Here in Germany (in Bavaria, to be exact, the school systems differ in our 16 states), we have to do a so-called "Facharbeit" in Gymnasium (Gymnasium is, in a nutshell, the school you go to if you want to go to university. It's a bit more complicated than that, of course, it's Germany after all), which is roughly equivalent to what they call an "extended project" in the UK. I did mine in maths, on the topic of "Nonlinear Optimisation and its Applications". It is, alas, in German, (Nichtlineare Optimierung und ihre Anwendungen) although I might translate it at some point in the future.
So, a brief summary: I explain how minima and maxima of multidimensional functions can be found with multi-variable calculus and also with the Nelder-Mead method, a numeric algorithm. Then, I name a few real life applications of it and explain Molecular Modelling - a method to determine the three-dimensional structure of molecules by minimising its calculated energy. Because I was extremely bored at the time, I also wrote a very simple implementation of it in Java and C. (using the Nelder-Mead method, which is not perfect for this and therefore a bit slow, leaving out the "simulated annealing" part and, because I only had data for carbon and hydrogen, it only works with saturated hydrocarbons and, with a trick, for fullerenes) I used it to calculate structures of propane, butane, dodecane, cyclohexane, decalin and a C60 fullerene with surprisingly good results, as you can see on the pictures below.
The paper and all the illustrations in it that were made by me (see table of figures) are licenced under CC-BY-SA. (Should you, for some reason, need to use something from it, but not under the terms of the licence, you can of course just send me a mail and ask me)
