## The maths and physics of walking in the sand

I love this – which I picked up from Ian Stewart’s now slightly out-of-date (e.g., pre-proof of Fermat’s Last Theorem) and out-of-print The Problems of Mathematics (but a good read and on sale very cheaply at Amazon) – because it demonstrates the harmony of physics with maths, is based on a common experience and is also quite counter-

intuitive.

Most of us are familiar with the experience – if you walk on damp sand two things happen: firstly the area around our foot becomes suddenly dry and secondly, as we lift our foot off, the footprint fills with water. What is happening here?

Well, it turns out that the sand, before we stand on it, is in a locally optimised packing state – in other words, although the grains of sand are essentially randomly distributed they are packed together in a way that minimises (locally) the space between the grains. If they weren’t then even the smallest disturbance would force them into a better packed state and release the potential energy they store in their less efficiently packed state.

This doesn’t mean, of course, that they are packed in the most efficient way possible – just as they are randomly thrown together they fall into the locally available lowest energy state (this is the physics) which is the locally available best packing (this is the maths).

But this also means that when we stand on the sand we cannot actually be compressing it – because that would actually imply a form of perpetual motion as we created an ever lower energy state/even more efficient packing out of nothing. In fact we make the sand less efficiently compressed – the energy of our foot strike allowing the grains to reach a less compressed  packing – and, as a result, create more space for the water in the surrounding sand to rush into: hence the sand surrounding our foot becomes drier as the water drains out of it and into where we are standing.

Then, as we lift our foot, we take away the energy that was sustaining the less efficient packing and the grains of sand rearrange themselves into a more efficient packing (or – to look at it in the physical sense – release the energy stored when we stand on the sand). This more efficient packing mean less room for the water in the sand and so the space left by our foot fills with water expelled from the sand.