[zyklu5]

The guy being interviewed (Daniel Litt) has a great expository video on Hilbert's third problem at Numberphile. (Here)[https://www.youtube.com/watch?v=eYfpSAxGakI]

Hilbert's Third was the first of his 23 problems to be solved. It asks whether volume is enough to determine whether one can transform one polyhedron into another via cutting into finite pieces and re-arranging with rotation and translation (such polyhedrons are called scissor congruent). The situation is true in 2 dimensions.

Dehn showed that's not the case for polyhedron. He created his eponymous invariant in order to solve the problem: two polyhedrons with the same volume but differing dehn invariants are not scissor congruent.

I love the result because it was one of the first bits of serious math I read as an undergrad and it was also the first time I got a glimpse into how tensor products could be used to package information.

A pretty clear account can be found in Harthshorne's Geometry: Euclid and Beyond.