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.
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.
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.
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.
If you aren’t used to staring at math, Poisson’s equation looks a little intimidating:
∇2feven 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.
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.
The Tower of Hanoi is a famous mathematical puzzle. A set of disks of different sizes are stacked like a cone on one of three rods, and the challenge is to move them onto another rod while respecting strict constraints:
Only one disk can be moved at a time.
No disk can be placed upon a smaller one.
Many years ago, I read in Ripley’s Believe It or Not column that, in a Buddhist temple in India, there is a large room with three stout pillars on which 64 golden disks are stacked. A team of monks work tirelessly, transferring these disks between the three pillars. With one movement each second, the time required to solve the puzzle is about 585 billion years, at which time the world will either end, or enter another cycle.
However, Ripley was not always a reliable source: in fact, the puzzle was invented in 1883 by the French mathematician Édouard Lucas. The minimum number of moves required to solve the puzzle with disks is , the -th Mersenne number. There are algorithms to ensure a solution in the minimum number of moves (see Wikipedia page referenced below).
The “essence of mathematics”, which we have tried to capture in these problems is mostly implicit, and so is often left for the reader to extract. Occasionally it has seemed appropriate to underline some aspect of a particular problem or its solution. Some comments of this kind have been included in the text that is interspersed between the problems. But in many instances, the comment or observation that needs to be made can only be appreciated after readers have struggled to solve a problem for themselves. In such cases, positioning the observation in the main text might risk spilling the beans prematurely. Hence, many important observations are buried away in the solutions, or in the Notes which follow many of the solutions. More often still, we have chosen to make no explicit remark, but have simply tried to shape and to group the problems in such a way that the intended message is conveyed silently by the problems themselves.