Author Archives: apgoucher

It is straightforward to position a cube such that its vertices are all integer coordinates. For instance, we could choose its eight vertices to be (±1, ±1, ±1). Similarly, the vertices of a regular octahedron can be positioned at cyclic permutations … Continue reading →

Suppose you were cast away on a desert island. You’re only allowed to take a maximum of eight known theorems with you, along with rudimentary results such as ZFC axioms together with ‘boring stuff’ such as mathematical induction over the naturals, commutativity … Continue reading →

In the not-too-distant future, people are going to be celebrating the 100th birthday of Bill Tutte. He’s not quite as well-known as Alan Turing, which is a shame since they were both equally invaluable in the cryptanalysis at Bletchley Park. You … Continue reading →

The Perrin sequence is defined by a recurrence relation, and is qualitatively similar to the Lucas sequence. The initial terms are and subsequent terms are defined by , and summarised in the following spiral of equilateral triangles: As you can … Continue reading →

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 →