Metabolic rate scales with body mass, but not linearly. A mouse doesn't burn energy ten times faster than a creature one-tenth its weight — it burns energy roughly 5.6 times faster. The scaling exponent is approximately 3/4, not 1. This three-quarter power law (Kleiber's law) has been observed across organisms spanning twenty orders of magnitude in mass, from bacteria to whales. The regularity is astonishing. The explanation is contested.
Physical approaches to metabolic scaling attempt to derive the 3/4 exponent from first principles. The most influential — West, Brown, and Enquist's fractal network model — argues that the exponent reflects the geometry of resource distribution networks (blood vessels, bronchial trees) optimized to service a three-dimensional body through a space-filling fractal. The 3/4 appears because 3/(3+1) = 3/4, where 3 is the spatial dimension and 1 is an additional dimension from the branching hierarchy.
But competing physical models produce the same or similar exponents from different assumptions. Surface-area models predict 2/3 (body heat loss scales with surface area). Cell-level models connect metabolic rate to the rate of molecular diffusion within cells. Thermodynamic models derive the scaling from entropy production constraints. Each framework has its own geometric or energetic argument, and each predicts an exponent close to 3/4 — but for different reasons.
The review of physical approaches reveals that the 3/4 law may be multiply determined: several independent physical constraints all push toward the same exponent. The regularity isn't one mechanism's signature — it's the intersection of several constraints, each of which would produce a similar scaling on its own.