Bigger means faster

More than a body

A resting adult gives off about 80 watts, roughly the output of a bright light bulb. That is the body’s metabolic rate, and it is not far from the 100 watts or so of food energy you eat in a day. Yet no one lives on 80 watts. We heat our homes, cook, drive, fly, run factories, and keep servers cool, and all of that burns fuel too. In 2025, according to the Energy Institute’s annual statistical review as compiled by Our World in Data, the average American used about 75,000 kilowatt-hours of primary energy, which works out to a continuous 8,600 watts. The average person in China used about 3,600 watts, and the world average was about 2,300 watts. These figures count traded fuels and electricity and leave out traditional wood and dung, so they are rough.

Put 8,600 watts into Kleiber’s law run backwards, and you get a thought experiment: the animal that burns that much at rest would weigh about 35 tonnes. The world-average person, at 2,300 watts, is a six-tonne elephant. Of course people are not elephants. But the comparison makes a point that the second lesson only hinted at: a society is more than a collection of bodies, and it has a metabolism of its own, with fuel flowing in and waste and heat flowing out. This lesson asks whether the rules of scale found in animals carry over to those larger flowing structures. The answer is yes and no, and the difference is the interesting part.

Cities speed up

Animals slow down as they grow. Cities do the opposite. In 1976 the psychologists Marc and Helen Bornstein timed how fast pedestrians walked in towns and cities of different sizes (15 places in six countries, as later summaries report it), and found that the bigger the place, the faster its people walk. Later work fitted this as a power law in which walking speed rises with population to the power of about 0.09, a rise of about 6 percent for every doubling of size. The pace of life really does quicken with city size, and the effect is not limited to walking.

In 2007 a team led by Luís Bettencourt and including Geoffrey West, who had also worked on the animal scaling in Lesson 2, gathered data on cities in the United States, China, and Europe and plotted dozens of urban measures against population. They found two different patterns. Things that people create, such as wages, patents, inventors, and economic output, grew faster than population, with exponents between about 1.1 and 1.3. Total wages in US metropolitan areas scaled with an exponent of 1.12, patents 1.27, and gross output 1.15 for Chinese cities, 1.13 for German ones, and 1.26 for European ones as a whole. Things that a city builds to hold itself together, such as roads, cables, and gas stations, grew slower than population, with an exponent of about 0.8: 0.83 for road surface in Germany, 0.87 for cable length, and 0.77 for gas stations in the United States. Housing and jobs, the needs of each household, scaled roughly in proportion to population, with an exponent near 1.

What do those exponents mean in practice? An exponent of 1.15 means that when a city doubles, the output per person rises by a factor of 2^0.15, about 11 percent, not 15. A city ten times bigger does about 41 percent more per person. An exponent of 0.85 means that a doubling brings about 10 percent less pipe, cable, and road per person: economies of scale. The same measure that counts wealth counts misfortune. Serious crime scaled with an exponent of 1.16 and new cases of AIDS with 1.23. Big cities pay more, invent more, and also make more trouble. Phone records tell the same story about people: in data from Portugal, the number of contacts a person has rises with city size with an exponent of about 1.12, roughly 9 percent more contacts per person for each doubling.

Cities do not die, companies do

Companies do not scale like cities. Studies of tens of thousands of American firms by West and collaborators found that a firm’s sales grow more slowly than its assets, with an exponent of about 0.9 (0.90 for US firms and 0.92 for Chinese ones), like the sublinear scaling of animals, not the superlinear scaling of cities. A growing company gets more economical, but it also gets heavier and slower to turn, and it does not accumulate the kind of increasing returns that a city does. And companies die. A 2015 study of more than 25,000 publicly traded North American companies from 1950 to 2009 found a typical half-life of about ten years, whatever the company’s age or business: a firm that is 50 years old is no safer than one that is five. “Death” here includes being bought, merging, or being liquidated, not only bankruptcy. Large companies are not exempt. Consulting firm Innosight has estimated that the average time a company stays on the S&P 500 index fell from 33 years in 1964 to 24 years in 2016, and projected 12 years by 2027. That figure is a rolling measure of how fast names turn over in an index, not a measured lifespan, but the direction is clear.

Cities, by contrast, rarely die. Individual cities have vanished, mostly in antiquity, but a modern large city almost never disappears. Detroit’s population fell from 1.85 million in 1950 to 639,000 in 2020, a drop of about 65 percent, and the city did not vanish; its metropolitan area, with more than 4 million people today, did not collapse. The same caution applies to Chinese “ghost cities.” The district of Kangbashi in Ordos, Inner Mongolia, was built in the 2000s for a planned population that early plans put as high as a million, and by 2010 it held only a few tens of thousands; by the 2020 census it had about 119,000 residents. It was built ahead of demand by planners and lenders, so its emptiness shows a construction boom outrunning its people, not a city that grew too large for its resources.

