Thursday, February 05, 2015

The Joy of Estimation

One of the tragedies of growing older is that just as one’s wisdom and insight expands and unfolds like some exquisite water lily then the willingness of other people to pay any attention to what you’re going on about takes a serious plunge. This leads to an increased use of the phrase ‘if I had my way’ to preface every suggestion. Sad really, but there it is.
Anyway, if I had my way, children would be taught the art of estimation from an early age. Not only is it fun but it promotes deep insight and, more importantly, develops a sense of proportion. A simple example might be something like: “Imagine each year as a 1cm mark on a ruler. Now cut a piece of string that matches your age; if you are 10 years old then it will be about as long as a pen. Now imagine a piece of string that matches the amount of time since the last dinosaur was alive (not counting our feathered friends). How long would it be? Would it stretch to the bottom of the garden, to the next town or from here to London? You’re allowed to know that it is 66 million years since the dinosaurs disappeared so it’s simply a matter of getting a feel for how long a 66 million centimeter piece of string would be. The answer is 660 kilometers or about the same as traveling from London to Paris and back. That’s quite a long way - or a long time ago.
Of course that’s a fairly easy one.
Slightly more difficult is to work out roughly how many grains of sand there are on all the beaches of the world and whether this is more or less than the number of stars in the universe. I can recommend this as an alternative to counting sheep when trying to fall asleep.
Let me take you through it: 
(arithmophobics are excused if they choose get off at this stop).
Let’s start with the easy bit — the number of stars in the universe. Our galaxy (the Milky Way) is said to consist of about 100 billion stars. That’s 1 followed by 11 zeros or 1011. The number of galaxies in the universe is about the same - 1011 so if we assume that our galaxy is about average that makes the total number of stars 1022 or ten thousand billion billion (as Brian Cox would put it). The point is not so much the number — it’s fairly meaningless — the point is to compare this number with the number of grains of sand on the beaches of the world and to find out how different the two numbers are.
The grain of sand calculation is a bit more difficult so let’s warm up by working out the number of pebbles on Chesil beach. Chesil beach is a magnificent shingle bank 29 km long, 200 metres wide and 45 metres high. If you haven’t been there, I can recommend (I believe the correct term is heartily recommend) a visit. You really have to sit on it to get a feel for just how many pebbles there are there.
So if we think of Chesil beach as a 29km long shoe box it’s easy to work out it’s volume - about 26 million cubic metres or, allowing for the fact that it isn’t square but is a sort of rounded hump, about half that - say 10 million cubic meters. That’s one of the tricks about estimating, by the way; as long as the number of noughts is about right you can be a bit free with the 6s and 7s.
Now the pebbles on Chesil beach are interesting because the natural action of the waves and currents have graded them so that at one end the pebbles are all about the size of frozen peas while at the other end they’re the size of tennis balls. This leads to the tale that local fishermen, landing somewhere along the beach on moonless nights were able to tell where they were from the size of the pebbles. So let’s choose an average sized pebble — say the size of a grape — and work out how many there are in a cubic metre. Thinking of each pebble as about 1cm in diameter and box-shaped rather than round gives the answer - about a million per cubic metre. And so the total number of pebbles in Chesil Beach is (very roughly) 1013 or 10 thousand billion. That’s about 100 times more than the number of stars in the Milky Way - which is disappointing as I was hoping that the two numbers would be about the same.
When it comes to working out the number of grains of sand on all the beaches of the world, we come up against what is known as the coastline paradox or the fact that measuring the length of a coastline will give different answers depending on the length of your tape measure. Not to be daunted, Wikipedia quotes the CIA’s World Factbook in giving the total length of the world’s coastline as 1,162,306 km. (or roughly 1 billion metres) so let’s take this to be the right figure. If we assume that this length consists of around 10% beach (as opposed to rocky cliffs, mangrove swamps and so on) then the total volume of sand — assuming an average beach is about 2 metres deep and 50 metres wide to the water’s edge — is around 1010 cubic metres.
Sand grains are smaller than pebbles - let’s say about 1mm diameter so there are 109 (or 1 billion) in a cubic metre. So the total number of grains of sand on all the beaches of the world is about 1019 or about a thousand times fewer than the number of stars in the universe (give or take a few).
In all of this, there’s one number that stretches the imagination more then any other — 1011 — the number of galaxies that are thought to exist in the universe — about the same as the number of pebbles in a 300 metre section of Chesil beach. When you consider that each galaxy is made up of thousands of millions of stars, it is a monstrous number that makes a mockery of the imagination.
Maybe I’ll go back to counting sheep.