World Flags Quizzer

A game to teach you world flags. 🇨🇳🇮🇳🇺🇸🇮🇩🇵🇰🇧🇷🇳🇬🇧🇩⋯ Read more

A Simple Example of a Connection With Torsion

While learning general relativity as an undergrad, I was uneasy with the idea of torsion of an affine connection. Familiar phrases like “torsion measures how a frame twists as it undergoes parallel transport” seemed too lofty to serve as a helpful mental model.

The following example is one which gave me an aha! moment.

A “rolling” connection on $ℝ^2$.

Consider a connection on the plane defined in such a way that a vector undergoing parallel transport rotates in the $xy$-plane in proportion to its motion along the $x$-direction. That is, define a connection so that the vector field defined by

Timeline of Scientists

Interactive timeline of famous scientists throughout history, using Wikidata. Read more

Explicit Baker–Campbell–Hausdorff–Dynkin formula for Spacetime via Geometric Algebra · arχiv

In brief. In $≤4$ dimensions, there’s a simple formula for the bivector $σ_3$ in terms of bivectors $σ_1$ and $σ_2$ such that $e^{σ_1}e^{σ_2} = ±e^{σ_3}$.

Abstract. We present a compact Baker–Campbell–Hausdorff–Dynkin formula for the composition of Lorentz transformations $e^{σ_i}$ in the spin representation (a.k.a. Lorentz rotors) in terms of their generators $σ_i$:

This formula is general to geometric algebras (a.k.a. real Clifford algebras) of dimension $≤ 4$, naturally generalising Rodrigues’ formula for rotations in $ℝ^3$. In particular, it applies to Lorentz rotors within the framework of Hestenes’ spacetime algebra, and provides an efficient method for composing Lorentz generators. Computer implementations are possible with a complex $2×2$ matrix representation realised by the Pauli spin matrices. The formula is applied to the composition of relativistic $3$-velocities yielding simple expressions for the resulting boost and the concomitant Wigner angle.

Mathematical One-liners

A collection of succinct but wonderfully satisfying mathematical results. Read more

Light installation for Gatherings Restaurant

Projector light show made for a multi-sensory dining experience. Read more

An Overview of the Strong CP Problem and Axion Cosmology · PDF

Literature review supervised by Dr. Jenni Adams at the University of Canterbury.

I wanted to learn more about particle physics after my Bachelor’s, so a year of part-time study culminated in this literature review. I learned basic classical (and a little quantum) field theory, and read about the “strong $CP$ problem” of quantum chromodynamics and its solution via “axions”. The report is aimed at the mathematically-inclined graduate who is uninitiated in particle physics.

The Many Faces of Stokes’ Theorem

Interactively tabulate the special cases of Stokes’ theorem, \( \int_Ω \dd ω = \int_{∂Ω} ω \). Read more