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I am am work on my mathematics extended essay, and I want to do it on:

1. Coming up with a general formula for the volume of the five Platonic Solids (I need to know from other mathematics higher level students if this is two simple).

2. The proof(s) for Euler's polyhedron formula.

3. The Travelling Salesman Problem (not sure how much math is involved as opposed to computer science).

I am most interested in the first but willing to drop it if it is too simple. I am also open to suggestions of other interesting topics in geometry (the subject I have the most interest in).
I want to get some work done before the summer break ends in about 4 weeks, so I could really use some advice (especially considering my adviser has gone incognito).

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From doing some basic research, and as a Further Mathematics HL student, I think the volume for Platonic solids is too simple (v=rA/3) however Group Theory is a very interesting subject that you could apply to the symmetry of Platonic Solids. This might be too complex but maybe have a look into dual Polyhedra and their symmetry. 

The problem with Euler's polyhedron formula is that most of its beauty comes from its geometric proofs which might be hard to illustrate unless you're familiar with Illustrator it can be quite complex. This is at a mathematics HL standard, I just find a purely mathematical proof may lose some of the value to the proof and it's directly referenced in the option so it might be hard to go more in depth.

As for the travelling salesman problem, it's covered in the Discrete Mathematics part of the syllabus so that could be a link to the syllabus, however I think it's important that if you are using something directly from the syllabus you add something original, like an application of the problem to something in your everyday life.

It seems like you have an interest in both CS and Geometry, just from your ideas, in which case I would consider looking into Riemannian Geometry, specifically their uses in Neural Networks which might be more slightly challenging. I myself am doing my Extended Essay on Spiking Neural Networks, and Machine Learning as a whole, from a mathematical perspective, was one of the most interesting subjects I discovered. I would advise watching the series by 3blue1brown on neural networks as that's where I started and then going onto blogs/articles. You'd be surprised at the uses of these networks from Fake News detection, to Netflix Recommendations, and Neutrino detection (for Geometric Neural Networks).

 

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Thanks a lot. I actually ended up changing my topic to the splitting necklace problem. I'm presenting the solution on 3blue1browns channel and presenting my own solution(to the base case 2 jewel types) and comparing the two. Coincidentally, I'm actually quite good with illustrator (designing poster and diagrams for the school for 2 years does that). I lost interest in Eulers problem for a math EE though I'm still working on it for my IA.

One question though statistics for a math IA seems to be very popular in my class. Is there something inherently easier about it?

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Statistics is probably one of the easiest subjects to tick the box for the 4 marks (I think or maybe 2) that show your interest in the subject. However especially when dealing with creating your own probability distributions, or looking at bivariate statistics (In one of the options) and can be made quite formal, especially using hypothesis testing however it is quite hard to show anything beyond that level, as that's typically going into 2nd-year undergrad statistics. I believe a more pure mathematics approach to statistics could be proving that Var(x) = E(X^2) - E(X)^2, a commonly quoted formula at IB, and deriving all the variations of the formula from that. Or proving the combinations/permutations formula (the book 'The Enjoyment of Math' has a great proof that pops out of nowhere along with many other proofs). Typically though, stats is a common math SL IA as its fairly easy to show and not much known outside the course is needed (It's a good insurance 5-6) but I don't think it will score highly on the mathematical knowledge part (6 marks).

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