One of those circulating email messages caught my eye recently. In a tone of some awe, it discussed interview questions asked at Google, Facebook and other such beguiling 21st century firms. These are, I assure you, quite different from that silly staple of the job interview, “Where do you see yourself in five years?” These are instead questions designed to provoke thought, assess common sense.
At least, the awe suggests as much. Considering I once fell asleep while being interviewed for a job—I swear this is true—I’m curious about intriguing questions. Like this one: if your honey suddenly shrunk you to a centimetre tall and flung you in a blender, how would you save yourself from turning into human chutney when the blades start spinning?
No clue what bearing this has on how you might further Google’s fortunes, if you got the job. But one answer draws on something you may have noticed for yourself: the smaller an animal, the greater its relative strength.
Our cat, for example, routinely leaps down to the parapet below our window, but then back up. That’s about 5ft, or five times her length, upwards. Can I jump five times my height—30ft—off the ground? Meaning, three storeys up? If I did, I’d be one celebrated dude: the world record in the men’s high jump is just over 8ft. Not even one-and-a-half times my height.
And let’s not even mention fleas, those insect maniacs which can leap up to 70 times their height.
So what’s going on here? Are littler beings just endowed with super-strength muscles? Oddly, the right answer to that is “yes and no”.
No, because small animals don’t have any special musculature. Yes, because with smaller bodies to propel, ordinary muscles perform extraordinarily. That’s why you can go up the scale and imagine elephants speculating enviously: how do these puny humans manage to jump higher than they are tall? Do they have super-strength muscles?
Underlying all this is a compelling mathematical idea: when you increase the size of something, its volume (and thus its weight) increases faster than its surface area. Or the larger a thing, the smaller its surface area, relative to its volume. (Conversely, the smaller a thing, the larger its surface area to volume ratio).
Take two balls, 10cm and 20cm in diameter. The surface area of the smaller one is about 300 sq cm; the larger one, about 1,200. But their volumes are, respectively, about 500 and 4,000 cu. cm. That is, when we increase the diameter of a ball by a factor of two, we increase its area by a factor of four, but its volume by a factor of eight.
The pertinent point: if they were made of the same material, the larger ball—its diameter, remember, only twice that of the smaller—would weigh eight times as much as the smaller.
Or consider: when my daughter was half my height, the skin area she had to soap in the shower was about a quarter of mine (she always was cleaner than me). On the scale, she clocked about one-eighth of my weight.
As so often with mathematical precepts, there are fascinating implications of this surface area to volume relationship. For example, in 1877 the biologist Joel Allen proposed Allen’s rule: animals from colder parts of the world have shorter limbs and ears (appendages), relative to their size, than counterparts from warmer areas. Why? Note first that larger the surface area of an animal, the faster it loses heat, because heat escapes via the skin. (Think of sweat, please think of sweat right now). In cold places, animals want to stay warm and, thus, they use every possible strategy to retain as much heat as they can. One strategy: their various appendages evolve to become shorter.
There is evidence for the truth of Allen’s rule. The polar bear, which lives in the icy surrounds of the Arctic, is one of the world’s largest land animals. Yet it has generally stubbier legs than other bears, tiny ears and an almost non-existent tail: thus a smaller surface area to volume ratio than its bear cousins do. In 1972, the anthropologist Kenneth Beals studied people in different parts of the globe. He concluded that in colder areas, their heads were closer in shape to spheres, with smaller noses and ears. That’s because for a given volume, the closer you can get to a sphere, the less surface area you need.
The less skin you need. The less heat you lose. The warmer you feel.
So, in Google’s blender, use your now powerful leg muscles to launch yourself free of the blades. But you had better do it fast. Being tiny, your much greater skin area to volume ratio means you will quickly lose heat and start shivering uncontrollably. Not conducive to jumping.
PS: I got that job. Then they laid me off.
Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. A Matter of Numbers will explore the joy of mathematics, with occasional forays into other sciences. Comments are welcome at email@example.com
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