(Revision 2024-12-09: corrected cos angle-sum formula and "rot" where I should have said "rot-angle") I cannot stand unmotivated steps in mathematics. Everything must make sense. Even if the historical development of a subject followed a haphazard path, due to "bad luck", I insist that there must exist a sense-making account of the field following an … Continue reading Challenge: Derive a 3D Rotation Calculus Without Magic Tricks
Category: Maths
Notes on single-var calculus
Calculus is really cool and rich. NB: This isn't what I've been researching for the past few months -- that's coming soon. I'm sick of having this 2-year-old draft sitting around so I'm finishing it. NB: I read this back and my eyes do the usual thing where all the equations turn into so I … Continue reading Notes on single-var calculus
Notes on Galois Theory
Or, with apologies to any French readers Réflexions sur la résolution des articles mathématiques incompréhensibles par blog posts. Since the year 2013, seemingly every springtime I become interested in the theory of algebraic equations / polynomials / Galois theory. Ever since I was a teenager I have really really wanted to understand why the general … Continue reading Notes on Galois Theory
Values, Diffs and Functions
We awaken in a world of "things" (e.g. points, states, versions). We call these values. From any ordered pair of values (a, b) we can form a diff from the first to the second. We write this as a→b, pronounced "a to b". (A better terminology could perhaps be simply absolutes and relatives, or abses … Continue reading Values, Diffs and Functions
Notes on the Differences Between Things
This series will deal with the following truly widespread concept: the absolute "state", and the relative "change between states". Some examples: Points and Vectors Certain areas of mathematics (e.g. homogenous co-ordinates in computer graphics) distinguish between points -- i.e. multiple origins -- and vectors, the "displacements between points". You can't really do any operations on … Continue reading Notes on the Differences Between Things
Antidote for the next time I forget how to do 3D math
--- EMERGENCY PROCEDURE 49-B SELF-RECOVERY MANUAL --- Welcome, Agent. We're very glad you stumbled across this document, intentionally or not. Now, you may think this is the first time you've been here -- and because of the nature of the subject we are about to discuss, this makes perfect sense. But I'm afraid I have … Continue reading Antidote for the next time I forget how to do 3D math
What are types? Part 2
WARNING: I've had this mainly completed article sitting around in my drafts for a long time. I am so unsatisfied with it, but I haven't been able to figure out how to improve it yet. So I may heavily re-work or just retract this sometime in the future. In the meantime, it's not benefitting anyone … Continue reading What are types? Part 2
The “proving too much” parlour trick
In mathematics and logic we are taught a nice trick. Even if you cannot prove a statement true, if you have a counterexample at hand, you can immediately prove it false. But in practice, this is only a trick; if something is so false that it cannot be refined into correctness, then a mere counterexample … Continue reading The “proving too much” parlour trick
What are types? Part 1
There are these things that, depending on your definition, many or all programming languages use: 'types'. There's also a rich mathematical study of types in Type Theory which, along with related disciplines, has many connections to logic and proof. Why? Often, they take the form of explicit 'annotations' to program artefacts, big and small. For … Continue reading What are types? Part 1
Inventing the Quaternions, Part 1: A Wizard Did It
When learning some mathematics, there are two main types of magic. The first kind is when you are presented with a fact, theorem, result–it may be surprising or mundane–but you do not yet have the means to prove it. For example, you might have just learned about complex numbers, and I might assure you that … Continue reading Inventing the Quaternions, Part 1: A Wizard Did It
Programming, Made Complicated 