Each triangle belongs to one of three groups about which membership its angles decide. These are the right, acute, and obtuse triangles. The law of sines holds within them all. Right triangles There is a right-angled triangle with the vertices A, B, C at which angles are α, β, γ, where α is the right […]
In physics, events characterized by a significant difference in size (for example, the behavior of waves in the oceans and of their individual water molecules) have a small influence on one another (they do not communicate). Therefore we can independently study the physical properties corresponding to each order of magnitude. This scale independence is precisely what allows us to […]
Graph Theory City Planning Using Graph Theory Strongly Connected Components Algorithm Pi Approximating Pi Using Needles A simple experiment to approximate π Euler From Derangement To Euler’s Number How e can rise from combinatorics Linear Algebra How Does Compression Work? SVD decomposition and a demo of an image compression. Open Problem The Simplest Open Problem […]
In 1915, mathematicians David Hilbert and Felix Klein came to Emmy Noether with a problem [1]. Einstein had published the field equations for his general relativity earlier that same year, and there seemed to be a worrying hole in the theory. Under certain circumstances, one of the most fundamental principles of physics was violated. Energy […]
Differential calculus was invented independently by Isaac Newton and Gottfried Leibniz and it was understood that the notion of the derivative of nth order, that is, applying the differentiation operation n times in succession, was meaningful. In a 1695 letter, l’Hopital asked Leibniz about the possibility that n could be something other than an integer, […]
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, […]