Other Studies of Topology

After having gone back and read the article associated with the diagram I posted I realize that the author, Yasushi Kajikawa, was a contemporary/collaborator of Buckminster Fuller and was researching "Synergetic Topology." Perhaps the field of Synergetics could be a side discussion in class, but basically the periodic table of the polyhedra that Kajikawa produced demonstrates that any Platonic or Archimedean polyhedron can be collapsed (folded) into three basic shapes: tetrahedron, octahedron, icosahedron.
Discussion of the donut and the coffee mug reminded me that the Poincaré conjecture had been in the news recently (2006) as the only one of the seven Millennium Prize Problems to be solved. The conjecture (now theorem) is slightly beyond my grasp, but a New Yorker article (Manifold Destiny) describes the general concepts:
"From a topologist’s perspective, there is no difference between a bagel and a coffee cup with a handle. Each has a single hole and can be manipulated to resemble the other without being torn or cut. Poincaré used the term 'manifold' to describe such an abstract topological space. The simplest possible two-dimensional manifold is the surface of a soccer ball, which, to a topologist, is a sphere—even when it is stomped on, stretched, or crumpled. The proof that an object is a so-called two-sphere, since it can take on any number of shapes, is that it is 'simply connected,' meaning that no holes puncture it. Unlike a soccer ball, a bagel is not a true sphere. If you tie a slipknot around a soccer ball, you can easily pull the slipknot closed by sliding it along the surface of the ball. But if you tie a slipknot around a bagel through the hole in its middle you cannot pull the slipknot closed without tearing the bagel."
Another use of polyhedra, more recently in the news, was originally studied by Lord Kelvin. From a New York Times article (A Problem of Bubbles Frames an Olympic Design):
"Lord Kelvin studied foams to try to understand the 'ether,' the medium through which he and others thought light propagated. In his work he wondered what would be the most efficient foam — how space could be partitioned into cells of equal volume that would have the least surface area.
True bubbles were not the answer, of course, because there would be gaps between the spheres. The answer Kelvin came up with used 14-sided polyhedrons."
In 1993 the Weaire-Phelan solution proved that two polyhedrons of equal volume, one of 14 sides and one of 12, that nest together in groups of eight was the configuration with the least surface area. The fact that the polyhedral matrix was sliced at an angle and used to construct the geometry of a space frame for the roof and wall structure of the Water Cube doesn't make the design topological, but does create a visually interesting yet repetitive pattern.