The important property of an egg timer from a day-to-day point of view is that it keeps time. That the time it takes for the sand to flow from the top bulb, through the throat, to the bottom bulb is reliable. Put another way, if you time a large number of eggs, the observed times will exhibit a normal distribution with a mean very close to three minutes and with a very small standard deviation. Your boiled egg will be just right nearly all the time.
Given that the dynamics of the large number of grains of sand in an egg timer are complicated, I suspect this simple but important property to be some consequence of the law of large numbers, a law which asserts that all kinds of repeatable things, never mind how complicated, will tend to exhibit normal distributions if you repeat them often enough. And, including as it does two of the most important numbers in the world, π and e, the normal distribution is clearly something rather special.
But the first pause is for the word repeat. Generally speaking, if you invert an egg timer, the sand will flow steadily from the top bulb, through the throat to the bottom bulb until the top bulb is empty. The number of grains per second will be very steady for most of the time and, more important, the flow will not stop. But sometimes, even if we assume that the sand is clean and dry, it will stop. The grains of sand at the bottom of the top bulb will interlock in some tricky way and make an arch over the throat, an arch which can take the weight of all the grains above. The equilibrium will be unstable, but it will be an equilibrium. The arch might persist for a long time, perhaps until some external shock disturbs it. In practise this does not happen very often, perhaps another manifestation of the law of large numbers.
We ignore the likelihood that a small number of grains of sand will stay stuck to the sides of the top blub, or perhaps around the throat, even when all the rest of the sand has fallen through to the bottom bulb.
While the relative size of the throat and the grains of sand is important, too small and the sand will arch. Maybe the throat being a magic seven times the diameter of that of the grains of sand is enough for stoppages to be very unlikely indeed. But is there actually any upper limit to the diameter of throat that can be bridged by grains of sand of any given size?
The second pause if for the word large. How many grains of sand are there in an egg timer? I guess my teeth brushing egg timer to contain about 2cc of sand. If we suppose the grains of sand to be cubes with 0.2mm sides and the fill level to be 100% we get to a quarter of a million grains, which strikes me as rather too many. So we raise the side to 0.33mm and drop the fill to 50% (devising a configuration of cubes which will give this fill is left as an exercise for the reader) we get to 27,000, not much more than a tenth of the earlier figure. So all we know so far is that we do not have much idea about how many grains of sand there are.
In practise, the grains are not going to be cubes, but sticking with the sort of sand one might get by crushing rock, how complicated are they going to be? How many complications do we need to allow in order to be able to model the behaviour of the sand in the egg timer?
Cubes and spheres are easy to describe, but neither is likely to result, not in large numbers anyway, from crushing rock. But we might suppose that the grains are convex polyhedra – that is to say ‘solids in three dimensions with flat polygonal faces, straight edges and sharp corners or vertices’. We might further restrict them by saying that the polyhedra cannot have more than so many vertices, that the minimum diameter must be more than some amount and that the maximum diameter must be less than some other, larger amount. The grains must be at least vaguely spherical, rather like, in shape at least, the stones which make up the ballast on which railway lines are laid. Bearing in mind that the sieves usually used to grade such materials can enforce minimum diameters but not maximum diameters.
We can then make piles of such polyhedra and calculate the forces at the points, lines & surfaces of contact. A bit long winded but straightforward enough, provided always that we can assume that the pile is not moving about. A problem in statics rather than dyanmics.
For present purposes we can perhaps neglect the possibility that over time, over many uses, the grains of sand in the egg timer will abrade. The grains will get smaller and more rounded, like the pebbles on the beach, and a much finer dust will start to accumulate around them. A dust which might, in principle, start to get in the way of the clean and simple workings of the egg timer.
However, when we move to dynamics, things still get rather difficult. I suspect that the problem of solving the equations of motion of numbers of polyehedra falling down through a glass tube while tumbling over each other, sliding & scraping across each other and bouncing off each other is computationally intractable. I seem to remember that solving the equations of motion of three idealised spheres moving around each under the force of gravity was bad enough – and that sounds almost trivial compared to the present problem.
But there is hope. We can change scale, we can move up a layer and rather than try and model things at the grain level, we can model them at the millimetre level, rather in the way that weather forecasters model at the kilometre level. Another trick is to chop time into lots of very small intervals and to move one’s system forward, one step at a time. A wheeze which might work, for example, with brains where, with a small enough interval one could proceed with the assumption that in any one such interval, each neuron is either firing or not. One can think about preservation of energy and of the various kinds of momentum. One can look at pretty pictures of currents and storms moving through the space in question. The authors of the paper at reference 1 do this with a version of the egg timer by means of watching the way that layers of coloured sand distort over time. Coloured sand of the sort which one can buy at Alum Bay. See reference 2.
In this way, I dare say that one can come up with equations which describe the gross behaviour of the egg timer with a reasonable degree of accuracy. Which is where I will let the matter rest on this occasion.
In the next episode we will move from the law of large numbers to the law of conservation of information, the IT version of the first law of thermodynamics, the one about the conservation of energy.
Reference 1: mechanics of the sandglass - A A Mills, S Day and S Parkes – 1996. This short and accessible paper contains some scientific background. Google will find it if you stick in the first sentence as a search term.
Reference 2: http://www.theneedles.co.uk/.
Reference 3: http://psmv2.blogspot.co.uk/search?q=egg+timer.
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