The problem of scale

Learning anything about physics always involves grappling with the problem of scale. Whether incredibly small or ridiculously large, the human brain hasn’t evolved to cope with the kind of extreme number necessary for investigation into nature’s workings. Science writers do their best to think up real-world analogies, but the nature of the number makes this extremely difficult to do.

If you created a scale model of the solar system in which the sun was a metre wide, Earth would be just under 1cm across, 100m away; Jupiter would be 10cm across, 560m away; Pluto would be just over 1.6mm across, 4.5km (2.8 miles) in the distance. This is already getting hard to grasp – the sheer amount of empty space boggles the mind. Outside of the solar system, the nearest star – Alpha Centaurai – would be 29km (18 miles) away. The centre of the milky way? 188,340,398 kilometres away – on our scale model it would be placed about a fifth of the way between Mars and Jupiter. Current estimates are that there exist around 100 billion galaxies within the visible universe.

Time’s just as bad. The best analogy I’ve heard came from Richard Dawkins: if you stretch out your arm and use the distance from the centre of your body to your fingertip to represent the time since the formation of Earth, brushing the edge of your fingernail would wipe away the dust representing the entire period of man’s existence.

At the other end of the size scale are atoms and molecules. The average small glass of water contains about 225 cubic centimetres of water. This works out at 7,525,000,000,000,000,000,000,000 molecules of water. This is far larger than the number of sand grains in the Sahara, and probably more than there are on the entire planet.

Richard Feynman came up with a good analogy for representing the size of an atom: if you enlarged an apple to the size of the Earth, the atoms would be roughly the size of the original apple. What’s amazing is that we can directly view atoms, in some cases, using electron microscopes. But that’s only half of it, because atoms themselves are almost completely empty, to a staggering degree. This website is a scale model of a hydrogen atom (on most computer monitors). Electrons and protons are close enough in size that they can be visualised in reference to each other, but the distance between them cannot. Make the electron the size of a pixel, and the proton about 1000 pixels across, and there’s still eleven miles of emptiness between the two.

The most insane analogy I ever read was from string theorist Brian Greene, when talking about the relative sizes of the theoretical strings which comprise all matter. He says that a string is to an atom as a tree is to the known universe. Unsurprisingly, nobody’s yet come up with a way to verify the existence of something so small.

Large numbers are sometimes useful for debunkage. Homeopathic ‘remedies’ supposedly increase in potency as they are diluted. A relatively standard amount of dilution is ’30C’, which means that the amount of ‘remedy’ is to reduced 1 in 100, thirty times over. When you figure it out (I’m going by what somebody tells me on this one, since I keep messing up the calculations) this is equivalent to placing one cubic centimetre of substance into a sphere of water with a radius of 800 light years, stirring until randomly distributed, then taking a cupful. It’s highly unlikely there’ll even be one molecule of the original substance in the cup, and that’s amongst the aforementioned 7 million billion billion molecules. And there are other remedies which claim potencies of 200C.

Hopefully the maths in the above stands up to scrutiny – let me know if I’ve made any silly mistakes! For some reason these numbers are astonishing me even more than usual, today.