## AA1 and thinking like a mathematician

This week I’ve been dabbling a bit in one of the other M208 units available on OpenLearn, AA1: Numbers. I think this unit is supposed to be studied after all the Group Theory A and Linear Algebra books, but it seems pretty straightforward so hopefully studying it out of sequence won’t do me too much harm. In a way, the material in AA1 is quite familiar but at the same time it seems like we’re looking at these subjects in a more precise, rigorous way than in MS221. It’s nice, but a bit intimidating! I often get a bit anxious that what seems like a simple statement might have some deeper meaning which is sailing over my head. Still, if that’s the case I’m sure I’ll find out pretty quickly when it comes to the assignments.

## Getting back into group theory

I was kind of expecting M208 to be over my head, but so far I’m really surprised at how much material in the first few units of M208 is also covered in MS221, and at how gentle an introduction the Intro Block actually is. Very pleasantly surprised indeed! I’ve been working on Unit I3: Number systems and GTA1: Symmetry this week, and there were only really two new ideas introduced in these, so I feel like I’m getting a nice, steady warm-up session before the really hard stuff starts.

The topic in I3 that I’d never encountered before was equivalence relations. It took me a little while to get my head around the idea (and to stop thinking of the phrase “equivalence class” in its software testing context from M255), but I think I’ve got a fairly good grasp of it now. I even managed to give a coherent explanation of it to Alex, with the help of the “coloured blocks” example from the h2g2 Equivalence Relations page.

The unit about Symmetry introduces the two-line symbol notation for describing symmetries, which I’d never come across before, and to be honest at first I couldn’t really see the point of doing it this way. If I’m going to describe a reflection of a square in the vertical axis, for instance, I’d much rather write ${q_{\pi /2}}$ than

$\left( {\begin{array}{*{20}{c}}1 & 2 & 3 & 4 \\4 & 3 & 2 & 1 \\\end{array}} \right)$.

But later on in the unit, I watched the video section about symmetries of Platonic solids, and it certainly sounds a lot easier to write down the symmetries of a tetrahedron in a two-line symbol than trying to figure out how to represent it in the form ${q_\theta }$!

Speaking of Platonic solids, I absolutely love the animations of them on the Wikipedia Platonic solid page. I could watch this octahedron spinning all day!

Octahedron image by Cyp, under the Creative Commons Attribution ShareAlike 3.0 License

## Attack of the Cycloids

This week I’ve been wrapping up Unit I1: Real functions and graphs, and making a start on I2: Mathematical language. The last few sections of I1 introduce some really interesting-looking graphs – I particularly like cycloids and cardoids. Partly because the graphs remind me of Spirograph pictures, and partly because the names sound like monster robots from a classic sci-fi film!

Cycloid

Cardoid