Topology Films Project Nelson Max

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)

This video defines the meaning of a regular curve (a smooth closed curve) in the plane, and the rules for a regular homotopy between regular curves. It then defines the rotation number of a regular curve, and proves that there is a regular homotopy between two regular curves only if they have the same rotation number. This proof constitutes the first half of the Whitney Graustein theorem (see Hassler Whitney, "On regular closed curves in the plane" Compositio Mathematica, 4 (1937), p. 276–284.)

This video completes the proof of Whitney Graustein theorem, by showing that if two regular curves have the same rotation number, then there is a regularly homotopy between them. The construction of the regular homotopy proceeds by the same algorithm that was used to produce the computer animation in these two films.

This video presents two fractal curves which fill up a square. It starts by defining the concept of the limit of a sequence of approximation curves, and illustrates this by the circle, and the fractal Koch "snowflake" curve, which is shown to have infinite length. Then two limit curves that fill up a square are shown, (a) a variant of the Peano curve, and (b) the Sierpinski curve. The fractal self-similarity of these curves is illustrated by magnifying the approximations as they get more complicated, producing a repeating cycle.