Mass in relativity via cubic polyhedra

Recently Stern has discovered a formula that relates scalar curvature to the level sets of harmonic maps. Prompted by Stern’s formula, we find that the mass of an asymptotically flat 3-manifold has a geometric interpretation if evaluated along faces and edges of a large coordinate cube. In terms of the mean curvature and dihedral angle, the resulting mass formula relates to Gromov's scalar curvature comparison theory for cubic Riemannian polyhedra. In terms of the geodesic curvature and turning angle of slicing curves, the formula realizes the mass as integration of the angle defect detected by the boundary term in the Gauss-Bonnet theorem