A natural extension of splines into three-dimensional space, subdivision surfaces were first explored in 1978 by Edwin Catmull & Jim Clark. They are a recursive means of approximating the limit of a surface. Their canonical use is in CGI rendering for animated films, as they allow artists to produce complex and smoothly shaded geometry from very simple piecewise linear polygonal control structures (demonstrated below). They have been used in virtually every Pixar Animation Studios film since A Bugs Life (1998), Toy Story 2 (1999), Cars (2006), Brave (2012), and so on.

Additional manifold-with-boundary test control meshes can be downloaded courtesy of Carnegie Mellon University: http://www.cs.cmu.edu/afs/cs/academic/class/15864-s04/www/assignment1/

Loop Subdivision
Loop Subdivision Wireframe