Scaling: The Surprising Mathematics of Life and Civilization

In late 1995 Mike Simmons, then-SFI vice president, introduced me to Jim Brown. At that time I was overseeing the high energy physics program at Los Alamos National Laboratory while Brown, who had recently moved to the University of New Mexico’s biology department, was developing an ecology program at SFI. Serendipitously we had both been thinking about a longstanding problem in biology, namely the origin of so-called “quarter-power allometric scaling laws.” I will elaborate on what this means later but, roughly speaking, it refers to the surprising observation that across the entire spectrum of life, almost all physiological variables and life-history events scale with size in a remarkably simple, systematic, and predictable fashion.

Sandwiched between quarks, Higgs, strings, and dark matter, I had been struggling with developing a physics-inspired network theory for the origin of these scaling laws, while Brown and his then-student, Brian Enquist (now at the University of Arizona), had been speculating that nutrient transportation through the bloodstream was a key ingredient.

Simmon’s intuition that we might have something to say to one another changed our lives and marked the beginning of what became known as the “scaling program” at SFI. The implications of scaling phenomena later expanded beyond biology, ecology, and biomedicine to embrace human socioeconomic systems such as cities and companies – even extending to the challenge of global sustainability. Thus began a beautiful relationship with Brown, Enquist, and SFI and, by extension, with the ensuing cadre of wonderful postdocs, students, and faculty who have since worked on the Institute’s scaling and cities programs.

Our sustained collaboration has been enormously productive, extraordinarily exciting, and tremendously fun. Beginning in 1996, initially with Brown, Enquist, and me, and later with the expanded group, we met every Friday at SFI from 9 a.m. to around 3 p.m. This continued almost uninterrupted until just the last couple of years. At the outset this was a huge commitment as both Brown and I ran large research groups elsewhere.

Once the ice was broken and some of the cultural barriers were crossed, we created a refreshingly open atmosphere where all questions and comments, no matter how elementary, speculative, or seemingly stupid, were encouraged, welcomed, and treated with mutual respect. There were lots of arguments, speculations, and explanations; struggles with big questions and small details; lots of blind alleys; and an occasional aha! moment – all against a backdrop of a whiteboard (and sometimes Institute windows) covered with equations, graphs, and illustrations. Brown and Enquist patiently acted as biology tutors, exposing me to the world of natural selection, evolution, adaptation, fitness, physiology, and anatomy, all of which were embarrassingly foreign to me. For my part, I tried to reduce complicated mathematical equations and technical physics arguments to relatively simple, intuitive calculations and explanations. In other words, we were engaged in a typical transdisciplinary Santa Fe Institute experience!

Recent research at SFI considers the city not only as people and infrastructure, but as a network of interactions in space and time. In this nighttime satellite image, populated areas branch out from the New York City metropolitan area. (NASA)

Recent research at SFI considers the city not only as people and infrastructure, but as a network of interactions in space and time. In this nighttime satellite image, populated areas branch out from the New York City metropolitan area. (NASA)

So what is “scaling”? In its most elemental form, it simply refers to how systems respond when their sizes change. What happens to cities or companies if their sizes are doubled? What happens to buildings, airplanes, economies, or animals if they are halved? Do cities that are twice as large have approximately twice as many roads and produce double the number of patents? Should the profits of a company twice the size of another company double? Does an animal that is half the mass of another animal require half as much food?

Asking such seemingly innocuous questions has had remarkably profound consequences across the spectrum of science, engineering, and technology and has impacted almost every aspect of our lives, even including how we perceive our place in the universe. Over the past 50 years, scaling arguments have led to a deeper understanding of the dynamics of tipping points and phase transitions (how, for example, liquids freeze into solids), chaotic phenomena (the mythical flapping of a butterfly’s wings in Brazil stimulating a hurricane in Florida), the discovery of quarks (the building blocks of matter), the unification of the fundamental forces of nature, and the evolution of the universe after the Big Bang. Three Nobel prizes have involved discoveries related to scaling during the past 30 years.

In a more practical context, scaling plays a critical role in the design of increasingly large human engineered artifacts, such as buildings, bridges, ships, airplanes, and computers, where extrapolating from the small to the large in an efficient, cost-effective fashion is a continuing challenge. (Indeed, simply doubling all dimensions of a bridge in order to traverse a river twice as wide would very likely lead to it collapsing under its own weight.) Even more challenging, and of perhaps greater urgency, is to understand how to scale organizational structures of increasingly large and complex cities, corporations, and governments, where underlying principles are typically not well understood because these – like living systems – are continuously evolving and adapting.

Scaling from small to large in an efficient, cost-effective manner plays a critical role in the design of increasingly large human engineering projects, such as the effort now underway to replace the 16,000-ft. cantilevered Tappan Zee Bridge that spans New York’s Hudson River. (Artists rendition of future bridge design, New York State Thruway Authority)

Scaling from small to large in an efficient, cost-effective manner plays a critical role in the design of increasingly large human engineering projects, such as the effort now underway to replace the 16,000-ft. cantilevered Tappan Zee Bridge that spans New York’s Hudson River. (Artists rendition of future bridge design, New York State Thruway Authority)