The Dehn invariant of a polyhedron is , where is the length of the edge , is the corresponding dihedral angle, and is an additive functional defined on a certain finite-dimensional vector space of reals over the rationals for which [1]. A polyhedron has Dehn invariant 0 if and only if it is equidecomposable with a cube of same volume.

This Demonstration calculates Dehn invariants for disjoint unions of Platonic and Archimedean solids with icosahedral symmetry and edge length 1. In this case, dihedral angles are supplementary to angles between suitable pairs of adjacent axes of rotational symmetry. The disjoint union of an icosahedron, a dodecahedron, and an icosidodecahedron is an example of a combination of polyhedra with Dehn invariant 0. Find some other combinations with Dehn invariant 0.