## 18 December 2007

### Scaling arguments

There's a classic argument, supposedly due to Galileo, for why mice and elephants are shaped differently, and there are no organisms that have the general mammalian body plan but are a hundred feet tall.

The argument goes as follows: assume that all animals have roughly similar skeletons. Consider the femur (thigh bone), which supports most of the weight of the body. The length of the femur, l, is proportional to the animal's linear size (height or length), S. The animal's mass, m, is proportional to the cube of the size, S3. Now, consider a cross-section of the bone; say the bone has cross-sectional area d. The pressure in the bone is proportional to the mass divided by the cross-sectional area, or l3/d2. Bones are probably not much thicker than they have to be (that's how evolution works), and have roughly the same composition across organisms. So l3/d2 is a constant, and so d is proportional to l3/2 or to s3/2. SSimilar arguments apply to other bones. So the total volume of the skeleton, if bones are asusmed to be cylindrical, scales like S7/2 -- and so the proportion of the animal taken up by the skeleton scales like S1/2. What matters here is that if S is large enough then the entire animal must be made up of bone! And that just can't happen.

Unfortunately, the 3/2 scaling exponent isn't true, as I learned from Walter Lewin's 8.01 (Physics I) lecture available at MIT's OpenCourseWare. It's one of the canonical examples of dimensional analysis... but it turns out that it just doesn't hold up. I suspect, although I can't confirm, that this is because elephant bones are actually substantially different from mouse bones. It looks like for actual animals, d scales with l (or perhaps like lα for some α slightly greater than 1), not with l3/2. Lewin uses this as an example of dimensional analysis; he also predicts that the time that it takes an apple to fall from a height is proportional to the square root of that height, which is true, but that's such a familiar result that it seems boring.

(Watching the 8.01 lecture is interesting. The videotaped lectures available on OCW are from the fall of 1999, which is two years before I entered MIT; occasionally the camera operator pans to the crowd of students, and a few of them look vaguely familiar.)

P. S. By the way, this blog is six months old. Thanks for reading!

Edited, Wednesday, 9:14 AM: Mark Dominus points out that the argument is in fact due to Galileo, and can be found in his Discourses on Two new Sciences. This is available online in English translation.

Anonymous said...

i've been listening to his electricity/magnetism course - he has aged between the two tapings.

Anonymous said...

in science-fiction circles
this is called the square-cube law.
the wikipedia article i just linked
led me to j.b.s. haldane's entertaining
on being the right size.
happy half-birthday; long live GPD!
v.

Althras said...

Sorry to dig this up after so long, but I'd like to point out that he wasn't doing dimensional analysis for Galileo's theory. He was doing scaling only (the act of reducing the number or related variables in order to create an easier to handle model of the world).

He only started doing dimensional analysis for the apple falling part.

I am now trying to figure out why he purposely showed a "wrong" example of scaling. Perhaps he is trying to point out that dimensional analysis is more strict and is thus a more reliable means of approaching modelling the real world.