Posts
Jan 14
Jan 14

Paul Nelson has solved the subconvexity problem, bringing mathematicians one step closer to understanding the Riemann hypothesis and the distribution of prime numbers.

Dec 19, 2021
Dec 11, 2021
Nov 27, 2021
Nov 27, 2021
Nov 20, 2021

Lagrange multipliers are used in multivariable calculus to find maxima and minima of a function subject to constraints (like “find the highest elevation along the given path” or “minimize the cost of materials for a box enclosing a given volume”). It’s a useful technique, but all too often it is poorly taught and poorly understood. With luck, this overview will help to make the concept and its applications a bit clearer.

Nov 11, 2021

What the heck is a Chu space? And whatever it is, does it really belong with all the rich mathematical structures we know and love?

Say you have some stuff. What can you do with it?

Maybe it’s made of little pieces, and you can do a different thing with each little piece.

Nov 3, 2021
Oct 13, 2021

Sometimes you discover a nice-looking path through the forest, and you follow it thinking it’ll lead you to some new, exciting place. But as you walk and walk amongst the trees, you come to realize it’s actually leading you somewhere you already knew. This article is a story about (re)discovery, and how new things can make us understand old things better.

Before we start, though, I want to make clear I’m in no way an “expert mathematician”. Every single piece of mathematical knowledge I’ve used in this article, I was taught in high school. And I have forgotten so much of it, in fact, that for the integrals in this article I had to resort to Wolfram Alpha every single time. ¯_(ツ)_/¯

This also means this article is likely to have innacuracies, and a fair share of things an actual mathematician would consider silly. For any such things (or anything else you want), you can reach me by email at jon at this domain, or on Twitter at jon_valdes.

Oct 12, 2021
Sep 10, 2021

Michael Freedman’s momentous 1981 proof of the four-dimensional Poincaré conjecture was on the verge of being lost. The editors of a new book are trying to save it.

Aug 31, 2021
Aug 18, 2021
Aug 12, 2021
Jul 31, 2021
Jul 31, 2021
Jul 17, 2021

In my recent post on mapping Perlin noise to angles, I was put on to the subject of Curl noise, which I thought I understood, but did not. I figured out what Curl noise really was in a subsequent post and then posted my earlier incorrect (but still interesting and perhaps useful) concept of Curl noise in yet another post. Although I kind of understood what Curl noise was at that point, I wanted to give myself a more complete understanding, which I usually do by digging into the code, making sure I understand every line 100% and seeing what else I can do with it, trying to make multiple visualizations with it to test my understanding, etc.

Jul 16, 2021
Jul 14, 2021

Mathematician: That depends on what you mean by “universe.”  Here’s a framing:

A circle of radius R centered at a point P is the set of all points in the plane with distance R from P.  The diameter D of this circle is twice the radius, but can also be thought of as the longest possible straight-line path from a point on the circle to another point on the circle.  The circumference C of the circle is its arc length.  By definition, Pi = C/D.

Jul 9, 2021

The Collatz conjecture is an allegedly-unsolved problem in mathematics.

I say allegedly there because, reading those words, I feel like mathematicians are going to jump out from my closet yelling, “You got punk’d!” for being duped into believing it’s unsolved.

Jul 6, 2021

If you aren’t used to staring at math, Poisson’s equation looks a little intimidating:

∇2f=h

What does ∇2f even mean? What is h? Why should I care?

In this post I’ll walk you through what it means, how you can solve it, and what you might use it for.

Jun 25, 2021

We’ve all heard of integers, rationals, reals, even complex numbers, but what on earth are surreal numbers? They are a beautiful way of defining a class of numbers which includes all reals, but also ordinal numbers; i.e. all the different infinities and even infinitesimal numbers. Not only this but we get a full system of arithmetic for all these numbers. Ever wondered what (∞−1) is, or √∞ ? Before we get stuck into that, let’s learn some history.

It all started a long, long time ago in a galaxy far, far away (Cambridge in the 1970s). A man by the name of John H. Conway was playing Go, an ancient Chinese game that’s very elegant in itself. After much thought, he realised that the later stages of the game could be thought of as the sum of many smaller games. Conway then applied his ideas to other games like Checkers and Dominoes. It seemed that these games were behaving as if they were numbers.