This is a story about a brilliant man that became so obsessed with a mathematical question that he went insane and eventually died alone in a sanitarium. This article is also meant as an introduction to the concept of mathematical infinity and the transfinite world. Introduction Think of the natural numbers 1, 2, 3, 4, […]
In 1983, Gerd Faltings proved a long-standing conjecture of Mordell from 1922. The statement and proof of this theorem illustrate just how connected many disparate branches of math are. I won’t state it in its original form, but there is an easily understandable modification of it that will probably come as a surprise to many […]
Last time, we looked at IBVPs for the heat equation in which forcing was present and the boundary conditions were homogeneous. This article will cover how to solve IBVPs for the heat equation with more complicated boundary conditions, in which the problem has the form: In this problem, f(t) and g(t) are smooth functions, meaning that they are […]
A couple of years ago I was thinking about how to generalize many of the infinite series that we know so well. It was clear to me that it was possible to generalize them since some of my studies into the field of Fourier series had led me to some of these generalizations in the […]
In this article, I will consider the problem of finding the shape of the free surface of the water inside a half-full bucket rotating with constant angular speed. Since I will work with systems moving with speeds much smaller than the speed of light, relativistic effects can be safely neglected. The derivation will be based […]
Geometry The Hidden Life of Circles Everyone knows that circles are quite simple figures, but from the perspective of number theory they have many secrets inside. Group Theory The Power of Invariants We can find invariants in many branches of mathematics. Let us try to use the theory of invariants to solve a few popular […]
Topology Theorema Egregium and Pizza Eating How can Gauss improve your pizza eating technique? Geometry To 3D and Beyond How many spatial dimensions do we live in? How many can we directly perceive? Are the two answers the same? If they are different, what effect, if any, do any higher dimensions have on us? Knot […]
I recently had the privilege of interviewing Ken Ono (1968-), one of the most brilliant mathematicians in the world. Not only has he made huge contributions to the field of number theory and was also a mathematical consultant for the Hollywood film The Man Who Knew Infinity, but his unique, inspirational story can provide valuable lessons to anyone, […]
Length contraction is the relativistic phenomenon where the length of a moving object is measured to be shorter than in its rest frame. It occurs only in the direction of motion, and its effect is significant only when the object is moving at speeds close to the speed of light. The Miracle Year In 1905, […]
There are a lot of reasons why I’m writing this, one of them is I could never understand Borel Sets, I got introduced to them on day one of my adventures in Probability Theory, there was something about the way it was portrayed, it didn’t make sense to me, like at all. So I broke […]
I love these three math problems! The first one requires very little math skill — but it is maybe the most difficult one. The second one only requires basic geometry. For the third one, you need to know elementary algebra. Problem 1: The 316 problem I call this problem “the 316 problem” because 316 is […]
In this article we explore how the principle of relativity may apply to curvilinear movements. Our aim will be to achieve a transformation analogous to that obtained by Einstein on his paper On the electrodynamics of moving bodies [1], the Lorentz Transformation, but referred to systems on rotational movement respect each other. For that sake we will first […]
Here’s a conjecture with a lot of really smart people working on it that has a fairly intuitive statement, and yet almost no one knows about it. Seriously, there isn’t even a Wikipedia page. The idea is to quantify how weird 1-dimensional sets can be. We’ll make these notions more precise in a bit, but […]
Puzzle This Math Trick Only Works for Everyone at the End of the Year So it’s the perfect way to usher in the new year Field Theory A Finite Number Of “Numbers” What exactly is arithmetic on a field of numbers which are merely elements in some set? Babylonian A Modern Look at Square Roots […]