Three segments: A) "Volume Filling Surfaces": three dimensional analog of the Sierpinski curve in the "Space Filling Curves" video, a limit surface which fills up a cube. The viewpoint moves closer to the center of the cube as the approximating surfaces become more complicated, producing a repeating cycle. B) "The Alexander Horned Sphere": topological sphere deformed to grow an infinite binary tree of intertwined horns, so that the space outside the Horned Sphere is not simply connected, that is, a rubber band wrapped around a horn of the limiting shape could never be removed. The viewpoint moves closer to one of the infinitely tangled limit points as the deformation proceeds, producing a repeating cycle. C) "Sierpinski's Curve Drawn as a Function of Time": Sierpinski curve of the "Space Filling Curves" video being drawn as it covers up the square, triangle by triangle. (The film-to-video transfer was aligned improperly so the starting point in the lower left corner of the square is not visible)