Perspectives on scissors congruence

In this paper we give a short introduction to the different theories of scissors congruence. We begin with classical scissors congruence, which considers equivalence classes of polyhedra under dissection. We then move to multi-dimensional scissors congruence along the lines of McMullen’s polytope algebra and then to the Grothendieck ring of varieties. Tying our discussion together is the question of whether algebraic invariants are sufficient to distinguish scissors congruence classes.