Summer of Math Exposition

This article explores the tension between discrete steps and continuous flow—two fundamental ideas in mathematics. Through the familiar setting of queues and waiting lines, it shows how these perspectives shape our world, examining their implications, motivations, and realistic extensions—before concluding with interactive simulations for readers to explore.

Data visualization-- the translation of data into images-- has become the de facto tool for scientists to explore, summarize, and communicate data. But what if we turned data to sound instead? Beyond helping make math and science more accessible to the visually impaired, data sonification-- the translation of data into sound-- could help us develop an entirely different intuition of data, and give us new ways of understanding it. This audio is a short introduction to data sonification that aims to demonstrate some of the possibilities it opens up. It takes the listener through the sonification of basic mathematical functions, vectors, and recreates a rather famous real-word data visualization. Most if not all of the content should be accessible to students in middle-school or above. Note: You should be able to access the soundcloud link without a login. If it asks you to log in, you should be able to close out of the login prompt by clicking the "X" button on the top right corner. But just in case, here is another link to the audio file: https://drive.google.com/file/d/1WEEEcCPO6ky3Z93EiDwwO-EWkIPNtBsz/view?usp=drive_link

I have attempted to help the reader discover how the epsilon-delta definition of continuity arises naturally. This article is intended to bridge the gap between intuition and rigour about the idea of continuity. When I first encountered this definition, I was encouraged to generalise it further, as much as I can. I quite enjoyed that feeling, and so I have attempted to give the reader a taste of the same feeling with this article. This article assumes familiarity with functions and is aimed at advanced high school/early undergraduate students.

Deep dive to some methods of proving theorems with an introduction to Lean theorem prover

In this blog entry, we visualize the Riemann surface (graph) of a trigonometric function, in order to get the reader to see a bigger picture when it comes to dealing with inverse trigonometric functions and give a concrete example where a continuous surface is patched together from "standardized" complex functions with ugly (but necessary) discontinuities. The premise of the entire blog is totally awesome visualizations, and hopefully this totally awesome visualization will get the reader more excited about math.

This article is a beginner-friendly journey into the elegant mathematics hidden within the Rubik's Cube. I talk about everything from the mind-boggling combinatorics of its 43 quintillion states to the powerful group theory that makes solving it possible. It's aimed at curious high school students just meeting permutations, university students looking for a tangible example of abstract algebra, and anyone who has ever wanted to understand the beautiful rules that govern the cube's chaos.

We go on a playful journey through one of the most famous — and mysterious — rules in maths: why multiplying two negatives gives a positive. Using examples with debts, colourful squares, and the number line, we visualise and explain the rule with intuitive activities. The aim is to show that maths rules are grounded in logic, balance, and symmetry — not just convention. The content is suitable for learners still mastering negative numbers, but encourages deeper questions for those interested in mathematical structure and reasoning. Recommended for curious readers, teachers, and anyone who enjoys a touch of monkey business in maths explanations.

This blog post explains how AI's learn, with a focus on gradient descent. I explain the intuition behind gradient descent using analogies, drawings and animations, structured in the form of a dialog with a (fictional) student. The entry starts out with arguing why AI is overall learning the same way humans do, by means of trial and error. Then follows an example of how one can measure the accuracy of the AI model using a cost function, followed by a description of how gradient descent can be used to efficiently find the optimal model parameters. Details such as step size, stopping criteria etc. are also briefly discussed. My goal with the entry is to make the topic of how AI's learn understandable for a target audience of people who are curious towards AI and technology, but without a requirement on them having any mathematic foundation in calculus, linear algebra or optimisation theory. My aim is to keep things very high-level, as to not scare readers away by mathematical nitpicking of details. The target audience could therefore be e.g. highschool students or students starting university - but it is intended for any curious adult.

In software development, programmers mostly use packages or libraries, which are existing pieces of code (that provide useful features) instead of making everything from scratch. These packages may also depend on some other packages, making a large web of dependencies, creating an extensive list of constraints. All the constraints must be satisfied for a package to get installed and work correctly. That's where package managers come in. Package managers are software that ensure all the constraints are met so that all the packages work together. Navigating via such a large web of dependencies can be daunting. This blog explores how package managers translate these dependencies into a Boolean formula and solve it as a SAT (satisfiability) problem.

