1. Introduction In plane geometry, the idea of an “angle” has long been built into our intuition. The image of two rays meeting at a point sparks a vague notion of how “far apart” the two rays are: this notion is made rigorous with the definition of a , the angle required to intercept a circular arc with equal length and radius. But what happens when we leave our sheet of paper and enter the real world? How can we…
Preface: Inspired by the countless imaginative works of science fiction I’ve come across, I decided to engage in some of my own science-based speculation. I wrote the following after pondering about what the Earth would be like in a two-dimensional universe, and I thought it might be an interesting read! This is quite different from my usual posts, in the sense that most of my claims aren’t followed with rigorous equations to back them up; I did, however try to…
Note: The blog-embedded version of this post will be coming very soon; in the meantime, enjoy the PDF version below!
1. Introduction Previously, I had written a post detailing the Fermat point of a triangle: the unique point which minimizes the sum of distances to the triangle’s vertices. Recently, I realized a small oversight I had made in my proof of the case where all of the triangle’s angles are less than . In today’s post, we will be patching this oversight, so it is recommended to read through the original post before continuing (as a challenge, try to find…
1. Introduction The binomial theorem famously allows one to fully expand out expressions of the form , where is a natural number. In these cases, the justification seems intuitive enough: if we write out a product of with itself a total of times, each individual term is produced by choosing either an or a from each factor, and multiplying them together. Then, for any , the term can be produced in ways,…
1. Introduction While browsing YouTube one day, I came across a video covering a problem from the 2016 JEE Advanced physics exam, an engineering entrance exam in India. The problem had several parts connecting to the following physical setup: In its simplest form, we can consider a circular wheel of radius connected by its center to one end of a rigid rod of length , with the other end fixed on the ground. The wheel is able to roll without…
Note: this post will soon be updated with diagrams, thank you for your patience! 1. Background Opposites attract, likes repel. Straight off the first page of Electrostatics 101, this forms the ground rule for all basic interactions between charged objects. We can be more precise: the exact force between two objects with charges and separated by a distance of equals This is known as Coulomb’s Law. Here, is an important physical constant related to how well electric field…
1. Introduction HHTTTHHHHHHTTTTHHTTHHHTHHHTHTTHHHTTTHHHHTTHTHHTTHH Displayed above is the result of consecutive random coin flips. In total there are 29 heads and 21 tails, not too outlandish given our relatively small sample size. We can ask several interesting questions about strings of random coin flips like the one above, and I was working on a mathematical research project recently when I encountered one such problem: Given a string of independent flips of a fair coin, what is the expected length of the…
1. Introduction Today’s post will be more elementary in nature, as it is something that I first thought of many years ago but haven’t gotten around to sharing until now. Nevertheless, I think it is good to know and may come off as surprising to those that are not familiar with it. It has to do with the properties of a special set of complex numbers: the roots of unity. We can extract lots of information from these numbers; today’s…
1. Introduction Today’s post will be a little different from usual: I wanted to share a few of my exploits from long ago with the online graphing calculator Desmos. Specifically, I tried to use its features to design a few interactive tools that you can try out. I will supplement the link to each tool with a brief description of what it is and how it works. Enjoy! 2. The Cycloid Drawer https://www.desmos.com/calculator/81brskzoje The cycloid is a famous curve that…