A pizza problem
You walk into a pizzeria and order a 9-inch pizza. The server comes back: sorry, we are out of 9-inch, but we have 5-inch pizzas. Could you take two of them? Two fives make ten, and ten is more than nine, so it seems like a good deal. It is not. A pizza’s size is quoted as its diameter, but you do not eat the diameter. You eat the area, and area grows with the square of the diameter. The 9-inch pizza has about 64 square inches. Two 5-inch pizzas have about 39. You would get roughly 40 percent less pizza.
The pizza problem
Change the size of the big pizza, the size of the small ones, and how many you get. Compare the diameters you add up with the areas you eat.
Most people get this wrong the first time, and the reason is general. Our minds extrapolate in straight lines: twice the size, twice the stuff. But the world does not scale that simply. When you enlarge an object while keeping its shape, lengths grow in proportion, areas grow as the square, and volumes, and so weights, grow as the cube. Galileo noticed the consequence in 1638, in Two New Sciences. A giant built with the same proportions as a man would have bones that could not hold him up, because his weight would grow as the cube of his height while the strength of his bones grew only as the square. Bones for a giant would have to be disproportionately thick. Scaling is the study of what changes, and how, when a thing gets bigger.
Mouse and elephant
Living things are a good place to look for scaling rules, because they come in sizes from a bacterium to a whale. One thing every animal does is burn fuel to stay alive. At rest, even asleep, the body is spending energy to keep warm, circulate blood, and rebuild its parts. That resting energy use is the metabolic rate. It is measured in a quiet, comfortable chamber, as the oxygen the animal takes in or the carbon dioxide it gives off, and then converted to watts or to food calories.
In 1932 Max Kleiber, a Swiss-born scientist at the University of California, Davis, gathered such measurements for animals from a ring dove and rats to a man, a woman, sheep, dogs, and steers, a range of body weights from about 150 grams to nearly 700 kilograms. Plotting the metabolic rate against body mass on a chart where both axes are logarithmic, each step along an axis a multiplication by ten, he found that the points lay on a straight line. A straight line on such a chart means a power law: metabolic rate is proportional to mass raised to a fixed power, and Kleiber’s best fit was about 0.74. The usual rounding, three-quarters, gives the common form of Kleiber’s law: resting energy use is about 70 × (mass in kilograms)3/4 food calories a day, or about 3.4 watts times mass3/4. For a 70-kilogram person that comes to about 82 watts, around 1,700 food calories a day: a little less than a bright light bulb.
From mouse to elephant
Pick an animal or slide through the sizes. The red dot sits on the trend line; the readout gives the energy use, the energy per kilogram, and the resting heart rate the same family of rules predicts.
What does the exponent mean? An elephant of about 4,000 kilograms is some 200,000 times heavier than a 20-gram mouse. If metabolism simply rose in proportion to weight, it would use 200,000 times as much energy. By Kleiber’s law it uses about 9,000 times as much. So per kilogram, an elephant spends only about a twentieth of what a mouse does. A mouse is a furnace: it burns roughly 9 watts for every kilogram of its body, an elephant less than half a watt. Bigger animals are more economical, gram for gram.
This is a statement about species, not about individuals. It does not say that heavier people are more efficient; within humans, carrying extra body fat is linked to more health problems, not fewer. It says that across the animal kingdom, size and metabolism are tied together in a precise way.
An elephant is about 200,000 times as heavy as a mouse. According to Kleiber’s law, about how much more energy does it use at rest?
Why three-quarters?
The number is a puzzle. Heat leaves a body through its skin, and an animal’s skin area grows as length squared while its weight grows as length cubed. That reasoning gives an exponent of two-thirds, not three-quarters, and in 1883 the German physiologist Max Rubner measured dogs and concluded that their heat output fit the area of their skin. For decades the “surface law” was the textbook answer. Kleiber’s data pointed to a different exponent, and the difference matters: for the elephant and the mouse, two-thirds would give a ratio of about 3,400, three-quarters about 9,400.
