If variety is the spice of life, the uniformity of quarter-power scaling is its mystery. For decades, scientists have puzzled over the fact that a wide range of an organism's traits--from its average life span and number of offspring to the typical duration of its pregnancy and its pulse--vary according to its mass (M) raised to some multiple of 1/4, regardless of its outward design.

Indeed, in a mouse or blue whale, the respiratory rate is proportional to M^-1/4, and the metabolic rate to M^3/4. So too, the diameters of tree trunks and human aortas both scale as M^3/8. All of this explains why elephants breathe more slowly and live longer than, say, chickens. But it doesn't answer why nature should use one ruler to draw all of life's plans.

Image: SANTE FE INSTITUTE POWERHOUSES. Geoffrey West (left) and James Brown (right), together with Brian Enquist, have offered an exiting explanation for the all-pervasive quarter-power laws in biological scaling.

At last, the mystery might be solved. Researchers from New Mexico recently published a compelling explanation in Science. To construct their argument, James H. Brown, an ecologist at University of New Mexico, Albuquerque, Brian J. Enquist, a post-doc in his lab, and Geoffrey West, a particle physicist from Los Alamos National Laboratory, refined ideas they had put forth in a 1997 paper. In essence, they invoke fractal geometry to make an end-run around what had been one of the most baffling aspects of quarter-power scaling laws--namely, why they aren't cube-roots instead.

The German physiologist Max Rubner first made the case for cubes in 1883. Drawing on Euclidean geometry, he reasoned that if an animal is N times taller than another, then its surface area should be N² greater and its mass N³ greater. Its total metabolic rate, then, which depends on the amount of heat it sheds, should vary according to its surface area, N², which is proportional to M^2/3. And the energy it burns per unit of mass--its specific metabolic rate--would be proportional to M^2/3 divided by M, or M^1/3. (A decade later, Rubner discovered a key fact about metabolism: the energy the body produces from food equals the energy a fire releases burning that food.)

Among very similar species, this cube-root law seemed to hold. In 1932, though, Max Kleiber proved it wrong by methodically plotting animal masses versus metabolic rates. He found correlations closer to quarter roots, measuring an exponent of 0.74 for total metabolic rate and -0.26 for the specific rate.

Image: UC DAVIS MAX KLEIBER. This scientist made the first systematic measurements of the correlation between an animal's size and its metabolic rate in 1932, correcting the belief in cube-root scaling to quarter-powers.

For 50-odd years, the problem lay more or less where the Maxs' left it. Then in 1995, Brown, long interested in biological scaling, and Enquist, who uncovered quarter-power scaling laws in plants, went in search of someone with a heartier math background to help them out. Mike Simmons, a retired vice president of academic affairs at Santa Fe Institute, put them in touch with West.

For their first crack at the problem, the trio homed in on what all animals and plants share: some sort of vascular network for distributing nutrients. They assumed that these networks are invariably fractal and that the end branches are of a fixed size. An elephant's capillaries are the same width as a mouse's, and a redwood's match an orchid's. They further guessed that nature had tuned circulatory networks to use as little energy as possible. And from there, using the physics of capillaries and hydrodynamics, they created a model that predicted a number of quarter-power scalings, including a mammal's aorta size, capillary density and heart size, with remarkable precision.