A visual proof of the Gauss Bonnet Theorem for triangles on spheres! Spherical geometry is a beautiful, and very visual, area of mathematics, with weird properties (such as that the angles of triangles don’t sum to 180 !!!). Understanding a little bit of Spherical Geometry The Gauss-Bonnet Theorem for triangles on spheres is a special, […]
Philosophy Is ‘Philosophy of Mathematics’ somehow useful? What the philosophy of mathematics is useful for? Max Black, the author of The Nature of Mathematics (1933), thought the main task of the foundation of mathematics (and, consequently the main task of any philosophy of mathematics) would be to elucidate “and analyze the notion of integer or […]
Before we begin, you should know that I’m not actually going to present a proof of the Riemann Hypothesis. This article is about a fictional object known as the field with one element, sometimes denoted Fᵤₙ. You can probably guess why: F is for field and “un” is the French word for 1. When I first […]
The year 2020 might be the year in which I’ve received the most math problems from friends and seen the most math tricks shared on social media. It turns out that sharing math tricks with one another is one of the things people do most when they’re under lockdown. That fact is tempting me to […]
This article will introduce a powerful class of numerical analysis techniques called relaxation methods in the context of electrostatics and fluid dynamics. Let ϕ be a function of x and y and let A, B, …, G also be functions of x and y. Any second-order partial differential equation with ϕ as its solution may be written in the form: The subscripts […]
A few days ago, I explained the content of my previous article on Benford’s Law to a friend, who’s one of the few people who would humor my enthusiasm with maths. And, being the brilliant person she is, she asked how we might observe Benford’s Law and see it in action. It’s one thing to create charts […]
From nothing but the definitions of derivatives and limits, we’ll prove one of the first derivative rules you’ll learn in undergrad calculus. All pictures made by me with LaTex, tikz, and GIMP unless otherwise specified. If you wish to make an apple pie from scratch, youmust first invent the universe. — Carl Sagan When most students first see […]
Every physicist would agree that the greatest paradigm shift of the last century was the quantum. Quantum mechanics has changed the face of science. It can explain everything. Electricity? Yes. How birds fly in one direction? Yes. Gravity? No. But, today we won’t get to that. Instead we will talk about another conflict in the […]
“There is a rumor in America that there are two intelligent races on Earth: Humans and Hungarians” — Isaac Asimov The Martians of Budapest”, sometimes referred to as simply “The Martians” is a colloquial term used to describe a group of prominent Hungarian physicists and mathematicians who emigrated to the United States following the Great Purge […]
Chess Chess and Mathematics At first glance, these two disciplines may seem unrelated. However, mathematics and chess have a large amount of unexpected connections. Computer Science Babbage, Lovelace, and The First Computer Long before Alan Turing revolutionized the field of computation, there was Charles Babbage and his difference engine. Babbage was a man of many skills, most notably in machinery […]
Set Theory My theory stands as firm as a rock; every arrow directed against it will return quickly to its archer. — Georg Cantor 06 Nov 2020 — 12 min read Image source My theory stands as firm as a rock; every arrow directed against it will return quickly to its archer. — Georg Cantor As quoted […]
Linear Algebra SVD decomposition and a demo of an image compression. Photo by Daniel Plan on Unsplash I structured this article in a way that you are gonna need your focus at the first section when I explain about the SVD decomposition and just after that you’ll get your reward which will be watching how it works in […]
The Dartmouth Summer Research Project on Artificial Intelligence was a summer workshop widely considered to be the founding moment of artificial intelligence as a field of research. Held for eight weeks in Hanover, New Hampshire in 1956 the conference gathered 20 of the brightest minds in computer- and cognitive science for a workshop dedicated to the conjecture: […]
History The Limit of Logic and The Rise of The Computer AI since Aristotle (Part 2) Analysis Cantor’s Diagonal Argument “Diagonalization seems to show that there is an inexhaustibility phenomenon for definability similar to that for provability” — Franzén… Philosophy Platonist Roger Penrose “Sees” Mathematical Truths What does Roger Penrose mean when he claims to “see” mathematical […]