Just something which I was looking at recently, by folding a rectangle from corner to corner you obtain a pentagon (albeit not a regular one). This got me thinking, what is the closest you could get to a regular pentagon? My approach (which I won't post yet since I haven't written it up neatly) focused on minimising the standard deviation of the side lengths of the pentagon and I thought was pretty interesting since it combines geometry, statistics and calculus. I'd be interested to see anyone else's approach since there is no real "correct" answer to this question it's really about how you interpret the idea of getting as close to a regular pentagon as possible (another obvious choice would be to minimise the standard deviation of the angles yet the calculus would probably get a bit hairy which is why I opted for the side lengths).

Very, very interesting Tobe. I will be sure to have a go a report back anything I find, I encourage you to post your solution when you're done with it!

PlanckTime - Amin