Net (unfolded) — fold along edges
Assembled polyhedron
Alexandrov's theorem (1941):
Given any polyhedral metric on S² with non-negative curvature at every point (angle sum ≤ 2π), there exists a unique convex polyhedron realizing it.
Angle defect: K(v) = 2π − Σθᵢ
By Descartes: Σ K(vᵢ) = 4π
Vertices: —
Faces: —
Edges: —
Euler χ: —
V − E + F = 2 (sphere topology)
Given any polyhedral metric on S² with non-negative curvature at every point (angle sum ≤ 2π), there exists a unique convex polyhedron realizing it.
Angle defect: K(v) = 2π − Σθᵢ
By Descartes: Σ K(vᵢ) = 4π
Vertices: —
Faces: —
Edges: —
Euler χ: —
V − E + F = 2 (sphere topology)