The word scale, when it refers to an object, may refer to a simple ruler. However, the act of measurement is the start of a deep relationship between geometry, physics and many important endeavors of humanity.

Scaling, in the geometrical context, is the act of resizing an object while preserving certain essential characteristics such as its shape. It indicates a relationship that is robust under certain transformations.

One uncovers scaling relationships by measuring and plotting key variables against each other. The simplest scaling relation is a linear one—yielding a straight line on a X-Y plot—denoting proportionality. The number of miles you can travel before refueling scales roughly linearly with the amount of fuel left in your gas tank. Twice the gas—double the miles that can be driven!

More complicated scaling functions include power-laws, exponentials and so on each of which are reflective of the underlying spatial geometry or dynamics of the system.

The circumference of a sphere scales proportionally with its radius. The surface area and volume scale, on the other hand, as the square and cube of the radius respectively. These power-law relationships show up as straight lines on a log-log plot. The slopes reflect the dimension of the object being measured demonstrating that length is one-dimensional, area is two-dimensional and volume is three-dimensional. This is true regardless of the shape of these solids—spherical, cubical and so on - and so plots using other solids will show the same robust scaling relationships and scaling exponents.

But sometimes length does not scale with an exponent of one—what then?

Richardson’s pioneering measurements of the length of coastlines across the world showed that measuring with increasing precision gave ever-increasing answers for their lengths. The various data sets (measurements with different precision) for each coastline scaled using an exponent greater than one.

Benoit Mandelbrot, in his classic paper “How long is the coast of Britain?” provided the insight needed to understand this result. As one measures the coastline with ever-increasing precision, one measures and adds the lengths of ever-smaller fjords, nooks and crannies. The coastline is not a smooth object; over a certain range of length-scales it is rough and undulating enough that it behaves like a self-similar object with a dimension between one and two. This insight led to the development of fractal geometry—the geometry of irregular, fractured objects: coastlines, mountains…broccoli. Systems that possess the property of self-similarity often show power-law scaling—alluding to a beautiful link between geometry and physics. Incorporating fractal geometry also turns out to be important for computer algorithms to generate the realistic-looking artificial worlds in movies and video games that we now take for granted.

More generally, a variety of natural phenomena appear scale-invariant (or look the same) when an important underlying physical parameter is changed (or rescaled) in a specific way. This physical parameter in many cases is a carefully constructed dimensionless quantity e.g. the ratio of two key length scales in the system being studied. The equations describing the phenomena must then obey and thus reflect the observed scale invariance.

Richardson’s delightful ditty captures the scale invariance of the Navier-Stokes equations of fluid turbulence—which remain unsolved to this day.

Big whirls have little whirls that feed on their velocity,
and little whirls have lesser whirls and so on to viscosity.

The laws of physics embody scaling. Newton’s law of gravitation states that the attractive gravitational force between two bodies scales linearly with each of their masses, and inversely with the square of the distance between them.

If you get large enough or small enough, almost any scaling law observed in the real world breaks down. This makes us think about the range of applicability and what causes the break down. The scaling relations implied by Newton’s laws break down at very small distances—giving way to quantum mechanics. The same happens at very large velocities - giving way to Einstein’s theory of relativity.

Thus it is often useful to refer to a scale to specify the range over which a theory or observed phenomenon is applicable: one refers to the quantum or sub-atomic scale, the human scale, the astronomical scale and so on.

Scaling concepts have found broad applicability in unexpected areas such as the study of social networks.

Technology companies are often not constrained by geography and can easily scale up their user base. Often early stage technology startups show exponential increases in usage that are then reflected in exploding valuations. Supply-demand problems that require resources to scale exponentially are simply not sustainable. As companies mature the exponential scaling and valuation must level off. On the other hand, power-law relationships and solutions indicate a business that may be scalable for a sustained period and are sought after by investors with a longer time horizon.

During the first internet bubble (circa 2000), companies rushed to get users and valuations were based on the number of people using their web-sites.

Today, social network companies value users on the strength and quality of their interactions with other users. One can argue that valuations of these companies should scale quadratically with the number of users. It remains to be seen if such scaling arguments used to justify higher valuations hold up, compared to more traditional measures such as revenue and profitability of a company.

Scaling relationships inevitably break down beyond a certain range, providing an important clue that other effects either ignored or not yet considered are now important. One should not ignore these deviations. While solving for persistent social problems that appear at all scales one should remember—what works for a family may not work for a business. What works for a business may not work for a nation.