I'm always really impressed by the "Japanese" diagonal wrapping technique.

Professional at work: https://www.youtube.com/watch?v=1qi8ZXUH_wY

Technique explanation: https://www.youtube.com/watch?v=BQM8KKP-3Oo

here's the real hard one, the cylinder, wrap it around like this aaaaaaaand skip!

we'll never know how to close cylinders sides I guess

(ok isn't hard but annoying as hell https://youtu.be/dH_gpSfaA6E?t=1m55s )

Neat. I wonder how this problem could be better posed mathematically. Minimize both the number of folds and surface area of wrapping used subject to the constraint the surface of the gift is completely covered? Yikes. It seems like predicting how a fold in the wrapping would affect the available wrapping left to cover the gift would be difficult.

That would be a research subject in the mathematics of origami.

That field is fairly advanced (See for example http://www.langorigami.com/science/science.php), but I am not aware of people studying 'boxes'.

I imagine you could get more usage out of a roll of wrapping paper if you could minimize any paper overlap. But it would take for more time to cut the paper.

I wrapped some presents last night that did not turn out great. Measuring the paper to minimize overlap makes a pretty big aesthetic difference.