Summer of Math Exposition

I am making an interactive essay that explains the numerical ratios that underpin musical notes, scales and chords.

You can measure the diameter of a circle by measuring chords (no need to align across the middle of the circle) and doing calculations. Here, I describe the method to do so, and the derivation behind it.

If I tell you that I see a finite set of points, which are characterized by a special propriety; each line that joins two points of this set will inevitably reach another point of this set. What do I see? Are these points co-linear? well, if they WERE co-linear, this propriety WOULD be satisfied. but how could you be certain that this is the only positioning that would assemble such propriety? What I learned solving this is how to turn intuition and what seems "obvious" into a rigorous argument.

Scuba divers know that the pressure experienced at depths increases by one atmosphere every 10 meters down. That’s just serendipitous; an easy conversion that isn’t actually part of the metric system and is due to specific physical properties of Earth that are “just right”. It didn't have to be this way; these "spark joy" for physicists.

In this interactive article, we will explore one way to generate plants procedurally, learning common problem-solving techniques for open-handed tasks. There are no particular requirements to enjoy this interaction-based experience.

A C# implementation of the Split-Void numbers, a number system in which division by zero is allowed.

FRACTRAN is an esoteric programming language created by John Conway. This interactive will guide you through a series of programming problems to teach you how to program in FRACTRAN and hopefully convince you that the language is Turing complete. Early levels require only basic math problem solving, the last few levels require familiarity with theory of computation topics such as automata theory and Turing machines.

Identifying a stable point as a maximum, minimum, or saddle point in a multivariable case. Visualizing the eigenvalues and eigenvectors of the Hessian matrix.

Short example of how impactful a nice parameter configuration can be.

This lesson is meant to introduce kids to the equation for the volume of a sphere, via discovery methods. It also serves to get them thinking intuitively about limits (in the context of integrals), many years before calculus would normally be introduced.

An exploration into how you might discover generating functions in an intuitive way. We felt that they are often presented in a way that doesn't lend itself well to understanding where they come from. This blog aims to give a different approach using linear algebra.

A package for Python for faster math on floats using IEEE 754 "mathematical properties". I started this project before I saw the post about SOME so it's not very explanatory at the moment, so you should already know about IEEE 754, C and Python to understand how it works...

Discovering the beauty of twine string by observing natural patterns. Starting with making cordage from natural fibers, we continue our research using crochet models to end up making discoveries about these intriguing surfaces.

In this interactive lesson, learn about the strategy of the game of Nim. Along the way, you will learn about the principles of combinatorial game theory, and come to a surprising conclusion about the usefulness of Nim in seemingly unrelated games.

This essay explores the correspondence between configuration spaces, algebraic data types, and positional number systems, covering a lot of ground while assuming very little prior knowledge in any one area. Then we explore the geometry undergirding the various kinds of positional number systems, a cone of continuous moduli, and demonstrate how a deeper understanding of this space may be translated across our correspondence.

This blog post is a walk-through on a seemingly geometry-related question that exposes (high school) students about a type of differential equation.

I have designed a website dedicated to the graphic representation of functions and mainly parabolas. Throughout the website, we can discover a unique way of designing curves by approximating them. As soon as we make sure that some of those curves can be defined as functions, we take a deeper look at a bunch of their hidden properties, as well as some interesting uses in our daily lives. It doesn’t require any deep mathematical knowledge, just curious readers who are enthusiastic about learning.

I work through one of the arguments that quantum mechanics has no room for local hidden variables, the Greenberger-Horne-Zeilinger-Mermin thought experiment, using baked goods.

We illustrate techniques of Geometric Dissection using a child's jigsaw pieces (Chinese Tans). They are, in every sense, our Toy model.

This blog starts with the premise of 'how can I evaluate if a method is shuffling is good enough?'. The deck used is from the boardgame Ticket to Ride, due to its more simple structure, and we look at two methods: riffle and pile shuffling. We visualize the patterns that arise while shuffling, we quantify the randomness in those patterns, and at the end, we reach a definitive conclusion on which of those methods is better than the other, and why.

I demonstrate the power of switching perspective to abstract spaces by comparing a purely calculus-based solution of a physics problem with a slick geometric proof. The latter has underlying connections to the seemingly unrelated brachistochrone problem.

In this short demo, Generative Einstein will provide a glimpse into his thoughts on special relativity.

A grounded approach to understanding Euler's formula by preferring explicit computations.

A magic trick where the assistant need to flip 1 bit to hint the magician to the correct binary number.

This paper presents a compelling case that the exact equation for the circumference of an ellipse is π × √(a2 + b2) × 2 . This case is supported by various types of mathematical reasoning. That mathematical reasoning includes geometric analysis and empirical examples. An exact equation for the circumference of an ellipse has implications across disciplines where ellipses are encountered. The goal of the paper is to provide a clearer understanding of an ellipse’s circumference in accessible terms