Posts related to mathematics
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…
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 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…
1. Introduction This post will continue my investigation into the “method” of small geometrical displacements that I discussed in my previous post. I will apply it to a famous problem regarding triangles: the Fermat point. The Fermat point of a triangle is defined as the point in the plane of the triangle that minimizes the sum of the distances to each of the vertices. As we learned in the last post, the method of small displacements can be very useful…
1. Introduction One of the most important things about calculus and its subfields is the ability to think about “very small” changes in certain variables and pull out information from them. The derivative of a variable with respect to another variable essentially tells us the relation between a very small change in (termed ) and the very small change in (termed ) that results. But how do we define “very small?” Limits are a powerful way of doing so: we…
1. Introduction We have all heard of the familiar Pythagorean theorem, relating the side lengths of a right triangle. But, have you heard of De Gua’s Theorem? An interesting generalization of the Pythagorean theorem into 3 dimensions, this theorem is very similar result for “right tetrahedrons” where three of the edges are mutually perpendicular. Suppose, in such a tetrahedron we have Denote the area of as . Then, In today’s post, we will investigate whether a…