We have developed a website to help users visually understand different types of optimization algorithms, such as Gradient Descent, Conjugate Gradient, Newton’s Method, Golden Section, Powell’s Method, and the Nelder–Mead algorithm. Users can either choose from existing benchmark objective functions or provide a custom one. After selecting an initial guess and the appropriate algorithm-specific settings, the user can start the optimization process and interactively track the path of the optimizer.

Starting with a tiny, simple recursive formula for counting primes, this website works through a dozen different variations of that formula, exploring a large spectrum of fundamental number theory ideas along the way. And it's all intertwined with vis.js animations of the recursive functions, informative d3.js animating graphs, live, runnable JavaScript code, and a dialogue format. Along the way, factor wheels, the logarithmic integral, the Dirichlet hyperbola method, the Dirichlet eta function, and the Chebyshev function will each show up organically. I had in mind both students of math and programmers who find primes interesting in mind when writing the site, but I think the striking animations, simple runnable code, and dialogue format could well broaden its reach.

The most ubiquitous symbol in mathematics often gets taken for granted. The humble equality forms the backbone of all mathematics and this is a call to pay it closer attention. Acting as the bridge between representations, equivalences point to the abstract, deeper patterns and behaviours that aren't easily written down.

Demonstrates how higher order numerical methods can be discovered by manipulating Talyor series. Prerequisites: Talyor series, concept of a differential equation Includes hyperlinks to further reading (not part of the entry).

'Number to the Pain of Mathematical Operations' introduces the Trachtenberg System of Speed Mathematics, which I believe is useful everywhere from primary school classrooms to pure mathematics university lectures via the meal deal checkout in Tesco and the medication dispenser at the pharmacy. Writing this article was extremely difficult and immensely rewarding. I challenged myself to paraphrase formidable mathematical expressions using comprehensive English complemented by humorous cartoons, which I hand-draw using GoodNotes on my iPad. I hope this article facilitates engaging maths lessons in schools, philosophical reflections about number theory, and easier day-to-day life

On why it is so rewarding to let your mind wander and chase rabbit holes

Linear Programming is one of the most useful mathematical tools whose application ranges from engineering, operations and research to business management. Hence, there is a considerable literature devoted to the topic. Here, we particularly explain the Simplex Algorithm, which is one of the prominent algorithms for solving the Linear Programming Problem. Simplex Algorithm, being purely procedural, also has a beautiful geometric flavor to it. However, in our high school and undergraduate exposition, we are usually introduced to the topic in an unintuitive tabular manner. In this interactive blog, we try to explain the geometric interpretation of it, which we believe will introduce you to a different viewpoint for looking at it. We have used the usual textbook example to keep the exposition simple. And we request that anyone use the PC Web browser, as the graphing has some responsiveness issues. (We are planning to incorporate that soon.) Assumed pre-requisite: We assume the reader is introduced to the notion of Linear Programming via the usual plotting method and the Simplex Algorithm in tabular form. This exposition is mainly to give the geometric perspective of the tabular algorithm, which we have studied. It’s our first attempt to make mathematics interactive and fun. We value your critical feedback. Thank you.

An Introduction to Computational Neuroscience: A Simulation-Based Approach

Attention mechanism is generally explained as a handed down concept. But the best way to learn is to try to come up with the modeling yourself. Here we try to navigate the choices and see where it lands us and evaluate those to finally reach the original conception.

Vandermonde determinant is an elegant but weird result in the field of special determinant. Why do we need to calculate it? How do people discover it? And how do we get the result with our stunning attention (All things proven by mathematical induction are like this, so I have a natural aversion to mathematical induction.)? If you are a little more inquisitive, you may ask why the commonly used notation is x instead of other letters? I used to be fooled like this, but now with enough knowledge, a flash of inspiration has come to me, and I have finally found a natural way to prove it.

Visualize the geometric series terms as the areas of trapezoids that aggregate into a single right triangle whose area represents the geometric series sum. With an image of the geometric series to go along with the equation, the concept of geometric series is multimodal and therefore more memorable.

An article building a definition of horn fillers and ∞-categories from scratch, going through monoids, graphs and categories in the process.

This is a short piece directed toward middle-school / lower secondary school on graph theory specifically Eulerian circuits and paths. I wanted to highlight the artistic side of maths whilst also demonsating practicle applications in real life.

This is a short and hopefully fun text written by, and aimed at high school calculus student(s), with a focus on intuition. It covers two topics in a way that I've endeavoured to make engaging. All JS graphics are by me. Please enjoy! Additionally, please note that this is a desktop only website. The graphics which are an integral part of this submission do not render on mobile. I apologise for the inconvenience.

