Complexity_ A Guided Tour - Melanie Mitchell [124]
This is illustrated in figure 17.1, in which a mouse, hamster, and hippo are represented by spheres. You might recall from elementary geometry that the formula for the volume of a sphere is four-thirds pi times the radius cubed, where pi ≈ 3.14159. Similarly, the formula for the surface area of a sphere is four times pi times the radius squared. We can say that “volume scales as the cube of the radius” whereas “surface area scales as the square of the radius.” Here “scales as” just means “is proportional to”—that is, ignore the constants 4 / 3 × pi and 4 × pi. As illustrated in figure 17.1, the hamster sphere has twice the radius of the mouse sphere, and it has four times the surface area and eight times the volume of the mouse sphere. The radius of the hippo sphere (not drawn to scale) is fifty times the mouse sphere’s radius; the hippo sphere thus has 2,500 times the surface area and 125,000 times the volume of the mouse sphere. You can see that as the radius is increased, the surface area grows (or “scales”) much more slowly than the volume. Since the surface area scales as the radius squared and the volume scales as the radius cubed, we can say that “the surface area scales as the volume raised to the two-thirds power.” (See the notes for the derivation of this.)
FIGURE 17.1. Scaling properties of animals (represented as spheres). (Drawing by David Moser.)
Raising volume to the two-thirds power is shorthand for saying “square the volume, and then take its cube root.”
Generating eight times the heat with only four times the surface area to radiate it would result in one very hot hamster. Similarly, the hippo would generate 125,000 times the heat of the mouse but that heat would radiate over a surface area of only 2,500 times the mouse’s. Ouch! That hippo is seriously burning.
Nature has been very kind to animals by not using that naïve solution: our metabolisms thankfully do not scale linearly with our body mass. Max Rubner reasoned that nature had figured out that in order to safely radiate the heat we generate, our metabolic rate should scale with body mass in the same way as surface area. Namely, he proposed that metabolic rate scales with body mass to the two-thirds power. This was called the “surface hypothesis,” and it was accepted for the next fifty years. The only problem was that the actual data did not obey this rule.
This was discovered in the 1930s by a Swiss animal scientist, Max Kleiber, who performed a set of careful measures of metabolism rate of different animals. His data showed that metabolic rate scales with body mass to the three-fourths power: that is, metabolic rate is proportional to bodymass3/4. You’ll no doubt recognize this as a power law with exponent 3/4. This result was surprising and counterintuitive. Having an exponent of 3/4 rather than 2/3 means that animals, particularly large ones, are able to maintain a higher metabolic rate than one would expect, given their surface area. This means that animals are more efficient than simple geometry predicts.
Figure 17.2 illustrates such scaling for a number of different animals. The horizontal axis gives the body mass in kilograms and the vertical axis gives the average basal metabolic rate measured in watts. The labeled dots are the actual measurements for different animals, and the straight line is a plot of metabolic rate scaling with body mass to exactly the three-fourths power. The data do not exactly fit this line, but they are pretty close. figure 17.2 is a special kind of plot—technically called a double logarithmic (or log-log) plot—in which the numbers on both axes increase by a power of ten with each tic on the axis. If you plot a power law on a double logarithmic plot, it will look like a straight line, and the slope of that line will be equal to the power law’s exponent. (See the notes for an explanation of this.)
FIGURE 17.2. Metabolic rate of various animals as a function