Using Möbius Maps to Manipulate the Complex Plane
Audience: high-schoolundergraduate
Tags: complex-analysis
As with all olympiad addicts, when I started university maths, I had to deal with a massive wave of depression when I realised that all those years I spent learning olympiad tricks, theorems and techniques, were useless. "I'll never need cyclic quadrilaterals again", I thought. Even worse, Euclidean geometry was my favourite subject - but it seems to vanish off the face of the earth as soon as high school finishes. Well, this all changed in my second year of university, when I took a complex analysis course that BLEW MY MIND. It made me realise that olympiad geometry has a use after all! There's a beautiful concept in complex analysis called a Möbius map, and it turns out to be very similar to olympiad inversion (if you know what that is). Today I'll do my best to build the intuition from the ground up. I'm going to teach you how to use Möbius maps to beat up the complex plane like it owes you money.
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Pretty nice interactive elements. Also, this aligns with my interest in CGA, so I guess I’m the target audience:)
Really liked the interactivity of the diagrams. The clear demarcation of proofs, definitions, and examples was really good for breaking up the text and making it more digestible. I think this would have been improved by fully working the example that was presented at the outset (and maybe leaving a different problem as an exercise). I also think it would have been improved by explaining a bit more why this is important. At the beginning there was mention of continuous deformation of shapes but it was never mentioned again. The explanation and visuals were very clear throughout but at the end I still felt myself missing an “aha” moment.
I love the interactive diagrams!
I like how there are interactive elements to help along the way. It might be nice to have practice questions that go beyond ABCD and I would personally prefer it if there was a more tangible example but it was an enjoyable experience overall!
I love the format and how clean everything looks. Very informative and interesting, just if the grid lines could be extended it would be INCREDIBLE.
++ diagrams / interaction. Felt very cohesive, perhaps end is slightly rushed?
Webpage is very good, well written and explanations are easy to follow
Very interactive and well-written
Easy to follow for the theory, but would be better if the final example had steps to trace through. After reading the text, I still wasn’t able to think of how to begin finding the parameters for the mobius map. I believe the hint probably gives some parameter (something related to b and d, possibly).