Growth that outruns its limits

The mathematics that gives cities their increasing returns also carries a warning. Bettencourt and his colleagues wrote a simple equation for a city’s growth, and the version described here is a simplified, “toy” form of it. The people produce resources in proportion to a power of the population, N^β, and the city must spend some of those resources on upkeep for each person, in proportion to N itself. Whatever is left over goes to growth. The change in population is then given by a term that grows as N^β minus a term that grows as N.

Three things can happen, depending on the exponent. If β is less than 1, as for animals and companies, the upkeep eventually catches up with the income, and growth levels off at a limit. If β equals exactly 1, growth is steady and exponential. If β is greater than 1, as for cities, the income outgrows the upkeep and growth speeds up faster and faster. Growth is then faster than exponential, so fast that the equation reaches an infinite population in a finite time, a “finite-time singularity.” Or it can go the other way: a settlement that starts below a certain size is not self-sustaining and shrinks away. Try the sliders.

The singularity in the equation is a mathematical fact, not a forecast. The paper says so plainly: the resources behind growth are limited, so the singularity is never reached, and unlimited growth is unsustainable. What can postpone collapse is reinvention. Major innovations, in the authors’ words, “effectively reset the initial conditions and parameters,” and a new cycle of growth begins; this can repeat, “pushing potential collapse into the future.” But, they argue, each cycle must be shorter than the last, so innovation has to arrive faster and faster, like a treadmill that keeps speeding up. For a large city, they estimate the time to the singularity at a few decades, so cycles of reinvention have to be shorter than that.

Popular summaries often add a list of dates, in which each technological revolution follows the last more quickly: the steam engine, electricity, computers, the internet, mobile phones. The list is a good illustration, but it is not a measured series. There is no agreed way to count and date “revolutions,” and the pattern depends partly on what the lister chooses to include. The only data in the 2007 paper are the growth cycles of the New York metropolitan area.

Doomsday on a Friday

Forecasts that run into a finite-time singularity have a history, and a funny one. In November 1960 the journal Science published a paper by Heinz von Foerster, Patricia Mora, and Lawrence Amiot with the title “Doomsday: Friday, 13 November, A.D. 2026.” They had fitted 24 estimates of world population, from the year 0 to 1958, to a curve that explodes in finite time. The best fit was N = 179 billion ÷ (2026.87 − t)^0.99, where t is the year, which reaches infinity at 2026.87, a date the authors rendered as Friday, 13 November 2026, with an error bar of more than five years. The authors were partly joking: a date precise to the day is absurd on such an error bar, and 13 November was one author’s birthday. The paper ends on a serious note: our great-great-grandchildren, it says, will not starve but will be “squeezed to death.”

For a few decades the curve fitted well. Then reality changed course. World population growth peaked at roughly 2.2 percent a year in the early 1960s, and has since slowed to under 1 percent. The world had about 8.2 billion people in 2025. The formula gave about 6.9 billion for 2000, close to the real 6.2, and then ran away: about 27 billion for 2020, when the real figure was 7.9. Try it.

The lesson is not that population forecasts are foolish; it is that a curve that fits the past, especially one that explodes, tells you the past trend, not what limits will intervene. Population growth turned down because families chose to have fewer children as societies grew richer and healthier, an effect the 1960 curve had no way to include. The United Nations now projects that world population will peak at about 10.3 billion in the mid-2080s.

The singularity, a claim

“Singularity” has a second meaning, and the two are often confused. The mathematical one is the blow-up in an equation, as above. The technological one is a hypothetical point at which machines surpass human intelligence and progress becomes impossibly fast. The idea has a lineage. In 1958 the mathematician Stanislaw Ulam recalled a conversation with John von Neumann, whom you will meet in the next lesson, about “the ever accelerating progress of technology,” which “gives the appearance of approaching some essential singularity in the history of the race.” That is Ulam’s memory of a conversation; it mentions no date and no artificial intelligence. In 1965 the British mathematician I. J. Good wrote that a sufficiently intelligent machine could design better machines, producing an “intelligence explosion” that would leave human intelligence “far behind.” In 1993 the science-fiction writer and computer scientist Vernor Vinge predicted that “within thirty years” we would have the means to create superhuman intelligence. Ray Kurzweil’s 2005 book The Singularity Is Near put the Singularity at 2045, and his 2024 sequel kept that date.

Notice what each claim is. The 2007 equation is a model of cities, and it says that a growing city must keep reinventing itself faster and faster. It is a warning about an innovation treadmill, not a prediction of artificial intelligence, and its authors do not forecast a technological singularity. Kurzweil’s dates are forecasts, which may turn out right or wrong. How should we weigh them? One data point is a survey. In late 2023 the research group AI Impacts asked 2,778 researchers who had published at leading AI conferences about the future of the field. The group’s combined forecast gave a 50 percent chance by 2047 that machines could “accomplish every task better and more cheaply than human workers,” and 10 percent by 2027. The spread of answers was wide, the respondents were experts in building AI, not in forecasting, and the survey’s authors caution that the numbers depend on how the question is asked.