Summer of Math Exposition

An interactive guide linking sound waves to electron orbitals.

At this very moment, billions of devices are running these three lines of code:

t ^= t << 23;
t ^= t >> 17;
t ^= s ^ (s >> 26);

This algorithm, and the others like it, are the metaphorical dice that your computer rolls every time it makes a decision. What does it do, and how does it work?

Join this deep dive into one of the most deployed families of pseudorandom algorithms: XORshift generators.

And the journey will start with a very simple question: what makes 23, 17, and 26 special?

This is post is a simple introduction to stream ciphers, and how the same basic concept is used for constructing ‘advanced’ and more widely used ciphers, with a focus on ChaCha20-Poly1305.

A turtle wanders an infinite plane. At every step, it turns by an amount that grows with the square of its step number. This simple rule leads to paths that merge chaos with symmetry.

What happens if we give our neural network more than just data? Through this article, we try and explore something called “Physics-Informed Neural Networks” which can help us solve differential equations. We define the architecture and build one to solve the 1D heat equation. We also dive into the math of how PINNs work, where they work well and where they don’t.

Losing weight may seem like a task that requires an enormous amount of effort, but with a bit of basic math and a touch of linear optimization, it will feel like child’s play.

An interactive article that explains how Transformers work by training a very small model to play fading Tic-Tac-Toe.

Instead of using text prediction, the article uses a simple game where the model can still learn real strategies. This makes it possible to clearly show how the main parts of a Transformer work together to produce a prediction.

Each visualization can optionally show the exact math behind the model, so readers can follow the matrix steps as the Transformer processes a sequence.

The article also includes experiments showing what happens when different parts of the model are removed.

The intended audience is students, programmers, and anyone who wants a simple and visual introduction to how Transformers work internally.

A Desmos notebook explaining PRNG basics, Lehmer RNGs, and randomness testing targeted towards middle/high school students

In this blog post I cover Bezout’s identity, how to compute Bezout coefficients, and how they can be used to quickly settle a question about all the common divisors between two numbers.

My aim was to make the mathematical content relatively light, but framed within a concrete story, which should make it accessible from high-school level onwards. I also included some questions throughout the text, for readers who want to check their understanding, or curious to be pushed into directions that go slightly beyond what is covered in the post itself.

An unexpectedly long and at times crazy adventure spurred on by a simple question of probability. The realization that the sum of two independent uniform random variables on $[0, 1]$ is NOT a uniform random variable on $[0, 2]$ may initially be surprising. But then I asked: if you can’t sum uniform to uniform, what two independent random variables CAN you sum to it? We explore, and actually construct an answer, which winds through a lot of math territory, with whacky manipulations involving Laplace transforms, power series, and a fractional integral!

My intended audience is probably upper division undergraduate, definitely one who’s had a course in probability theory and differential equations. Possibly understandable by engineering students, as those delta functions and Laplace transform calculus very much were the bread and butter in a course I took called Signals and Systems.

This article explores the question of how to stretch audio waves without distorting their frequencies. It focuses on an example audio signal of two interfering sine waves, and aims to show that there is no single “perfect” way to stretch an audio signal: some form of distortion is inevitable. It briefly introduces/motivates modern audio stretching algorithms towards the end.

This essay is a nine-day diary of rediscovering Vladimir Arnold’s topological proof that the general quintic equation cannot be solved by radicals. Instead of relying on heavy Galois theory, Arnold’s proof works by continuously moving the roots of the polynomial around each other in the complex plane. I didn’t invent this proof, but I’ve written it as an accessible journey so that you don’t need a semester of abstract algebra to uncover its geometric secrets. You can find it at: https://aleksandarliden.com/nine-days-with-the-quintic/

This game describes an algorithm for shuffling cards fairly. It’s used in statistics, cryptography and computer games. It has to be implemented carefully because a common mistake produces a shuffle that’s catastrophically biased. It’s worth studying how bad the biased shuffle is, and some mathematical results about it are unexpected. For example, once we have at least 18 cards, the most likely permutation is the identity, where none of the cards change position.

I’ve chosen the format of an Interactive Fiction (“Choose Your Own Adventure”) game rather than an article because I can use games, random events and quizzes, and because you can skip sections if they don’t interest you. Please explore the islands at your own pace and don’t feel required to visit every section.

