The energetics of polycrystalline solids at high temperatures is treated using topological methods. The theory developed represents individual irregular polyhedral grains as a set of symmetrical abstract geometric objects called average N-hedra (ANH's), where N, the topological class, equals the number of contacting neighbor grains in the polycrystal. ANH's satisfy network topological averages in three-dimensions for the dihedral angles and quadrajunction vertex angles, and, most importantly, can act as "proxies" for irregular grains of equivalent topology. The present analysis describes the energetics of grains represented as ANH's as a function of their topological class. This approach provides a quantitative basis for constructing more accurate models of three-dimensional well-annealed polycrystals governed by capillarity. Rigorous mathematical relations, derived elsewhere, for the curvatures, areas, and volumes of ANH's yields quantitative predictions for the excess free energy. Agreement is found between the analytic results and recently published computer simulations. (C) 2005 Springer Science + Business Media, Inc.