Catmull–Clark subdivision
The most common subdivision rule for quad meshes, converging to a smooth surface related to bicubic B-splines.
Edwin Catmull and Jim Clark described the method in 1978 — the same Catmull who later led Pixar. Each step creates one new point per face (the average of its corners), one per edge, and moves every old vertex by a weight formula, then connects them into four times as many quads.
The limit surface is smooth (C²) everywhere except at extraordinary vertices, where it is C¹. It handles arbitrary topology, which made it the film industry's workhorse; Pixar's OpenSubdiv is the reference implementation. Don't confuse it with Catmull–Rom, an interpolating spline from the same pen.