I want this blog post to feel like a natural exploration of digital wireless communications, and for you to feel like you could’ve come up with these concepts yourself.

I’ve tried to make the blog post as understandable as possible for those without a background in signal analysis, but it would probably help if you were at least comfortable with the idea of complex exponentials and representing signals in frequency space. You don’t have to fully understand these topics, but a surface level comfortability will help!

At the end of this blog post, you’ll know the exact technology that enables high-speed wireless communications, such as WiFi and Cellular, and hopefully feel that it’s the most natural solution to an otherwise complicated problem!

Coding theory forms the backbone of modern digital communication. This article is a playful introduction to some of the most important and impactful ideas in its history. While many of us are familiar with topics like error-correction and cryptography, I wanted to focus this post on a relatively less discussed topic, at least in my experience: source coding, which we will explore through an interesting problem setup from Charles Petzold.

Coding theory is usually taught at the undergraduate level, but I have tried to keep the jargon to a minimum and build things up using mostly elementary algebra. My hope is to make this accessible to everyone, from undergraduates and high schoolers to curious middle schoolers. More than anything, I want this to feel like a mathematical journey where you get to discover the ideas for yourself, step by step.

An exploration of a novel integer sequence related to triangular numbers, the triangular grid, and minimal polyiamonds.

Visual comparison of four geometric constructions for QR factorization: Gram–Schmidt orthogonalization, Givens rotations, Householder reflections, and Direct Rotations.

The first three are standard QR constructions. Direct Rotations is a much less commonly presented construction that we arrived at independently while developing these visualizations; we later found an earlier related formulation in the literature.

The same full-rank matrix and the same flat rank-deficient matrix are used for all four constructions. The animations show how Gram–Schmidt reshapes the columns of A into Q, while Givens rotations, Householder reflections, and Direct Rotations transform A into the upper triangular matrix R.

In 2D, Direct Rotation coincides with the single Givens rotation. In higher dimensions, Givens uses successive coordinate-plane rotations, while Direct Rotations uses one plane tailored to each active column and its target axis. Direct Rotations is also closely related to Householder QR, despite using proper rotations rather than reflections.

The entry is intended for students learning linear algebra, mostly at the undergraduate level, as well as instructors of those courses.

Created by Eteri Byazrova and Yuri Morozov for GraphMath / Linear Algebra World.

If you are reviewing entries for SoME, you have probably heard of the triangle of power. This is an attempt to introduce the concept to students who have not yet encountered logarithms. The tone is intended to be humorous, but it likely doesn’t work for the target readership. I hope you enjoy it.

I made a balanced ternary calculator. You can use it to add, subtract, multiply and divide just like any other calculator, only it works with a different number system. This takes a little getting used to, but it opens up an interesting new perspective on arithmetic.

In particular, it provides an answer to the question: how do balanced number systems affect the way arithmetic works? When positive and negative numbers are on completely equal footing, does it still make sense to say that a negative times a negative equals a positive? The answer is: yes, it does - but it depends on the direction from which you approach it. This, along with other aspects of balanced ternary calculation, is explored in the ternary calculator. It’s a tool with a story to tell, and the story unfolds in the calculator’s info screen.

This article presents the derivation of the Euler-Lagrange equations and their connections to Lagrange multipliers, conserved quantities, and classical mechanics. It includes generalization to higher-dimensional manifolds when possible, such as conserved currents and classical field theory. Prerequisites: optimization, partial derivatives, the concept of differential equations, and wave motion (optional). Includes hyperlinks to further reading (not part of the entry).

Many STEM YouTubers recommend popular Math(s) books as a way to explore the ideas behind their videos in greater depth. But with so many excellent books available, where should you begin?

Recommendation engines help us discover music, films, and even sporting talent by quantifying the attributes of our favourites. Could we do the same with Math(s) books?

Inspired by the radar charts commonly used to compare modern athletes, this proof of concept gives each Math(s) book its own “fingerprint”, with each spoke corresponding to a broad mathematical topic.

Overlaying multiple fingerprints allows readers to compare books visually, identify complementary reads, or discover the surprising contrast in books with similar titles.

The current bookshelf contains just a small sample. But because it’s built in Desmos, anyone can duplicate the graph, edit it, and build their own mathematical library!

