Posts tagged with "Mathematics"
After watching the excellent piece of exposition
on mathematical exploration by 3Blue1Brown guest Ben Syversen: “What was Euclid really doing?”, I discovered the fantastic web and mobile game Euclidea.
Geometry was never my favourite branch of mathematics, I always felt more drawn to algebra and more symbolic and abstract forms of reasoning. But Euclidea really does a masterful job of capturing the joy of exploration and problem solving through the lens of scaffolded puzzles of compass-and-straightedge geometric constructions.
Sometimes the mere existence of a question is dangerous. A colleague recently asked me whether it’s possible to validate credit card numbers in regex using the Luhn algorithm, and, well, I was thoroughly nerd-sniped. I’ll outline the problem briefly and we’ll try to explicate a solution together.
For what follows, it’ll be helpful if you’ve encountered concepts such as Deterministic Finite Automata (DFA), Regular languages and modular arithmetic before. If you haven’t, I think a quick skim through the Wikipedia pages should be sufficient to follow. I’ll also pair the mathematical formalisms with Python code in the exposition to make it easier to follow.
About a decade ago, whilst meandering through a late-night Wikipedia rabbit hole, I stumbled across the page for the Lambda Calculus.
A small footnote linked to An Introduction to Lambda Calculus and Scheme, a transcript of a short talk presented by Jim Larson, and, well, I fell in love. I loved the fact that there was this beautiful correspondence between foundational mathematics and models of computation; and that you could describe it through these symbolic wrappers called Lambda terms. There was something deep and abstract about it that I felt immensely drawn to.
There’s a lot of unsolved problems in mathematics. A lot. It’d be a bit imprecise to define a measure on the set of open problems, but I’d wager that it is much larger than the set of solved problems. In fact, for every solved problem, I’d conjecture that the process of solving it simply creates more open problems.
One such open problem that I find deeply unsettling is the Hadamard conjecture. Let’s define a Hadamard matrix and explain the conjecture:
MathJax v4.0 has been released!
I just integrated MathJax v3 with Hugo last night for this blog to start writing some maths posts, and v4 is released today…
I’m hoping it improves rendering though, I was seeing some issues with SVG output for inline formulae in Safari. KaTeX is neat and fast, but I find the quality of the typesetting to be quite sub-par compared to MathJax.
Manim is a Python animation engine designed to build animated mathematical explainer videos (examples here) created by Grant Sanderson of the superb YouTube channel 3Blue1Brown.
A couple years ago I wanted to write my own mathematics explainer videos with Manim, but found the learning curve to be pretty steep at the outset.
The user experience has since improved dramatically - in large part because of the Manim Community Edition, but also because there’s now enough samples in recent LLM training runs that they can produce workable Manim code!
I’ve been reading through Step-by-Step Diffusion: An Elementary Tutorial by Apple (arxiv). The mathematics underpinning diffusion language or image models is quite complex, but this walkthrough strikes a nice balance between concrete mathematical grounding and intuition.
My friend, Dania Freidgeim, created a probabilistic model of First Preference votes for the 2025 Australian Federal Election. Australia uses a Preferential voting (Instant-runoff) system, which makes this a really interesting exercise! Please go check it out!
TIL that the dimensionality reduction algorithm UMAP has support for inverse transforms!
Naturally, these are lossy inverses, but being able to generate novel examples of, for example, handwritten digits (yep, classic MNIST once again) is exceedingly cool.
The example images of handwritten digits here are sampled from the compressed planar space, and “inverted” into the original “image space”.
On a related note, the whole explanatory article for UMAP is a beautiful work of exposition, full of rich ideas that make one ponder deeply. I only wish I understood everything in here end-to-end.