To pop back into math-speak: The general form of the allometry equation is y = axb. If b is greater than 1, the trait gets proportionately larger, like deer antlers; if b is less than 1, as is the case with metabolic rate or babies’ heads, it gets proportionately smaller. A good analogy is two bank accounts with different rates of compound interest. Over time the amount of money in each account will diverge, but each account’s growth rate will remain the same. If b is zero, the y variable remains constant. If b is less than zero, the trait declines with increasing size. This is what happens with relative, or cellular, metabolic rate. If metabolic rate is proportional to body mass raised to the power of 3/4, then relative metabolic rate will be proportional to this figure divided by simple mass (mass to the power of 1, in other words)—which equals body mass to the power of −1/4. Kleiber helped show that the idea of allometry could extend beyond solid anatomical features such as horns and into the body’s processes, such as metabolism. Biological growth, like financial growth, is a compound process—another reason logarithmic axes are more useful than arithmetic ones when comparing different-sized animals. Cells arise from cells, so the amount of new material added in any time period depends on how much there was before.
Allometries are an example of something called a power law, so called because the y variable depends on the x variable being raised to some power. Power laws spread far beyond biology. The frequencies of earthquakes and landslides of different sizes follow them, as do the sizes of air bubbles in a breaking ocean wave or the length of the waking periods during a night’s sleep. They also apply to social patterns, such as the length of time patients in the United Kingdom wait for a hospital operation, or the size of changes in financial markets, the frequency with which words appear in a language, and the popularity of people’s names. Power laws also relate the size of an event to its frequency. Crudely put, in systems that follow negative power laws, the probability of events such as quakes or stock market crashes declines at a constant rate as those events get bigger. Small earth tremors are common; city-destroying catastrophes are rare. As for sleep patterns, most of the time you will wake up long enough only to turn over and go back to sleep, but occasionally you’ll lie awake for what seems like half the night.