Scale Analysis

There is a well-known saying: dividing the universe into things that are linear and those that are non-linear is very much like dividing the universe into things that are bananas and things that are not. Many things are not bananas.

Non-linearity is a hallmark of the real world. It occurs anytime outputs of a system cannot be expressed in terms of a sum of inputs, each multiplied by a simple constant — a rare occurrence in the grand scheme of things. Non-linearity does not necessarily imply complexity, just as linearity does not exclude it, but most real systems do exhibit some non-linear feature that results in complex behaviour. Some, like the turbulent stream from a water tap, hide deep non-linearity under domestic simplicity, while others, weather for example, are evidently non-linear to the most distracted of observers. Non-linear complex dynamics are around us: unpredictable variability, tipping points, sudden changes in behaviour, hysteresis are frequent symptoms of a non-linear world.

Non-linear complexity has also the unfortunate characteristic of being difficult to manage, high-speed computing not withstanding, because it tends to lack the generality of linear solutions. As a result we have a tendency to try and view the world in terms of linear models — much for the same reason that looking for lost keys under a lamppost might make sense: because that is where the light is. Understanding — of the kind that "rests in the mind" — seems to require simplification, one in which complexity is reduced where possible and only the most material parts of the problem are preserved.

One of the most robust bridges between the linear and the non-linear, the simple and the complex, is scale analysis, the dimensional analysis of physical systems. It is through scale analysis that we can often make sense of complex non-linear phenomena in terms of simpler models. At its core reside two questions. The first asks what quantities matter most to the problem at hand (which tends to be less obvious than one would like). The second asks what the expected magnitude and — importantly — dimensions of such quantities are. This second question is particularly important, as it captures the simple yet fundamental point that physical behaviour should be invariant to the units we use to measure quantities in. It may sound like an abstraction but, without jargon, you could really call scale analysis "focusing systematically only on what matters most at a given time and place".

There are some subtle facts about scale analysis that make it more powerful than simply comparing orders of magnitude. A most remarkable example is that scale analysis can be applied, through a systematic use of dimensions, even when the precise equations governing the dynamics of a system are not known. The great physicist G.I. Taylor, a character whose prolific legacy haunts any aspiring scientist, gave a famous demonstration of this deceptively simple approach. In the 1950's, back when the detonating power of the nuclear bomb was a carefully guarded secret, the US Government incautiously released some unclassified photographs of a nuclear explosion. Taylor realized that, while its details would be complex, the fundamentals of the problem would be governed by few parameters. From dimensional arguments, he posited that there ought to be a scale-invariant number linking the radius of the blast, the time from detonation, energy released in the explosion and the density of the surrounding air. From the photographs, he was able to estimate the radius and timing of the blast, inferring a remarkably accurate — and embarrassingly public — estimate of the energy of the explosion.

Taylor's capacity for insight was no doubt uncommon: scale analysis seldom generates such elegant results. Nevertheless, it has a surprisingly wide range of applications and an illustrious history of guiding research in applied sciences, from structural engineering to turbulence theory.

But what of its broader application? The analysis of scales and dimensions can help understand many complex problems, and should be part of everybody's toolkit. In business planning and financial analysis for example, the use of ratios and benchmarks is a first step towards scale analysis. It is certainly not a coincidence that they became common management tools at the height of Taylorism — a different Taylor, F.W. Taylor the father of modern management theory — when "scientific management" and its derivatives made their first mark. The analogy is not without problems and would require further detailing than we have time here — for example, on the use of dimensions to infer relations between quantities. But inventory turnover, profit margin, debt and equity ratios, labour and capital productivity are dimensional parameters that could tell us a great deal about the basic dynamics of business economics, even without detailed market knowledge and day to day dynamics of individual transactions.

In fact, scale analysis in its simplest form can be applied to almost every quantitative aspect of daily life, from the fundamental timescales governing our expectations on returns on investments, to the energy intensity of our lives. Ultimately, scale analysis is a particular form of numeracy — one where the relative magnitude, as well as the dimensions of things that surround us, guide our understanding of their meaning and evolution. It almost has the universality and coherence of Warburg's Mnemosyne Atlas: a unifying system of classification, where distant relations between seemingly disparate objects can continuously generate new ways of looking at problems and, through simile and dimension, can often reveal unexpected avenues of investigation.

Of course, anytime a complicated system is translated into a simpler one, information is lost. Scale analysis is a tool that will only be as insightful as the person using it. By itself, it does not provide answers and is no substitute for deeper analysis. But it offers a powerful lens through which to view reality and to understand "the order of things".