Fascinating new scientific papers suggest how elementary geometry involving animals' physical dimensions is sufficient to shed light on some very basic biological phenomena. In particular, the papers attempt to determine the metabolic pace of all life and, in the process, help resolve a problem in evolutionary time measurement.

Hold on tight.

Let's start by considering why an animal can't be, say, five times its normal adult size. To understand that we can't simply multiply physical dimensions by a factor of five, imagine what would happen if a 6-foot, 160 pound man were scaled up to a height of 30 (6 x 5) feet. His weight, like his volume (in cubic feet), would increase not by a factor of five, but by a factor of 5^3 and thus would rise from 160 pounds to 20,000 pounds (160 x 5^3) -- 125 times as great as his original weight if he were proportioned similarly.

And what would hold up such a behemoth? The supporting cross-sectional area of his thighs, say 2 square feet originally, would increase not by a factor of five, but by a factor of 5^2 and would thus rise to 50 square feet (2 x 5^2) -- 25 times as great as the normal area if he were proportioned similarly. (The same would hold for his spine, knees and so on.)

But the pressure on his thighs -- his weight divided by the area of a cross-section of his thighs, i.e. 125 times his original weight divided by 25 times the original area -- would be five times as great. This would be a crushing pressure and the man would collapse. This is why heavy land animals like elephants and rhinos have such thick legs.

Mathematical considerations not too dissimilar to these also lie behind various scaling laws in biology relating animals' metabolic rates -- heart, breathing, twitching, etcetera -- to their surface areas and masses. Small animals' hearts, for example, beat faster than large animals' hearts and, more generally, they live faster and die younger than do large animals who measure out their energies at a more lumbering pace.

Since areas, including animals' surface areas, scale up with the 2nd power of their relative dimensions, and their masses or volumes scale up with the 3rd power, such considerations long ago led scientists to the belief that animals' metabolic rates were proportional to the surface areas of their skins or, equivalently, proportional to their masses to the 2/3rd power.

Much evidence suggests, however, that metabolic rates are proportional to animals' masses to the 3/4th power, not the 2/3rd power, and recent papers by ecologists Brian Enquist and James Brown and physicist Geoffrey West explain why this and other "quarter power scaling laws" make sense theoretically as well as empirically. (The short explanation is that the metabolic rate is affected not by how fast heat dissipates through the skin, but rather by how efficiently nutrients reach the body's cells and for this latter quantity a broader definition of "surface" area reflecting internal structure is needed.)