Trust scientists to ruin all our fun. The spectacularly beautiful 2014 film reboot of *Godzilla*, it turns out, is anatomically misleading. At 350ft tall, such a beast would simply collapse under its own weight, because an animal’s mass cubes with a doubling of its size, while the strength of its supporting limbs only squares. The basic principle was known to Galileo, and it turns out that the simple observation that things do not scale linearly can tell us much else besides, as this quite dazzling book amply demonstrates.

Geoffrey West is a theoretical physicist who became interested in biology and then in cities, and (with colleagues at the Santa Fe Institute) developed ways of looking at such apparently disparate subjects that uncovered surprising similarities in the mathematics of the networks involved. The branching of the blood-circulatory system in humans obeys the same rules as the branching of trees. Cities have a quantitatively predictable ‘metabolism’, in terms of how they process energy, just like living things do. There are, in general, rules to how things scale to different sizes, but they are not intuitively obvious. Where do such regularities come from, and what might we do with them?

The book proceeds by introducing one mathematical concept in each chapter (power laws, fractals and so on), and explaining it vividly through numerous examples drawn from biology, history, urban planning and many other fields. We learn, for instance, why BMI (body mass index) is a pretty useless measure: it ignores correct scaling laws and so gives inflated readings of obesity for people who are taller than average. We delve into the history of bridges and shipbuilding, areas in which the naive assumption that everything scales up at the same rate can lead to disaster. And we learn how much LSD you should give an elephant: not as much as you’d think.

At the core of this thinking is the insight that biological and human networks differ in one crucial aspect of their scaling behaviour. Biological systems profit, up to a point, from economies of scale: technically, they scale sub-linearly, which means when an animal’s size increases, for instance, it requires proportionally only three-quarters as much food. By contrast, cities scale super-linearly: make a city bigger and you get proportionally more in terms of wealth and innovation per head — but also more crime and disease — by a surprisingly reliable factor of about 1.15. This is basically like compound interest acting on the city’s ‘social metabolism’, as West calls it. This is great news in many ways — at least so far. What it means mathematically, however, is that cities tend to grow ‘super-exponentially’. And that is not ideal, because to keep civilisation on a treadmill of super-exponential growth, world-changing innovations will need to happen more and more rapidly, and resource demands will rocket skywards — indeed, they will become literally infinite within a finite time span.

This is something for humanity to think about, West suggests mildly. Happily, though, he is no misanthropic deep-green calling for the dismantling of civilisation. The future of energy, for one thing, looks pretty good. (The annual energy needs for the entire planet are delivered gratis by the sun every single hour: it’s just a matter of photovoltaic technology and delivery systems catching up.) Other resources, such as minerals, however, are not as practically inexhaustible. So West’s message is that there really are hard limits to the kind of super-exponential growth to which we have become habituated, but taming it and trusting in human ingenuity should ensure a bright future for us all.

This is all written with great joy and a disarming humility: unlike many scientists popularising their own theories, West is careful to point out the limits of what his mathematical models can explain, even as he implies that the future of such research will be crucial to the continuing existence of high-tech urban civilisation. He also relates charming scenes from his own life: I particularly enjoyed his story of working as a teenager in a Limehouse brewery, which brought on a pleasant thirst.

Just as you think he has abandoned monster-movie fans, he even generously holds out some hope for the next *Godzilla *reboot. Suppose, West says, that this beast (with a heart the size of a blue whale) could stand up: in that case his thighs would have to have a diameter of 100ft each: in other words, he ‘would have to be almost all leg’. Thus, a monster that is basically two giant legs supporting a tiny body and head that exhales blue laser-fire is, after all, eminently feasible. Let us hope Hollywood takes note.