Hodge-podge Theory
Remarkably, some of it is actually related to the Hodge Theory!

Fractal mountains

How to turn a 2-sphere in the 3-space inside-out

 Animation of the square in the plane.

The configuration space of planar realizations of the "square" linkage
(polygon which has four equal sides), is not a circle but the union of
three circles; each two of these circles share a common point.
The above animation illustrates three different irreducible components
of the configuration space. The realizations which look like parallelograms
correspond to one of these circles. The "folded" realizations correspond
to the two other circles. 

Look here for a beautiful animation of the Peaucellier linkage.

Animation of a buckyball (carbon nanostructure).