The topology of a mathematics book — What does the structure of a math book really look like? What theorems are the most important ones? Reading and trying to understand a mathematics book is a fun and usually exciting experience. Still, sometimes it feels like when you are deeply buried in the Wikipedia jungle, constantly being sent…

A million-dollar sudoku — Back in the 60s and 70s, people began to realize that it wasn’t always enough to make a program that could run error-free on their computers while solving a problem. They quickly realized that the runtime of the computer program was an essential factor; in the extreme case, a program…

A world where you can divide by zero — Something is going on with infinity that we don’t fully understand. Physicists have long been assigning finite values to infinite divergent series (infinite sums if you wish) such as 1 + 4 + 9 + 16 + ⋅⋅⋅ ≃ 0 and 1 + 2 + 3 + ⋅⋅⋅ ≃ -1/12…

Explaining the periodicity of the exponential function by a nice piece of infinite geometry — Why is the natural exponential function periodic with imaginary period 2πi? How come there are infinitely many (complex) logarithms of a number? If I told you that it has to do with points on an infinite circle, you would probably think that I am a nutcase! …

The Beauty of Mathematical Moonshine and Kissing Spheres — The topic of Modular forms is in my opinion one of the pinnacles of pure mathematical beauty. Unfortunately, the theory is only being taught to a tiny group of people because of the apparent inaccessibility to the lower levels. Through this article, I hope to change that. I think that…

A surprisingly beautiful blend of algebra, geometry, and number theory — These curves defined by very simple equations are shrouded in mystery and elegance. In fact, the equations describing them are so simple that even high-school students would be able to understand them. However, a ton of simple questions about them remain unsolved despite tenacious efforts by some of the greatest…

One of the most beautiful equations in mathematics and the symmetry it displays — I once saw a t-shirt with the following print on it called a functional equation: What part of

Dirichlet characters - an indispensable tool in analytic number theory — One of the most powerful tools for studying primes is the theory of Dirichlet characters — a beautiful invention by the “father” of analytic number theory. Before explaining what they are, however, we will try to understand why they were needed in the first place. A Big Idea In France in 1805, a…

Knowledge sharing is hard — Can we get serious now?… I was once at a concert with Simon&Garfunkel and I clearly remember Paul walking up to the microphone saying “I need to say this”… All 40,000 spectators went pin-droppingly quiet and listened carefully to his next sentence. I always thought that this was a badass…