Two results in discrete geometry II

Last quarter I gave a kiddie talk about classifying polyhedra, and cutting up polygons into equal area pieces. Today we'll instead be cutting up polyhedra and guarding polygons.

1. Given any two polygons of equal area, one can cut the first shape into finitely many pieces and rearrange them into the second. Does this result hold in higher dimensions? This was Hilbert's 3rd problem (in his famous list of 23 problems), and we'll recreate Dehn's remarkably simple proof that the result fails in 3 dimensions.

2. You're in charge of the security for an art gallery but have a very limited budget. What's the minimum number of guards you must employ to ensure that every artwork (every region of the gallery) is being watched at all times? We'll solve the art gallery problem and discuss some other variations of it.