneath the bridge. We’ll start with the basics of SAR imaging before diving into the physics behind these three reflections.

The goal of my channel is to excite more people about SAR. In my opinion, it’s one of the coolest technologies of our time, yet not many people know about it. I want to explain the signal processing and math behind SAR imagery in a simple way, so that we can “see what radar sees”.

I’ve put together the specific explanations that have helped me understand the foundations of special relativity in the past, and attempted to do it in a way that would connect seamlessly to the deeper concepts of differential geometry to clear the path to general relativity, as best as I could!

We start with the relativity of simultaneity, get to this important effect through the light rays-thought-experiment, and start drawing spacetime diagrams to have them lead the way from the start.

Then we connect this to length and time contractions, and find our way to the Minkowski metric, since the contracted lengths on the diagrams don’t look right at first, and demands a change in the way we look at their geometry!

A video about enumerating all of the solutions to a puzzle game, using the language of group theory.

The cellular automaton rule known as Rule 30 generates a universe whose overall behavior is complex. But it has highly structured regions hidden among the chaos. In this video, we’ll discover why these regions behave like they do. They’ll tell us quite a bit about Rule 30 as a fundamental law of physics, specifically about whether we can run it backward in time.

What does it mean to “factor” the sine function? We explore Euler’s brilliant infinite product for sine and show how he used it to solve the Basel Problem. Along the way, we see an infinite product expansion for pi/2 (the Wallis Product) as well as some genuinely beautiful math involving complex analysis, infinite products (and infinite series), Weierstrass/Hadamard factorization, and more.

Why do big exams like the SAT, GRE, ACT, LSAT, or GMAT often feel UNFAIR? Trick questions, ambiguity, and senseless difficulty make these tests frustrating. But to an extent, these tests HAVE to be this way.

This video looks at 4 example SAT or ACT questions that might be considered unfair. Through these questions, we’ll discover a simple formula that’s behind every exam question you’ve ever seen. The formula explains why its necessary to write hard questions that feel tricky or unfair to students. These questions often must have tricky but convincing answer choices (called distractors) or be brutally difficult.

Specifically, the video presents the 3-item characteristic curve for item analysis. This formula, and others similar to it, is used at large companies like College Board to design exams. The video is mostly targeted to high-school students who may be mathematically inclined (or angry about their exam scores), but educators should find value in understanding how to analyze test items in more detail.

The Hodge Conjecture has a daunting reputation as the Millennium Prize Problem so abstract that people have called it “impossible to explain” to a general audience.

My project is an attempt to unravel this math mystery through storytelling. In the video I build intuition from foundational geometry, through projective spaces and topology, all the way to the formal problem statement. This video is for math enthusiasts and lifelong learners with a baseline understanding of undergraduate math.

I used dynamic programming to find the optimal strategy for maximizing expected points in a game of Yahtzee. I then analyzed the resulting strategy to show what this optimal play looked like in various phases of the game, eventually getting to adjustments that would have to be made in multiplayer games.

My goal in this video is not to explain the math as much as it is to show a cool way that math can be used to solve a problem that people might care about. I’m trying to figure out the right balance of how much to talk about math versus just applications in my videos, so I’d love to hear people’s thoughts on this.

The Slope Problem asks a deceptively simple question: What’s the smallest number of different slopes of line that n points can define? Answering that question requires a surprising amount of ingenuity.

What’s prettier, rainbows or math? Good news, you don’t have to choose!

In this video, following many cues from Vladimir Arnold’s book Mathematical Understanding of Nature, I attempt to explain the rainbow in a self-contained way, without assuming any calculus or trigonometry. Arnold’s approach is remarkably simple. Accessible to anyone with a bit of patience, really. Yet it can explain many of the complexities of the rainbow. In this video, we see why the rainbow always appears at the same angle, why that particular angle, and why it appears to be lit from below but dark from above.

If you persevere, there’s also an Easter Egg at the end.

Supplee’s submarine paradox asks a deceptively simple question: what happens to a neutrally buoyant submarine when it starts moving relativistically?

In this video, we explain why the submarine sinks—and show how the apparent contradiction between different reference frames is resolved. We solve the paradox in two different ways, highlighting how relativistic density, force, and gravity fit together consistently.

A compact but surprisingly deep application of special relativity and the equivalence principle.

Double cube hides every square root from √1 to √6, but √7 is nowhere to be found. Why? The answer isn’t in the geometry of this solid, but in the algebra. It all boils down to a number theory question that took over 2000 years and the work of many great mathematicians like Euler, Lagrange, Legendre and Gauss to answer. This is the origin story of additive number theory, starting from the Pythagoreans’ polygonal numbers, through 18th century world-class brainstorm, to the final piece of Fermat’s [redacted].

It’s a topic close to my heart since I’be been working on it for roughly 3 years now. So… enjoy!

This video explains how probability problems can be solved by representing them as state diagrams, also known as Markov chains. Using this concept, we solve five probability problems, starting off easy and building up to hard.

Some of the most celebrated results in geometry and topology are often perceived as elegant, yet intimidating. Euler’s Formula or the Gauss-Bonnet Theorem are sometimes presented through abstract arguments that can feel inaccessible to many learners.

The goal of this video is to change that perspective.

Using a set of interlocking polygonal tiles, each proof becomes a hands-on process. My intent is to use these tiles to build polyhedra, allowing viewers to follow the mathematics. Quantities such as faces, edges, vertices, and angular defects become tangible objects that can be manipulated and observed throughout the proof.

I hope this approach offers a useful example of how physical models can help communicate deep mathematical ideas while preserving both the beauty and the rigor of the original proofs.

Starting with the natural numbers, we construct the integers, then the