Almost all interesting systems are chaotic, in a mathematical sense. Chaotic behaviour can be thought of as taking place in a phase space whose dimensions are determined by the variables of interest – for example, position, x, and momentum, p. As the position and momentum of the object changes, it curves through phase space. If objects that start close together in phase space rapidly diverge you can be sure you're dealing with a chaotic system.
There's a difference between counting and measuring. You can never measure anything with absolute precision. This means that you can't tell, exactly, where an object is in phase space. If you measure it a thousand times as precisely as you can, you'll get lots of very slightly different measurements. In the end, your measurements result in a cloud of points, and the object is probably somewhere near the middle of the cloud. The more precisely you measure, the more closely clustered will be the cloud.
No matter how closely clustered, that cloud means that the object may move off on any of the neighbouring phase-space trajectories that start that close together. We don't quite know where the object is, and however tiny our ignorance at the start, we soon find our guess at where the object will land up at some later point becomes rapidly less certain. The nice tight cluster explodes into a delocalised scatter.
People (theoretical physicists are people too) used to think that Heisenberg uncertainty principle meant that spacetime has no features at scales less than the unimaginably tiny Planck distance. At the turn of last century, however, a physicist called Zurek found that phase space can have features smaller than the Planck scale.
His work shows that what limits structure is interference, not the uncertainty principle, and that structure expressed at this scale represents how likely quantum states are to decohere.
Decoherence is an important characteristic of the quantum world, one of whose consequences is that you and I aren't subject to quantum mechanical weirdness in our daily lives. Zurek found out that when coherent quantum states interfere with one another they set up something conceptually rather like this photo — fringes with sub-Planck dimensions. And quantum states interfere inevitably when the system is chaotic, and it is very hard indeed to conceive of states at that scale that are not chaotic.
Not that this has anything to do with this photograph, which is ripples on the surface of a pool. But I thought you might like to know about sub-Planck quantum chaotic interference. It's quite a conversation stopper.