Introduction into basics of signal processing, beginning with complex numbers, the Fourier series and Fourier transform (1D/2D), and with more topics along the way, such as Taylor/Maclaurin series, convolutions, signal filtering and more. The tutorial script comes with over 2700 lines of R code, reproducing most of the discussed formulas and algorithms with hand-written code representations, entails three example images, including a magnetic-resonance image (MRI, magnitude image), and around 240 R plots. The tutorial is not written for absolute beginners and requires some basics in R (which we provide on our website though), but it can also be read without executing code, since all results are provided in the tutorial as well. It is mostly written for non-mathematician students, interested in or working with some form of applied mathematics and physics, and which are interested in engaging with interdisciplinary open educational peer-teaching projects.

In the public domain, black holes are generally understood as local astrophysical phenomena. This short article aims to move beyond this treatment and impress upon an enthusiastic (non-expert) reader a global understanding of black holes. By exploring relativity via causal structure rather than tensor calculus, our discussion covers advanced topics whilst remaining accessible.

The goal of this article is to invite algorithmists (and competitive programmers) to the fascinating theory of modular forms, which is a very deep tool in number theory. Here we consider the computational problem of counting the number of ways to represent an integer into sum of eight squares of integers.

Tanglegrams are a way to compare evolutionary histories of species in ecology. Drawings of hosts and parasites are compared to infer the amount of non-ancestral gene transfer between species. The related leaves in two evolutionary trees are connected by edges. The number of crossings between the edges represent the evolutionary "distance" and the amount of horizontal gene transfer needed to explain the coevolution of the hosts and parasites. However, this comparison depends on the drawings of the trees. Another drawing of the trees leads to a different number of crossings. Can we redraw the trees to have the least number of crossings? Will this change the biological conclusion? We introduce the problem as a game. Given a drawing, try to click on the nodes to create a drawing with the smallest number of crossings. After playing the game for a while, the player will be naturally curious about many aspects of the game --- what is the least number of crossings for a particular game, what is the smallest number of clicks to solve the problem etc. We will use this to explain the science and the open questions. 1. The evolutionary biology and why these shapes are of interest in science. 2. The mathematics of the number of drawings, number of crossings etc. 3. The algorithms, the complexity and the many approaches for the solution. This is a large area and there are many angles to the exposition. The submission has only the simple version of the game. There are many more versions of the game, which will be introduced as levels.

This interactive “2D Tessellation Explorer” turns tessellations into a hands-on investigation. Load a Shape Workshop with triangles, squares, pentagons, hexagons, octagons, or dodecagons, then drag and arrange pieces to see what locks perfectly with no gaps or overlaps. Hints scattered throughout help explain what to look for. Start with exploring what regular tilings are possible (like 6 triangles, 4 squares, or 3 hexagons) and then experiment with semi-regular mixes - e.g., squares with octagons - to uncover patterns and see what can't work. A Custom Builder lets you pick up to three shapes (20 of each) to search the space of possibilities, including the eight classic semi-regular tessellations. It’s exploratory first, proof-minded second - students feel the constraints before they formalize them. The exploration focuses on visual shapes so that it can be used with young students - older students can use it to try things out before confirming the angles manually, but the angles were mostly left out of this activity to avoid intimidating young pupils.

Other intuitive ways of visualizing and thinking about definite integration.

What are the connection between quaternions and rotation? We will explore this topic, building from the basics, connecting concepts from the real, imaginary and quaternions. Through this approach, I aim to explain in a way that even a high schooler with basic calculus knowledge can understand, and learn something.

Revisiting polynomial and integer division and extending it in ways that aren't typically covered in school, including a p-adic detour. Python code is provided for further experimentation and algorithmic clarity.

Mathekton is a singleplayer game about creating and solving math problems. To guide you, a deck of 100 legendary mathematicians is presented to you. They will inspire you with themes to explore, and tips to help you further develop you problem-creating and problem-solving skills. The game requires you to use pen and paper, or or a note-taking software like Obsidian, to write down your problems and solve them. It is recommended that you are familiar with math proofs. Nonetheless, the only prerequisite is curiosity ! To demonstrate how it is is played, we show an example of a game in the following video : https://youtu.be/pPWjWNjSnhk

A blog post about the reason we use algebra so often in math, its ties to the language of mathematics, and some helpful explanations of concepts in algebra that may trip students up.

