Those shapes look like they could be tiled with hexagons, which would create a hexagonal tiling of the sphere. Since the article mentions that this is impossible (and links to a proof), can anyone tell me what I'm missing in this image?
For instance, look at the border between the upper left and lower left islands. In this picture, those two islands are only touching along a third of the border that they share on the sphere. If you look at the leftmost point of where they are touching in this picture, that's a vertex that would have only two hexagons.
Edit: Basically he hasn't actually tiled a sphere with regular hexagons (which is what the proof said was impossible. He has titled a flat projection of a sphere with regular hexagons, which would have to be morphed to irregular hexagons if tiling a sphere when the projection was reversed.
Not taking away from the tiling, which is quite interesting in itself.