Geodesic Sphere & Dome

Geodesic Sphere: Buckminster Fuller popularized geodesic domes in the 1950s, earning a patent in 1954. Start with an icosahedron (20 triangular faces), then subdivide each triangle into n² smaller triangles and project vertices onto the sphere. At frequency n, each original edge is split n times, giving V = 10n² + 2, E = 30n², F = 20n². Euler's formula V − E + F = 2 always holds for any convex polyhedron — a topological invariant proved by Descartes (c. 1630) and Euler (1758).