The best-known explanation came in 1997 from the physicist Geoffrey West, the ecologist James Brown, and the biologist Brian Enquist. Their idea is that an animal is not a ball of cells that breathe through the skin. Every cell must be fed by a network that reaches it: arteries and capillaries, airways, and, in a tree, the vessels of the trunk and branches. These networks are branching tubes that fill the body, and they have two properties. The smallest tubes, such as capillaries, are about the same size in a mouse and a whale. And the whole network has to deliver the supply with as little waste as possible. Work through the mathematics, and the delivery network behaves as if the body had an extra dimension: a three-dimensional body that acts as if it were four-dimensional, so that the exponent is not 2/3 but 3 divided by 4. In their words, life acts as if it has a “fourth dimension.”
A picture helps. Think of a long line of people at an airport security checkpoint, folded back and forth by posts and tapes so that a line that would stretch for a block fits in one hall. A line that fills a surface behaves like a surface: it uses the whole floor even though it is only a line. A branching or folded network inside a body does the same thing in three dimensions, filling the whole volume with something that is effectively thicker than a surface. The authors argued that the same logic runs through the lungs, the gut, and the leaf veins of a tree.
Another view
Kleiber’s law is a robust pattern, but the details are contested. Metabolic rate certainly rises more slowly than body mass, by an exponent that most studies of mammals place somewhere between two-thirds and three-quarters. In 2001 Dodds, Rothman, and Weitz re-examined the data and found little reason to reject two-thirds; in 2003 White and Seymour, working with 619 mammal species and correcting for body temperature, found about 0.69. Defenders such as Savage and colleagues (2004) found quarter-power scaling in many other traits. Kolokotrones and colleagues (2010) found that on a log chart the mammal data actually curve slightly, with a local exponent of about 0.57 for small mammals and 0.87 for large ones, so no single exponent fits every size. And the network explanation is itself disputed: Banavar and others showed that other network models give two-thirds or three-quarters without needing a fractal design. The 1997 paper also claimed that scaling runs over 21 orders of magnitude in body size, and a larger claim extends it from whales to molecules; critics note that the best-measured data cover far fewer. What no one disputes is the main point: a mouse burns far more energy per gram than an elephant.
A billion heartbeats
If bigger animals run on less energy per gram, everything in their bodies should run more slowly. It does, and by a pattern as regular as Kleiber’s. Take the idea of a body as a tank: fuel flows in at a rate set by the metabolic rate, and the tank holds an amount proportional to the body mass. The time to cycle through the tank is then mass divided by metabolic rate, which scales as mass raised to the power one-quarter. That quarter-power time sets the pace of the body’s clocks. Resting heart rate falls with mass as the power minus one-quarter: a mouse’s heart beats about 600 times a minute, a person’s about 70 to 80, an elephant’s about 30. Lifespan goes the other way, rising roughly as a quarter power: a lab mouse lives two to four years, a person decades, an elephant 60 or more. Pregnancy lasts about three weeks in a mouse and nearly two years in an elephant.
Multiply the two, heartbeats per minute by years of life, and the exponents cancel. A mouse’s fast heart and short life, and an elephant’s slow heart and long life, give about the same number of beats: roughly a billion. Estimates range from about 700 million to 1.5 billion, depending on the study. It is a striking regularity: small animals live fast and die young, large ones live slowly and last, and the total beats at the end come out similar.
People are the big exception. A person who lives to 80 with a resting heart rate of 70 has had about 2.9 billion heartbeats, two to four times the mammal rule of thumb. Modern food, sanitation, and medicine have stretched a human life far beyond what size alone would predict. The longest verified human lifespan belongs to Jeanne Calment of Arles, France, born on 21 February 1875 and died on 4 August 1997, at 122 years and 164 days. (A 2018 claim that her record was the result of an identity swap has been rejected by most experts.) Other animals are exceptions, too: bats live several times longer than other mammals of their size, and the naked mole-rat lives about 30 years in captivity, against about four for a mouse. The heartbeat count is a rule of thumb across mammals, not a law that every species must obey.
Why do a mouse and an elephant have roughly the same number of heartbeats in a lifetime?
Live slowly, live long?
