Regge Calculus: Discrete Gravity on a Triangulated Sphere

n = 2
ε = 0.00
s = 1.00
Regge Calculus (1961): A discretization of general relativity on a simplicial complex (triangulation). Curvature is concentrated at vertices (in 2D) as deficit angles: ε_v = 2π − Σ_t θ_t(v), where θ_t(v) is the angle at vertex v in triangle t.

Regge action: S_R = Σ_v ε_v · A_v (area dual to vertex). For a triangulated S² the Gauss-Bonnet theorem demands Σ_v ε_v = 4π (Euler characteristic χ=2). As triangulation refines, the Regge action converges to the Einstein-Hilbert action ∫R√g d²x.

Vertex color encodes deficit angle: blue=negative curvature (saddle), red=positive (cone). A regular sphere has uniform positive deficit at every vertex.