Category Archives: Uncategorized

I’ve been intending to write about the Leech lattice for a while now. I wanted to ensure that I could actually do it justice, and in particular make it strictly more comprehensive than the Wikipedia and Mathworld articles. The authoritative source on … Continue reading →

(This post is completely unrelated to the category of cones. Apologies if you were mistakenly directed here by a fuzzy search engine.) In May 2013, the category theorist Dr Eugenia Cheng (whom you may recognise from her quotations on the Imre Leader Appreciation Society) wrote a … Continue reading →

Some mathematical objects are completely boring. Others have a few interesting properties. The Petersen graph, by comparison, has a plethora of quite remarkable features that, taken alone, would each qualify the graph as being interesting. I’ve mentioned this graph a few times … Continue reading →

Given a real number x, computing its continued fraction can reveal a lot of information. In the simplest case, the continued fraction terminates if and only if the number is rational. One particular example of this is the rational , … Continue reading →

Breaking news: Patrick Stevens (whom you may recognise from the cp4space cipher-solving leaderboard) has published a collection of poems pertaining to Sylow’s theorems. Anyway, on the real projective plane, there are a couple of geometric theorems worth mentioning. The thing that’s interesting … Continue reading →

A fascinating paper by Louis Kauffman establishes the equivalence of the four-colour theorem (the assertion that any planar graph can have its vertices coloured with one of four colours, such that neighbouring vertices have different colours) to the following seemingly unrelated combinatorial … Continue reading →

Usually, it is very difficult to find a reasonably naturally-defined sequence which is not present on the Online Encyclopedia of Integer Sequences. The OEIS contains everything from the ubiquitous Bernoulli numbers and the fascinating Golomb self-referential sequence, to such utterly pointless trivialities as A171056 (the … Continue reading →

In this rather popular video, Vi Hart experiments with the patterns that can be formed by connecting the vertices of a regular polygon with straight lines of equal length. [youtube=http://www.youtube.com/watch?v=CfJzrmS9UfY] In addition to the obvious number-theoretic interpretation, there is a nice … Continue reading →

This article can be regarded as a sequel to my 5 × 5 monotile. Since its announcement, several variants have been discovered. In particular, towards the end of this article we shall present a three-dimensional monotile with the following properties: Nonperiodic by … Continue reading →

On an infinite square lattice, the B36/S125 cellular automaton proceeds similarly to Conway’s Game of Life (B3/S23), but with different birth and survival conditions. Specifically, a dead cell becomes live if surrounded by 3 or 6 live neighbours, and a … Continue reading →