“Imagine the cow is a sphere…” is the punchline for many versions of a popular allometric physics joke. Allometry, by the way, is the study of how organisms scale. The canonical example is the 1950s horror film Them!, where giant mutant ants threaten a New Mexico town. The real horror is that the film writers didn’t understand that ants could not possibly grow to the size Shaquille O’Neal in the off-season.

Imagine an ant a set of spheres...

Why? Consider the heuristic ant above, which we imagine is composed of three roughtly spherical sections with rod-like connections. Now what happens when we double the scale, $\ell$ of the ant? The mass of the ant goes as its volume, i.e. $m \propto \ell^3$. Most of this mass is concentrated in the head, thorax, and gaster (the three round sections) which are held together with rod-like connections (neck and petiole). The shear strength of these rod-like bits to hold up the massive parts go as their cross sectional area, $s \propto \ell^2$. Or, said in another way, the ant’s exoskeleton scales roughly as the area.

Thus at some scale $\ell_0$ the ant becomes too massive for its support structure to keep it together. Lawrence Krauss opens his book Fear of Physics with this parable, explaining that (1) one cannot expect to grow arbitrarily large cows for uber-milk efficiency and that (2) this is why brontosauruses had such small heads relative to their bodies.

This kind of analysis known to physicists as dimensional analysis. While one might think that dimensional analysis is only useful for making back-of-the-envelope estimates, we will see in a subsequent post how it can be used rigorously to understand the renormalization group. Undergrads will already be familiar with `rigorous’ dimensional analysis, however, in the context of mechanics, where the use of similarity transformations. As a quick reminder, we can take the force law:

$m \frac{d^2\mathbf{r}}{dt^2} = - \frac {\partial U}{\partial \mathbf{r}}$,

and note that if we scale $t \rightarrow t'=\alpha t$ and $m \rightarrow m' = \alpha^2 m$ then the above equation is still true. Thus we can conclude that the velocity of a particle’s motion in a central force field is halved when its mass is quadrupled. (Update, 22 Dec: this, of course, only holds when $U$ is independent of $m$!)

The above example comes from chapter 2.11 of Mathematical Methods in Classical Mechanics by Arnold, which contains three of the neatest problems to be found in any physics textbook, which I reproduce here. (It’s worth noting that Arnold cites Smith’s Mathematical Ideas in Biology for these problems.)

1. A desert animal has to cover great distances between sources of water. How does the maximal time the animal can run depend on the size $L$ of the animal?

To be explicit, one can imagine the animal is a sphere (though the scaling holds even if it weren’t a sphere). The animal fills up on water at one lake and must run to the next lake, which is just at the limiting distance that it can run before dehydrating iteslf. The amount of water the animal can store goes as its volume, $W = \alpha L^3$ while the rate at which water is perspired away is proportional ot the animal’s surfac (e area, $R=\rho L^2$. We assume that the rate of perspiration $\rho$ is constant and hence over a time $t$ the animal perspires a volume $Rt = \rho L^2 t$. Setting this equal to the volume of water stored $W$, we see that the maximum time $t_m$ goes as its length $L$. To be exact, $t_m = \frac{\alpha}{\rho}L$, but we’re not interested in overall constants.

2. How does the running velocity of an animal on level ground and uphill depend on the size $L$ of the animal?

The trick here is to think about the power (energy per time) used. On level ground, the main resistance to motion is air resistance, which goes as the cross sectional area times the squared velocity $F \propto v^2L^2$. The power is obtained by multiplying by velocity, so that we have $P \propto v^3 L^2$. What is the power output of the animal? This goes as the heat output, which is proportional to the animal’s surface area, so that $P\propto L^2$. Setting these two equal we see that $v \propto L^0$. The running velocity of an animal on level ground does not depend on its size! (Well, it can be proportional to the logarithm of its size… as anyone who has done “Naive Dimensional Analysis” on divergent integrals would remind you.)

In the uphill case, however, the main resistance comes from gravity, with $F=mgh$. Taking the slope of the hill to be constant, the power is then $P \propto mgv \propto L^3 v$. Setting this equal to $P \propto L^2$ from the surface area argument above, we get $v \propto L^{-1}$. Arnold notes that a dog will easily run up a hill while a horse will slow its pace.

3. How does the height of an animal’s jump depend on its size?

The energy required to jump to a height $h$ is $mgh$, thus has the proportionality $E \propto L^3 h$. Muscles can produce a force proportional to $F\propto L^2$ (e.g. the strength of bones is proportional to their cross section), while the work accomplished by this force goes as $W \propto FL \propto L^3$. Thus setting the energy equal to the work, we find $h \propto L^0$. Arnold notes that “a jerboa and a kangaroo can jump to apporximately the same height.” (Tall and short basketball players have roughly the same leaping ability, but being taller makes it easier to dunk.)

I hope you enjoyed that as much as I did.