The first frosty morning of the year, around here. It is gloriously fresh and bright. Oblique sunshine, filtering through gaps in the curtains, picks out detail on the bookshelves opposite the bedroom window as we wake to the gentle sound of the builders across the way starting up their cement mixers. The frost remaining on the shaded roofs allows us to compare the quality of our neighbours’ loft insulation.



An hour later, as we cycle to her gymnastics class, my daughter asks “Daddy, why do your fingers get colder than the rest of your hand?”

The answer, of course, is surface-to-volume ratio. Following the spherical cow approach so beloved of we physicists, fingers are cylinders and the rest of the hand is more of a blob. A spherical blob, at a stretch. Anyway for a given volume (in which heat can be stored) a finger has more surface area (through which heat can be lost) than the rest of the hand. There are probably other contributory factors to cold fingers, such as the difficulty of circulating blood to extremities, but that’s biology, anatomy even, and therefore too hard not fundamental.

I like surface-to-volume ratio though, it’s an example of why scaling nature down, or up, in size doesn’t really work - or at least, it shows why you have to change a lot more than just the size of things if you want them to function properly. An ant the size of an elephant just wouldn’t work, its legs would snap because the strength they need to support the ant increases in proportion to its mass, and therefore its volume, whereas the support provided by the legs would increase roughly with their cross-sectional area. Before I outgrew the smug pleasure of debunking sci-fi films, I enjoyed using such arguments to demonstrate why Raquel Welch’s surgical task force was unfeasible in Fantastic Voyage. It was also disappointing to realise that atoms, even in the semi-classical picture of Niels Bohr where electrons orbit the nucleus like planets, could not really be mini-solar systems housing civilisations like our own.



Similar scaling effects mean that when special effects teams blow up small models, they have to film in slow motion or the size looks wrong (of course, there’s the added benefit that explosions look cool in slow motion). And this is why a speed-of-light cataclysmic explosion such as a supernova will nevertheless be observed as majestic expansion over many days, even years. Even at the speed of light, covering the distances involved takes a long time.



Both the Bohr atom and the supernova touch on another reason things don’t scale, of course. The speed of light is an absolute, so there is a limit to how much you can change the speed of things to account for differences in size. And in the case of the atom, we know that electrons really aren’t at all like planets, whatever else they are. Their behaviour is fundamentally different from that of a classical particle, and is entirely quantum mechanical. The scale at which quantum behaviour kicks in is set by Planck’s constant and, like the speed of light, it sets a limit on how far things can be arbitrarily scaled up or down before there is a qualitative change in their behaviour.

Although surface-to-volume ratio does that for ants, and fingers, even without help from Einstein or Planck.



Now, my daughter is significantly smaller than me, so overall she gets colder more quickly anyway. And, given that “Gloves that work” features on her Christmas list, an equally valid answer to her question could have been “Because you need better gloves, and I’m a bad dad.” But the physics happens too.

Jon Butterworth has written a book about being involved in the discovery of the Higgs boson, Smashing Physics, available here. Some interesting events where you might be able to hear him talk about it etc are listed here. Also, Twitter.