The Finite Shapes of Growth

We have the capacity to imagine infinity. Or at least, we think we do. One way we do this is to create an imaginary machine, a kind of software that we run in our minds. The program is designed to add one to the current count. We set our imaginary machine in motion and say, “it continues to work like this, adding one, and so on.” The machine creates an infinity. At whatever point we look in on it, it’s in the process of adding one to the set of numbers. The trick of infinity isn’t in making something that’s immeasurably large, but rather it’s in creating an algorithm that doesn’t have a defined stopping point. This process defines our idea of a certain kind of growth.

Geoffrey B. West on Why Cities Keep Growing,
Corporation Always Die, and Life Gets Faster
Why Cities Keep Growing, Corporation Always Die, and Life Gets Faster

Geoffrey B. West, of the Santa Fe Institute, gave a presentation at the Long Now Foundation entitled: “Why Cities Keep on Growing, Corporations Always Die, and Life Gets Faster.” The talk is filled with lots of interesting facts about statistically common features of cities and corporations. But it was the preliminary foundation of the argument that I found most interesting–in particular, the idea of sigmoidal growth patterns. This is the idea that animals begin at their smallest viable size and quickly grow to their optimal size and then stop. Living in an age with an excess of infinities, it’s a startling fact to contemplate. Most things in the universe grow to a certain size and then stop.

Here’s Stewart Brand’s summary of West’s discussion of scale and energy use:

Working with macroecologist James Brown and others, West explored the fact that living systems such as individual organisms show a shocking consistency of scalability. (The theory they elucidated has long been known in biology as Kleiber’s Law.) Animals, for example, range in size over ten orders of magnitude from a shrew to a blue whale. If you plot their metabolic rate against their mass on a log-log graph, you get an absolutely straight line. From mouse to human to elephant, each increase in size requires a proportional increase in energy to maintain it.

But the proportion is not linear. Quadrupling in size does not require a quadrupling in energy use. Only a tripling in energy use is needed. It’s sublinear; the ratio is 3/4 instead of 4/4. Humans enjoy an economy of scale over mice, as elephants do over us.

With each increase in animal size there is a slowing of the pace of life. A shrew’s heart beats 1,000 times a minute, a human’s 70 times, and an elephant heart beats only 28 times a minute. The lifespans are proportional; shrew life is intense but brief, elephant life long and contemplative. Each animal, independent of size, gets about a billion heartbeats per life.

We like to talk about exponential growth, especially when thinking about the Network. It’s as though abstract-thought machines had manifested in a mesh of connected computers growing without limit. Exponential growth is infinite, it doesn’t have an end point. While we like to use biological metaphors when discussing the Network, we seem to ignore the growth pattern of most biology. While it’s highly likely that the growth of the Network is sigmoidal in shape, we love the slightly naughty thought that it will expand geometrically ad infinitum. What we seem to be thinking of is the possibility of ungoverned growth patterns in bacteria and viruses.

“The mathematics of uncontrolled growth are frightening. A single cell of the bacterium E. coli would, under ideal circumstances, divide every twenty minutes. That is not particularly disturbing until you think about it, but the fact is that bacteria multiply geometrically: one becomes two, two become four, four become eight, and so on. In this way it can be shown that in a single day, one cell of E. coli could produce a super-colony equal in size and weight to the entire planet Earth.”

Michael Crichton
The Andromeda Strain

To the best of my knowledge, this hasn’t happened recently. According to Lynn Margulis, the last time was probably around 2.5 Billion years ago, when the earth’s atmosphere lacked sufficient oxygen to sustain humans.

Perhaps 2.5 billion years ago, a new group of photosynthetic bacteria evolved, the ancestors of today’s cyanobacteria. These advanced photosynthesizers split water to produce the hydrogen ions (H+) needed to build sugar molecules. A byproduct of this water-splitting reaction was oxygen gas. This was a catastrophic event in the history of life. Oxygen is such a reactive element that it easily destroys delicate biological structures. As the amount of oxygen in the atmosphere increased, most species of anaerobic bacteria were driven to extinction, victims of the earth’s first case of air pollution. Some survivors retreated to areas of brackish water or other oxygen-depleted habitats, where their anaerobic descendants still flourish today. A few prokaryotes became aerobic by evolving various mechanisms to detoxify oxygen. The most successful of these processes was respiration, which not only converted toxic oxygen back into harmless water molecules, but also generated large quantities of ATP.

According to the SET, the photosynthetic production of oxygen gas and the subsequent evolution of respiration set the stage for the evolution of all eukaryotic cells. This evolutionary process occurred in several separate symbiotic events. The first eukaryotic organelles to evolve were mitochondria–structures found in almost all eukaryotic cells. In Margulis’s theory, small respiring bacteria parasitized larger, anaerobic prokaryotes. Like some bacteria today (Bdellovibrio), these early parasites burrowed through the cell walls of their prey and invaded their cytoplasm. Either the host or the parasite was often killed in the process, but in a few cases the two cells established an uneasy coexistence. The mutual benefits to the partners are obvious. The respiring parasite, which actually required oxygen, would allow its host to survive in previously uninhabitable, oxygen-rich environments. Perhaps the parasite also shared with its host some of the ATP that it produced using oxygen. In exchange, the host provided sugar or other organic molecules to serve as fuel for aerobic respiration. Eventually, as often occurs with parasites, the protomitochondria lost many metabolic functions provided by the host cell. Similarly, as oxygen in the atmosphere continued to increase, the host became more and more dependent upon its pro-tomitochondria to detoxify the gas. What began as a case of opportunistic parasitism evolved into an obligatory partnership. The small respiratory bacteria eventually evolved into the mitochondria of eukaryotic cells.

The growth pattern from which we spend most of our time attempting to escape is the sinusoidal–the one that looks like a sine wave. We like the sine wave as it travels up, feeling as though it could go on forever. When it reaches its peak, we have a feeling of total mastery. And then suddenly, things begin to decay. We fall to earth as quickly as we ascended. The process begins again, but this time for our descendants. It’s this pattern that is expressed through evolution. Once Darwin’s thoughts had diffused through the atmosphere, we began to rebel. We woke from a long slumber to find we were inside a process of natural selection that would not bend to our will. Here we introduce the concept of “the fittest.” And through a simple slight-of-hand, we confuse ideas of physical fitness with the fact of just happening to fit with a particular state of the environment. It’s with this concept of “the fittest” that we stand on the bridge of evolution with our hands on the tiller. With our newly found powers, we design ecosystems that operate in both a perfect steady state and with unlimited growth. The downward slope of the sine wave is for other entities, not us.

Of course, there are many ways to frame the process of natural selection. I particularly like the phrasing of Richerson and Boyd in their 2005 work, “Not By Genes Alone.”

“…All animals are under stringent selection pressure to be as stupid as they can get away with.”

Their inversion of the idea of “fitness” does a nice job of puncturing our illusion of being able to move the odds to our favor. If we’ve only been allocated roughly a billion heart beats arranged in the shape of one oscillation of a sine wave, it’s a clear blow to our sense of self esteem. The infinity inside of us doesn’t seem to jibe with these finite patterns of growth. Of course, infinities are much easier to imagine standing on the shore and gazing toward the horizon. Once we’ve seen the satellite photo of the earth, we begin to understand that our finitude, while very large, still has edges. The earth grew to its optimal size, and then stopped.

Once the earth was within the surround of the satellite, Planet Polluto was in need of the attention of the ecologist…

Marshall McLuhan
On “The Dick Cavett Show”