Weird things in a bucket of water
That's all you need to punch a geometric hole in water. Take a look. When the bucket is rotated so fast that the depression in the central vortex reaches the bottom, it can develop a cross-section shaped like triangles, squares, pentagons and hexagons. My story about it is here.
Harry Swinney at Texas says that this isn't unexpected – a symmetry-breaking wavy instability is bound to set in eventually as the rotation speed rises. Harry has seen related things in rotating disk-shaped tanks (without a free water surface) created to model the flows on Jupiter (see Nature 331, 689; 1988).
The intriguing question is whether this has anything to do with the polygonal flows and vortices seen in planetary atmospheres – both in hurricanes on Earth and in the north polar circulation of Saturn. It's not clear that it does – Swinney points out that the Rossby number (the dimensionless number that dictates the behaviour in the planetary flows) is very different in the lab experiments. But he doesn't rule out the possibility that the phenomenon could happen in smaller-scale atmospheric features, such as tornadoes. Tomas Bohr tells me anecdotally that he's heard of similar polygonal structures having been produced in the 'toy tornadoes' made by Californian artist Ned Kahn – whose work, frankly, you should check out in any case.