The Triangular Face Theorem: Surface Energy and Structural Modes in Convex Polyhedra

For convex polyhedra with exclusively triangular faces, we prove that 2F = V + E − 2, where V, E, F are vertices, edges, and faces. This identity connects surface counting (faces) to structural counting (vertices and edges) through the Euler characteristic χ = 2. For the icosahedron, this gives 2F = 40, which combined with Klein's local mode count of 41 = (V − 1) + E reveals a fundamental 40/41 threshold: surface constraint (40) versus local accessibility (41). The one-mode difference reflects the distinction between global topology and local geometry. We generalise to surfaces of arbitrary genus and note applications to materials science.