The following article/website is mostly for high school students, though I have kept it very simple. So maybe anyone with enough knowledge of algebra and arithmetic can probably understand this. I wrote this article because this is the way I wanted to be taught mathematics in high school. But unfortunately, I didn't. I request the readers to read the entire article and not judge it based on one particular section. Though this article is still in development, there's a lot that I'm gonna add and probably change. But in its current form, it's enough to give the readers a fair judgment of this article. It is advised to be read on a laptop.

The blog unpacks Hierholzer’s Algorithm—a beautiful method to construct Eulerian paths and circuits in graphs. Aimed at beginners, it starts by demystifying what Eulerian circuits are, builds intuition for their existence conditions with crisp examples, and then walks through Hierholzer’s algorithm with clear visuals and step-by-step logic. It also includes an intuitive proof of correctness, helping readers appreciate why the algorithm works. Whether you're a student tackling graph theory for the first time or a curious learner, this post offers both clarity and rigor, making an elegant idea feel within reach.

A story about the bias-variance trade-off. It is told in terms of three magical entities, Bias, Variance, and Epsilon. These entities work together to overcome a struggle that affects the town they live in, casting the machine learning model spell. The target audience is individuals that have studied Linear Algebra, Statistics, and Calculus.

Some almost-formal proofs of how certain periodicity bulbs in the Mandelbrot set touch each other, with a connection to the Fibonacci numbers.

An article about why 22/7 became the standard approximation of pi.

Hi, thanks for viewing my entry! I created a one-picture illustration to help people understand how exponents work. The story behind this picture is that there was an old wives' tale about a king who wanted to reward a wise woman for her services to the kingdom. She asked for one gold coin on one square, twice that on the next square, and so on and so forth until an 8x8 grid of squares was filled. The king agreed, not realizing how much money that is! I created this picture as a way to both capture that story and visually see how much that is--each rectangle is a proportionate representation of the amount of money starting from a single coin, and of course the size cuts off because the numbers get too large. I do hope that this image is helpful in classrooms and such, and please do feel free to share it if you find it helpful or know someone who might! If you'd like to use it in a more professional capacity, I do kindly request you cite Ikonosophy under a Creative Commons license or reach out to [email protected] for a more direct request. Talking about it in an educational capacity i.e. to students is of course fine.

This is a solution to the Duoploy Problem using Desmos. The Duoploy problem is a famous problem in economics and game theory in which there are two firms competing against each other in order to maximize their profit. Each firm has to make the decision of how many items they should produce. But the optimal number of items that a company should produce does not simply depend on one firm. The optimal number of items that one firm produces is inherently dependent on the number of items the opposing firm produces. The more items produced decreases the price since the price is inversely proportional to the number of items produced. If the price goes too low, then the profit would also decrease even though more items are being sold because the cost of production would cancel out the revenue generated. So each firm should consider what strategy the opposing firm is using in order to maximize the profit. The optimal strategy is found by finding the point where both firms would be doing worse off if they were to slightly deviate from that point. This special point is called the Cournot Nash Equilibrium. In this project, you can input any function you want for the profit and the average cost of production to calculate the Cournot Nash Equilbirum.

A group of friends must use logic, not a scale, to solve their mischievous teacher’s nail puzzle, proving that chemistry is more than just calculations.

It shows the underlying unity of nature. I showed that different completely physical scenarios can be understood using same mathematical tool, i showed that analogy between steady ideal fluid and electrostatic field. Target Audience High School Students.

A thorough walkthrough, and a rigorous formulation, of the process of finding the molecular orbitals of a molecule using nothing but its symmetry. For chemists/physicists with a linear algebra background.

Learn how to find the determinant of a matrix step by step with this guide including 2×2 and 3×3 determinants, cofactor expansion, and row operations.

Generative AI tools like OpenAI’s ChatGPT and Google’s Gemini have captured — and held on to — public imagination for a few years now. But how do these systems work, and why do experts prescribe abundant caution? Read on to gain technical intuition of the maths that underlies generative AI, and hence appreciate the risks of this technology, in particular the hazard of it being used to expedite propaganda machines. This essay has been written with a senior high school or undergraduate audience in mind. Less advanced readers are welcome to skip over the formulae and refer to the supporting text for the essential insights. More advanced readers are requested to forgive the (over-)simplifications :)

This project explains the parallel axis theorem, a simple equation that connects the moment of inertia at different axes of rotation. Everyday examples motivate a step-by-step derivation, making the concept intuitive and clear.