Perspectives on scissors congruence (Q2797790)

Two geometric objects are called scissors congruent if the first one can be dissected into finitely many pieces that can be rearranged to form a dissection of the second one. The algebraic analysis leads to the general question if two objects are scissors congruent as soon as all their canonical invariants under dissection and rearrangement agree. The author comments in particular on the classical setting, where the given objects and pieces are polyhedra, dissections ignore boundaries and rearrangement is based on isometries of the underlying space, on the case where lower-dimensional intersections of pieces are no longer ignored (McMullen's polytope algebra) as well as on more general scissors congruence of varieties, where invariants are described by the Grothendieck ring of varieties.