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It was a deeply held belief of Al Bartlett that “the greatest shortcoming of the human race is our inability to understand the exponential function.”

Bartlett, a University of Colorado physics professor and Boulder environmental giant who died Sept. 7 at the age of 90, spent a lifetime trying to correct that shortcoming.

He shared his views on the importance of understanding the exponential function with anyone who would listen, and a lot of people did. He crafted a speech explaining the implications of the exponential function as it pertained to population growth and resource and energy consumption and gave versions of it 1,742 times. A YouTube video of the speech has been viewed nearly 5 million times.

I think the most fitting way to memorialize Al may be to share some of the thoughts that were in that remarkable talk, which he delivered under several titles, including “The Arithmetic of Growth” and “The forgotten fundamentals of the energy crisis.”

The essence of the talk was this: When the population of a country or the consumption of a resource, such as oil or gas, grows at a fixed percentage per year (say 5 percent a year), the growth rate is said to be “exponential.” And exponential growth rates can produce huge increases in the size of a nation’s population or the consumption of a resource, even when the annual growth rate is seemingly small.

If oil consumption increases at 5 percent a year, for example, annual global consumption of oil will double in 14 years. If a country’s population is increasing by 3 percent a year, the country’s population will double in 23 years.

If there was one single thing Bartlett would have wanted you to take away from his talk, it probably would have been the following equation:

T 2 = 70/P

The equation tells you how to calculate how long it will take for the annual consumption of a resource or population of a country to double, assuming the annual rate of growth of the consumption of the resource (or the annual growth rate of a population) is sustained. The number 70 divided by the annual percentage growth rate (P) will give you the doubling time (T 2 ).

For example, if Boulder’s population was growing at a rate of 5 percent a year (the rate it averaged during the decade of the 1960s) its population would double in 14 years. And that is pretty close to what happened. The 1960 Census put Boulder’s population at 37,718. Ten years later the 1970 Census put it at 66,870. In 1975, Boulder’s population was about 76,000; some estimates put it as high as 80,000. A growth rate of 5 percent may seem modest, but it is actually a pretty torrid pace.

During the 1960s countries like India and Pakistan were growing at a rate of more than 3 percent annually, and the prospect of global famine was an imminent threat.

Bartlett’s speech contained some cool examples that cast the implications of exponential growth into sharp relief. Here are two of my favs:

• Legend has it that after the inventor of chess presented his game to his king, the king asked him what he would like for his reward. He said all he wanted was some wheat, the amount to be determined by the following process: A single grain would be placed on the first square of a chess board, two grains would be placed on the second square, four grains on the third square, eight grains on the fourth square, and so on, with the number of grains on each square doubling through the 64th square.

Using this method, the total amount of wheat in the reward would come to 2 64 grains less one — or 18,446,744,073,709,551,615 grains, or about 300 times the 2012 global wheat harvest.

• The bacteria-in-a-bottle case illustrates the consequences of exponential growth in a finite environment.

Bacteria reproduce by division so that a single bacterium becomes two, the two divide to become four, the four divide to become eight, and so on. Thought experiment time: Consider a hypothetical strain of bacteria for which the division time is one minute. One bacterium is put in a bottle at 11 a.m., and it is observed that the bottle is full of bacteria at noon. Bartlett would point out that this scenario is “mathematically identical to the case of the exponentially growing consumption of our finite resources of fossil fuels.”

Now for a short quiz: At what time was the bottle half full? Answer: At 11:59 a.m. At what time was the bottle 1.5% full? Answer: At 11:54 a.m.

In other words, an exponential growth rate can occur with little or no noticeable impact during its early stages, but its impact during its later stages can be staggering and can burst upon us with little warning.

This arithmetic is one of the factors that in 1976 led me to propose the growth limitation ordinance colloquially known as the Danish Plan — and prompted Al to support it. (He agreed to be one of the five sponsors of the initiative petition that put the plan on the 1976 ballot. This took some political courage. Several then-prominent Boulder environmentalists chose not to get involved, but Al stepped up.)

From 1970 to 1975 Boulder’s housing stock grew at an annual rate of 3 percent a year, down from the 5 percent rate of the 1960s, but still fast.

If that 3 percent growth rate had continued (and the average number of people in a Boulder household had remained the same) the city’s population would have increased to more than 150,000 by 1998, from 76,000 in 1975.

Today, Boulder’s growth rate is under 1 percent and its population is about 97,000. This reduction in the city’s exponential growth rate was achieved by the adoption of a number of growth-related measures starting in the 1950s. Two of the most important were adoption of the Blue Line in 1959, which helped prevent Boulder’s mountain backdrop from being developed by preventing water mains from being extended above a certain altitude, and creation of the Boulder open space program in 1967, which has encircled the city with tens of thousands of acres of publicly owned greenbelts that prevented urban sprawl in the Boulder Valley and limited the city’s size. Al Bartlett was intimately involved in the creation of both. He also was a founder of the local environmental group PLAN Boulder.

Al spent thousands of hours arguing for the reduction of exponential rates of population growth and resource consumption, and in his civic life he worked for policies and programs that would further those ends.

To a large extent, Boulder as we know it today is his legacy.

*Respond: letters@boulderweekly.com*

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