- Scaling laws apply consistently to organisms, organizations, and cities. They are power laws that are either super-linear (exponent > 1) or sub-linear (exponent < 1) as a function of size; the exponent is usually a multiple of 1/4. In biology, these laws are remarkably strong predictors of traits like metabolic rate, mortality, heart rate, and more across an enormous range of scales. Some underlying phenomena explaining this regularity are the interplay between energy and entropy, and the mathematics of networks and complex systems.

- Definitions for orders of magnitude, exponents, and logarithms. In geometry, volume and area increase at a faster rate than length, and this has all kinds of implications for physical systems and the relationships between mass, strength, drag force, and other quantities. Innovation is defined by changing the characteristics of a system to maintain growth as it scales. Only in the past few centuries have we begun to apply rigorous scientific models to engineering problems, and even some modern practices in medicine don't take basic scaling arguments into account.

- All mathematical models are limited in their applicability and accuracy. The scaling laws described are a zeroth-order approximation of living systems—they describe average behavior across species and organisms. All living systems can be modeled as self-similar (i.e. fractal) networks, whether it's the circulatory system, plant vascular system, road networks, or mitochondrial networks. The principles of energy optimization, space-filling, and invariance of terminal units (e.g. the capillaries of all mammals are about the same)—combined with the basics of metabolism and fluid dynamics—provides the basis for the quarter-power scaling laws found in living organisms. Fractals are ubiquitous in nature, though their physical origins are as yet unknown.

- The scaling relationship between metabolism (the source of energy in organisms) and mass (the sink of energy in organisms) sets hard constraints on both growth and aging. Growth results when there is more energy available through metabolism than is needed for maintenance of existing cells. Aging is the result of damage and decay to structures in the body, mostly in the terminal units of networks (capillaries, mitochondria). Because the rate of damage is directly proportional to metabolic rate, life span increases as per-cell metabolism decreases. Metabolic rates are exponentially sensitive to temperature, so life span also increases with lowered temperatures.

- Exponential growth is when growth rate is proportional to the existing amount, i.e. $f'(x) \propto f(x)$. Also, doubling time is constant. Since the industrial revolution, human population has had faster-than-exponential growth, i.e. doubling time has been
*decreasing*. Since Malthus' predictions in the 19th century that population growth would overwhelm food production, there has been a continuous set of technological innovations supporting more humans expending more energy, with agricultural workers comprising less than 1 percent of the US population today. The debate about the limits of human resource use continues, but any such debate must include an understanding of energy and entropy. The amount of energy that arrives at the earth from the sun is much greater than what we need, but we've come to rely on limited earth-based stores of energy like fossil fuels.

- Cities are made up of physical and social networks. Much of urban planning has taken an "inorganic" approach—building cities out of brick and steel, smooth Euclidean shapes, and little concern for human interaction and community.

- Cities display superlinear scaling in socioeconomic metrics and sublinear scaling in infrastructural metrics. The same principles of space-filling, invariant terminal units, and optimization apply for both the physical and social networks. The exponents in the two networks are inversely coupled—the economies of scale in infrastructure feed the greater density of social interactions, and the social interactions foster innovations in infrastructure. Human social networks are constrained by the brain's ability to maintain up to 150 total connections. Rank-size distributions following a power law are prevalent in quantities like city populations and company sizes.

- We can use cell phone and internet data as a proxy for social interactions in a city, and their usage aligns with the 1.15 superlinear scaling found in traditional socioeconomic metrics. There's a systematic regularity of individuals' commute times and their travel to "central places" in cities – influx of people to a given location scales as an inverse of distance traveled and frequency visited, both squared.

- Companies are more similar to organisms than cities in that they show sublinear scaling relative to employee count. The science of companies is less complete and precise, both because there's been less "evolutionary" time for companies undergo optimization (a few hundred years versus thousands of years for cities and billions of years for organisms) and because there's less data on the internal dynamics of companies. Company mortality follows an exponential curve with a half-life of ten years, i.e. half of all companies on the public market will be dead in ten years. The major cause of company mortality is mergers and acquisitions.

- Mathematical concept of a singularity – vertical asymptote. Superlinear growth in socioeconomic metrics requires an increasing pace of innovation, which is physically impossible to maintain at some point. The possible outcomes are collapse, or a regression to sublinear growth.