This is a good article about Gödel’s incompleteness theorem.
Richard Feynman’s Integral Trick: I remember that in Surely You Are Joking, Mr. Feynman, Feynman talked about an obscure trick for doing integrals. What was he talking about? This article explains it.
An Infinite Universe of Number Systems - The p-adics seem to be a fanciful formalism of numbers. What’s interesting about it is that it provides a way to study an entire university of number systems.
Let's make some spirals
Disk point picking is an elementary problem in geometry. To evenly distribute points on a disk, an intuitive idea is to sample pairs of numbers in polar coordinate (radius and angle) randomly from uniform distributions. After all, “uniformly” is just another word for “evenly”, isn’t it? But it doesn’t work. When the points are sampled this way, the density of the dots fall off linearly with the distance to the origin.Read more
Gradient descent on a non-Euclidean surface
Is the salesman travelling on foot or on an airplane? This article describes an experiment to develop a version of the elastic net algorithm that works on spherical surfaces. I needed it for a computational neuroscience problem, but for those who are mainly interested in machine learning, it also serves a simple and intuitive demonstration of using gradient descent on non-Euclidian surfaces. The Tensorflow source code is available on GitHub. I also made a youtube video showing the algorithm in action.Read more