energy needs are the result of its particular physical proportions, chemical makeup, evolutionary history, and environment. Nevertheless, most measurements are close to Klieber’s 3/4, particularly if you look at a large number of species across a large range of body sizes—allowing the big picture to emerge from the fog. By 1960, species ranging in size from elephants weighing several thousand kilograms down to microbes weighing less than a millionth of a millionth of a gram, or about 20 orders of magnitude, had all found a place on Kleiber’s line.

Quarters, Quarters, Everywhere

It wasn’t just metabolic rate that caught the attention of allometry seekers. Biologists got to measuring and comparing just about everything they could. As the power law equations piled up, a pattern began to emerge. Other biological processes, such as mammals’ dietary requirements for nitrogen and vitamins, and the rate at which they produce urine, also scale to body mass to the power of 3/4. An animal’s heart rate is proportional to its body mass to the power of −1/4, the same as the amount of fuel burned per cell. For every 1,000-fold increase in body mass there is a nearly sixfold decrease in heart rate. Many biological times, such as life span, time spent in the womb, and time between birth and maturity were found to be proportional to body mass raised to the power of 1/4, so a 1,000-fold increase in body mass leads to a sixfold increase in life span. Walking speed also increases with the 1/4 power of body mass. Similar rules apply to biological structures. The cross-sectional area of the aorta, the largest blood vessel, scales as the 3/4 power of body mass. Plants joined in the fun: The area of tree trunks scales in the same way as that of aortas. Other features abandon 1/4 but stick with multiples of it; for example, hemoglobin’s ability to seize hold of oxygen declines slowly as animals get larger, with an exponent of −1/12.

The number 4—in 3/4, 1/4, −1/4—or multiples of 4, such as in the case of hemoglobin, came up again and again. It is not surprising that scaling laws should be related, because the different bits of an animal



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