From (a−b)² to x³+y³+z³−3xyz: A Geometric Exploration

I have been exploring geometric interpretations of algebraic identities taught in school mathematics. My aim is to see whether a suitable algebraic decomposition can make an identity easier to visualize and understand.

I first created pictorial geometric representations of the familiar identities

and

Instead of treating $2ab$ as one term, I split it as $ab+ab$:

Although algebraically equivalent, this decomposition gives a different way of viewing the geometry. This idea helped me explore $(a-b)^3$.

I experimented with two different representations of $(a-b)^3$: one using physical carrom coins, and another using FreeCAD to create a 3D visualization.

YouTube demonstrations:

$(a+b)^2$ — https://www.youtube.com/watch?v=G-mybx9ZI-I

$(a-b)^2$ — https://www.youtube.com/watch?v=lIpBRiWCxWk

$(a-b)^3$ using carrom coins — https://www.youtube.com/watch?v=On84n_m7o4A

$(a-b)^3$ using FreeCAD — https://www.youtube.com/watch?v=dtQB1yiPwTY

These explorations eventually led me to the more challenging identity

The difficult term was $3xyz$. Following the same idea, I split it as

and grouped the terms as

Similarly, I considered

This revealed an interesting sign pattern. For positive $x,y,z$ with $x\geq y\geq z$,

and

while $y^2-zx$ can be positive or negative depending on the values of $x,y,z$.

Let

Then

and

Therefore,

This resembles a weighted average, although the weights need not all be positive. This gave me another way to think about the factorisation and its geometric structure.

I created an interactive application in which students can enter different values of $x,y,z$ and explore the identity.

Interactive application: https://sravanam-keerthanam-smaranam.github.io/Algebraic-Identity-Visulization/x3_y3_z3.html

GitHub source: https://github.com/sravanam-keerthanam-smaranam/Algebraic-Identity-Visulization

AI assistance: I developed and refined the mathematical thought process through iterative discussion with ChatGPT. After finalizing the mathematical approach, I used ChatGPT and Claude AI through iterative prompting to generate the complete HTML, CSS and JavaScript implementation. I reviewed, tested and repeatedly directed modifications to tailor the generated code to my requirements.

My central idea is that geometric interpretation need not begin with geometry. It can begin by finding the right algebraic decomposition. Splitting $2ab$ into $ab+ab$ and $3xyz$ into three $xyz$ terms changed the way I could see these identities. I hope these visualizations encourage students to experiment with algebra rather than merely memorise identities.

An explanation of why using modal logic to encode vowel harmony is a bad idea. Not that this is a commonly held belief but it provides a good way to introduce phonological analysis and its connections with logical transductions as well as some of the model theory of modal logic to a general audience. Written as a Substack post with associated stylistic flourishes.

Over the last six months, I have created a system to recognize neat handwriting as an optional input method for the computer algebra system (CAS) I have developed over the last five years. The CAS is designed to work similarly to how we work by hand, but without all of the silly mistakes.

This article has two purposes. Firstly, it describes the process I went through to implement handwriting recognition. Secondly, it is my attempt at marketing—if it can even be called marketing when it does not cost anything.

On the second point, I would be grateful for your feedback because I am clearly terrible at it. To show you how low my reach is, here are some statistics for the entire 200-page site:

• An average of 4 clicks from Google search per day.
• An average search position of 19.6 (putting it on the second or third page of Google results).
• Ad revenue for the entire site is around 2 cents per day, or $7 a year. (My total earnings over 5+ years are still below the payout threshold.)

In any case, I hope you find the CAS and handwriting recognition useful.

UTF-8 is the world’s most used text encoding. It cleverly extends 7-bit ASCII into an 8-bit multi-byte code, equipped with many nice properties.

But what happens if UTF-8 runs out? Can we extend it infinitely whilst preserving those properties? The answer is yes! I call the extension that I have invented “UTF-8000”.

This website lays out the specification for UTF-8000, explaining Coding Theory terms like ‘self-synchronization’ and ‘self-punctuation’ as we go. It should be a good educational website that intuitively explains how one might derive these encodings oneself.

Some mathematical topics can be useful in various real life cases. Expected value is one of these that can inform mathematical thinking in many practical ways. This lesson shows how some simple math can improve your mathematical reasoning in finances, in business, and in health.