Category Archives: Uncategorized

In three dimensions, the familiar cross product is a bilinear function expressible in terms of the Levi-Civita alternating tensor. Specifically, a = b × c can be written as ai = εijk bj ck, which has the following beautiful properties: Bilinearity: … Continue reading →

Given a finite alphabet Σ of size s and an integer n, a de Bruijn sequence is a cyclic string of symbols, such that each n-symbol string over Σ appears exactly once. For example, taking Σ to be an alphabet of … Continue reading →

As usual, I’ll start this post by mentioning the current state of the bounded gaps between primes project. The current values are , with an unconfirmed result giving a value of H below 5000. It’s surprising how far these sieve methods … Continue reading →

Several geometric problems and Diophantine equations can be converted into the task of finding rational points on elliptic curves. The canonical example is to determine for which rational numbers n can a right-angled triangle with rational side lengths have an area of … Continue reading →

The competition known as the Mathematical Ashes was created by analogy with the better-known cricketing Ashes, and is an annual competition between Britain and Australia. At the moment, Britain is in the lead, with Australia attempting to reduce the gap. … Continue reading →

The von Neumann universe V is the hierarchy of all set-theoretic sets. It is itself too large to be a set, and therefore is considered to be a proper class. There’s a very simple systematic construction of the von Neumann universe, … Continue reading →

The abbreviation TFAE (the following are equivalent) is often used in the statement of various theorems. Of course, a completely synonymous phrase would be ‘any one of these implies the other n − 1′. This then admits a natural generalisation … Continue reading →

A particularly important function is Klein’s j-function. It is defined on the upper half-plane of the complex numbers, and is incredibly symmetrical. For example, it is periodic, and thus invariant under translations by integers: It is also invariant under a larger … Continue reading →

A graph is vertex-transitive if its group of automorphisms acts transitively on the set of vertices. For example, the skeletons of uniform polyhedral are all vertex-transitive, as is the complete graph Kn. It’s obvious that a vertex-transitive graph must be … Continue reading →

Some polyhedra have lots of symmetries. For example, consider the omnitruncated dodecahedron: The interactive demonstration constructs it kaleidoscopically, by reflecting a fundamental region (the Schwarz triangle) in three mirrors. The same fundamental region, however, can result in lots of different polyhedra and … Continue reading →