Abstract Algebra Adventures Part 1: Introduction

I'm one of those people who had good grades in Math back in school but who promptly forgot a lot about the subject as soon as it no longer really mattered. As an average non-mathematician adult nerd, I do occasionally read math-related articles on Wikipedia for fun, and I'm subscribed to several "MathTubers", but for most of my life I couldn't really say that math was a passion of mine, and it shows by how little of what I read/watch actually sticks to my head.

I think I'm starting to develop a taste for abstract algebra, though. You could say it became my pandemic hobby. 😆 Take note that I never took any abstract algebra classes back in college, so this was quite a journey for me, to say the least.

No, I still haven't taken any formal classes or anything like that. So far I've only been learning things bit by bit, not as part of a systematic abstract algebra course, but as stepping stones to answer questions that actually interest me.

For example, I once wondered how to position three points on the complex plane in the "nicest" way possible. At first "nicest" was pretty vague and I didn't really know exactly what I was looking for, but eventually I learned about the third roots of unity \(-\frac{1}{2} \pm i\frac{\sqrt{3}}{2}\) and their many cool properties. As I learned more, my questions became more informed and more specific, and soon enough I started having a better understanding and appreciation for certain math topics:

• roots of unity and cyclic groups in general
• quaternions, octonions, etc. (I already knew about \(i^2=j^2=k^2=ijk=-1\), but I didn't start playing with quaternion rotations and stuff like that until all this happened)
• Geometric/Clifford algebras
• other unital algebras over the field of real numbers
• regular and uniform polytopes and their connection to abstract algebra

I also learned way more linear algebra than I bargained for. As someone who's always had an irrational dislike for matrices, I have begrudgingly accepted their utility, especially since there are tools I could use to easily work with them without doing things by hand.

Speaking of tools, websites like Wolfram Alpha and matrixcalc.org, as well as the small programs that I wrote to study the things mentioned above, all gave me enough confidence to create my own algebraic structures despite my lack of mathematical talent.

As for the value/usefulness of those structures I created, well, I have no idea. I'll describe them later in this series of posts, then maybe actual mathematicians could take a look and provide their insight if they wish. But at this point it's all just fun and games to me.

Continued in Part 2.