When his dog passed away, a six-year-old reflected on why dogs live shorter lives than humans, “If we’re born to learn how to love all the time and dogs already know how to do that, then they don’t have to stay as long.” If love was counted in heartbeats, that kid was onto more than he thought. While our hearts beat the same number of times as a dog’s over the course of our lifetimes, dogs squish all of theirs into 10 – 15 years, while we spread ours out over 80.

Your heart will beat about 1.5 billion times in your lifetime. Every mammal, from the smallest shrew to the largest whale gets about 1.5 billion heart beats. The faster the heart rate the shorter the life. The slower the heart rate, the longer the life. 1.5 billion heart beats. This is the miracle of scaling, where simple mathematical relationships define the parameters within which life occurs, in all its diversity. Here we find ourselves as part of something much larger than ourselves.

We, mammals, exist at a controlled burn. We are an organized flow of energy. Metabolic rate describes how much energy flows through us. As an organism’s mass increases, its energy efficiency increases. The heavier the animal, the less energy (per gram) it uses. Animals increase their energy efficiency by slowing down. Heavier animals do everything slower. Their hearts beat slower. They breathe slower. They breakdown slower. By doing everything slower, they live longer.

One might suspect an animal 10,000 times heavier than another would require 10,000 times more energy to match the mass increase. But as it turns out, a 10,000 time increase in mass only requires 1000 times increase in energy. The relationship between mass and metabolism is not linear, but rather, a power law. As mass increases by the power of four (10^{4}), metabolism increases by the power of three (10^{3}). Many phenomena obey power or scaling laws–from moon craters, earthquakes, and solar flares, to frequency of word use–all become more prolific with decreasing size—lots of little words, fewer gigantic words; lots of little earthquakes, fewer earth-shattering ones. Power laws describe exponential variation across scale.

Since power laws cover many scales (from mice to whales), they are best illustrated on a logarithmic graph. Logarithms help us fit exponential increases into a finite graph size. Logarithmic axes count: 10, 100, 1,000, 10,000, flattening exponential changes to their exponent count: 10^{1}, 10^{2}, 10^{3}, 10^{4 }become each just one step apart. Counting by powers of 10 rather than by 1’s, reveals relationships across scales.

With linear axes, the slope of metabolism/mass is 10^{3}/10^{4}. Logarithmic scaled axes flatten the slope to 3/4. Since the slope is less than one-to-one (sub-linear), we know that metabolism becomes more efficient with increasing size rather than growing at the same rate, as a slope of 1 would indicate. Mass ^{3/4} = metabolism. This is known as the ¾ scaling law or Kleiber’s Law, named after Swiss chemist Max Kleiber.

(West, 2018)

Decreased energy needs per increase in body mass belie an “economy of scale.” The economy of scale is also at work when buying in bulk or getting more bang for your buck, in this case more mass for less energy.

One might suspect that this variation has to do with temperature regulation. Mass produces heat. Surface area distributes heat. Too much volume per surface area risks overheating. Too little volume to surface area risks heat production falling behind heat loss. The perfect ratio between the heat producing volume and the heat distributing surface area maintains a perfect operating temperature.

Smaller animals have less volume per surface area. Thus, smaller animals need to produce more energy per gram in order to keep up with a proportionally larger surface area to volume ratio. Larger animals produce less energy to prevent overheating with greater volume to surface area ratios. To prevent overheating, metabolism slows down to accommodate the changing ratio of heat production to distribution.

This, however, would suggest a 2/3 scaling law. An animal’s heat producing volume increases in three dimensions, by the power of three—length x width x height. Heat distributing surface area increases in two dimensions, by the power of two—length x width. Thus volume (heat production) increases faster than surface area (heat distribution). One might expect the slope of Kleiber’s law to measure 2/3, for area/volume–length^{2} / length^{3}.

Mysteriously however, Kleiber’s law is not a 2/3 scaling law, but a ¾ scaling law. While some arguments out there that suggest the ratio may change with scale, a deeper understanding of fractals might help us understand the role ¾ might play.

Like power laws, fractals also emerge when patterns repeat across scale. This repetition across scale generates an irregularity or crinkliness that move the phenomena beyond its topological dimension. For example, a sheet of paper is topologically two-dimensional, but when crumpled into a ball it begins to take up more three-dimensional space. Yet, it does not completely fill the 3-D volume, hence its fractal dimension fall *between* 2 and 3, like 2.7. The decimal illustrates how much of the 3^{rd} dimension the paper takes up.

In our bodies, our internal 2-D surface area of blood vessel walls branch fractally, filling our 3-D structure. As our internal 2-D surface area approaches a fractal dimension of 3, our 3-D volume approaches a fractal dimension of 4 by deepening into the 4^{th} dimension of time. Our bodies’ frequencies nest fractally within one another as well, from brainwaves to heartbeats. Faster frequencies subdivide time, deepening into it. We are organized flows of energy, through fractally branching vasculature, deepening into space and time, just like the watersheds, plants, and animals that build us. When we count by heartbeats, we know we’re home.

**References: **

West, Geoffrey. 2018. “Complexity, Regularity, Diversity and Scale” from “The Complexity of Diversity” Santa Fe Institute. March 22–23, 2018. San Francisco. https://www.youtube.com/watch?time_continue=1971&v=WcwqcKkkkcg