the angle defect
Cut a paper cone open and lay it flat: the triangles around the tip do not add up to a full turn — there is a wedge of missing angle. That missing wedge is the angle defect, and it is exactly where the sharpness of the cone hides. On a triangulated surface, every interior vertex has some triangles meeting around it; sum their corner angles at that vertex, and the amount by which the sum falls short of 2*pi is the curvature concentrated at the vertex.
Concretely, at an interior vertex v of a triangle mesh, let theta_1, ..., theta_m be the interior angles of the triangles incident to v at v. The angle defect is d(v) = 2*pi - (theta_1 + ... + theta_m). If the vertex were flat (the surface could be unrolled around it without tearing or overlap) the angles would sum to exactly 2*pi and d(v) = 0; a positive defect means a dome-like, sphere-like vertex, and a negative defect means a saddle, where the angles overshoot 2*pi. This single number is the discrete analogue of integrated Gaussian curvature: in the smooth world Gaussian curvature K is spread continuously over an area, and on a mesh the integral of K over a small region around v collapses to the lump d(v) at the vertex. For a boundary vertex one uses pi minus the sum of angles instead.
The angle defect is beautiful because it is intrinsic — it depends only on the edge lengths through the law of cosines, not on how the surface sits in space — and because it converges, in a suitable averaged sense, to smooth Gaussian curvature as the mesh is refined. It is the discrete curvature that makes the discrete Gauss-Bonnet theorem an exact identity rather than an approximation: summing d(v) over all vertices gives 2*pi times the Euler characteristic, on the nose, for any triangulation. The standard caution is that the pointwise defect divided by an area (to get a curvature density) is mesh-dependent and converges only weakly; the robust, convergent object is the integrated defect itself.
At a vertex of a cube where three squares meet, each corner angle is pi/2, so the angle defect is 2*pi - 3*(pi/2) = pi/2. The cube has 8 such vertices, and 8 * (pi/2) = 4*pi = 2*pi * chi, matching chi(S^2) = 2 — the corners alone carry all the curvature of the sphere the cube approximates.
All curvature of a flat-faced polyhedron lives at its vertices as angle defect; the faces are flat and the edges carry none.
Do not divide the defect by a dual-cell area and call the quotient 'the' discrete Gaussian curvature without saying so: that density depends on the chosen area assignment (barycentric vs Voronoi) and converges only weakly, whereas the bare angle defect is the canonical, exactly-summable quantity.