Author Archives: apgoucher

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 →

There are several items of news worth mentioning. Firstly, I have been informed that the current status of the ‘bounded gaps between primes’ effort has now reached H = 6966, which is impressively low for these sieve methods. According to … Continue reading →

The partition numbers (sequence A000041) count the number of distinct ways to partition n identical objects. For example, p(5) = 7, as there are seven distinct ways to partition 5 objects: This visualisation, where partitions are graphs whose connected components are … Continue reading →