An obvious question is whether the rule works inside a species: could one animal live longer by burning less? This course puts forward a simple idea, that the body produces entropy as it runs, so slowing the metabolism, in a healthy way, should slow the wear. The evidence is a mixture. In cold-blooded animals, lower temperature means a slower metabolism and a longer life, as it does in fish and flies. In warm-blooded animals the picture is murkier. A 2006 study in Science by Bruno Conti and colleagues made mice whose core body temperature ran 0.3 to 0.5 degrees Celsius lower, and they lived 12 to 20 percent longer, though it is a mouse result and not a recipe for people. Cutting calories lengthens the lives of many rodents, but in rhesus monkeys, two long studies disagreed: a Wisconsin study reported that monkeys eating about 30 percent fewer calories lived longer, and a similar study at the National Institute on Aging found no gain in survival, although health markers improved in both. A joint 2017 analysis pointed to differences in diet and study design.
The idea that people who exercise hard burn out young is not supported. A 2012 study in the BMJ of 15,174 Olympic medalists from 1896 to 2010 found that, on average, they lived 2.8 years longer than matched people from the general population, and the French riders who raced the Tour de France between 1947 and 2012 had about 40 percent lower mortality than the general population. Part of that is that athletes start out healthy, but the pattern is the opposite of “fast burn, early death.” The older “rate of living” theory, that every animal has a fixed lifetime supply of energy, has largely fallen out of favor; it fails across classes of animals, since birds and bats live far longer than their metabolisms would predict.
Closer to home
The Chinese history Records of the Three Kingdoms tells of a scale problem solved by a child. When Sun Quan sent the warlord Cao Cao an elephant, nobody could think how to weigh it. Cao Cao’s young son Cao Chong, then about five or six years old, suggested putting the elephant on a boat, marking the waterline, and then loading the boat with objects until it sank to the same mark; the objects could be weighed one at a time. Whether the tale happened exactly so is unknown, but the method is sound. Elephants appear in American life too: if you have stood beside one at a zoo, you know how slowly it moves, and the quarter-power rule says why. Its heart takes two seconds a beat and its steps are long and unhurried, where a mouse’s are a blur.
What to keep
- Scaling asks what changes when something gets bigger. Lengths, areas, and volumes grow at different rates, which is why two 5-inch pizzas are not a 9-inch one.
- Kleiber’s law: resting metabolic rate grows roughly as body mass to the power 3/4. An elephant is 200,000 times heavier than a mouse and uses about 9,000 times the energy, so it spends far less per gram.
- Biological clocks run on a quarter-power time: heart rate falls and lifespan rises with size, and for most mammals the lifetime total is about a billion heartbeats. People, at 2 to 3 billion, are an exception.
- The exact exponent (2/3 or 3/4) and the network explanation are still debated, and “live slow, live long” holds across species but not as simple advice for one person.
Think it over
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Sources and further reading
- Max Kleiber, “Body size and metabolism,” Hilgardia 6 (1932).
- Geoffrey B. West, James H. Brown, and Brian J. Enquist, “A general model for the origin of allometric scaling laws in biology,” Science 276 (1997), and “The fourth dimension of life: fractal geometry and allometric scaling of organisms,” Science 284 (1999).
- Geoffrey West, Scale: The Universal Laws of Growth, Innovation, Sustainability, and the Pace of Life in Organisms, Cities, Economies, and Companies, Penguin Press, 2017.
- Craig R. White and Roger S. Seymour, “Mammalian basal metabolic rate is proportional to body mass2/3,” Proceedings of the National Academy of Sciences 100 (2003); Van M. Savage et al., “The predominance of quarter-power scaling in biology,” Functional Ecology 18 (2004); Andrew Kolokotrones et al., “Curvature in metabolic scaling,” Nature 464 (2010).
- Herbert J. Levine, “Rest heart rate and life expectancy,” Journal of the American College of Cardiology 30 (1997).
- John R. Speakman, “Body size, energy metabolism and lifespan,” Journal of Experimental Biology 208 (2005).
- Bruno Conti et al., “Transgenic mice with a reduced core body temperature have an increased life span,” Science 314 (2006); Philip M. Clarke et al., “Survival of Olympic medallists,” BMJ 345 (2012).
- Galileo Galilei, Discourses and Mathematical Demonstrations Concerning Two New Sciences, 1638.