As a three-dimensional object is scaled, its surface area scales by its square (^2), but its volume scales by its cube (^3). This is a fundamental mathematical law, and extrapolated, it predicts hard limits to the growth or scaling of any object or system.
For example, imagine a wooden beam. The strength of that beam is determined by its cross-sectional dimension, the thickness of the beam. As this beam is scaled, its thickness (the short side) squares, so its strength squares as well. But its weight, determined by its volume, cubes. It is therefore inevitable that, scaled up to a large enough size, the weight of the beam will far exceed the strength it gained from scaling. At this point, the beam would collapse under its own weight.
A similar effect occurs if you scale up a mammal. The heat produced by that mammal is a function of its volume, while its ability to disperse heat is a function of the surface area of its skin. This is why elephants evolved large ears: they are essentially two-dimensional and give the elephant a disproportionately larger surface area relative to its volume.
This evolved feature also offers insight into the concept of innovation in complex adaptive systems. The ears, by providing more surface area, allow the elephant to grow larger than it otherwise could. You might think of this as an innovation that lets the elephant “jump” to a new scaling power.
The insight for companies, as an example, is that if they continue to grow without innovating, they will collapse under their own weight. Software, for example, is an innovation that allows information to flow faster and at higher volume through the people (nodes) of an organization, allowing businesses to grow larger than they could have prior to the digital age.
Connections
Entropy And The Second Law Of Thermodynamics
Link Explanation: This note lays the foundation for the thermodynamic understanding of why systems collapse under their own weight and why innovation is the only way out. The core idea is that local order costs a larger increase in disorder elsewhere, and that disorder scales with the size of the system, its volume, node count, transaction count, employee count, while the systems capacity to dissipate or export that disorder scales only with its surface area, the interfaces it has to the outside world or to itself. This is the square-cube law above in action.
A business, like a body or a beam, generates entropy as a function of its volume (employee, processes, internal information transactions) but can only shed it through its surface area (customer touchpoints, information channels, interfaces). Innovation, in the connected note is identified as whatever manufactures new surface area, new channels for energy or information to flow with less loss, faster than the system’s own growth generates disorder.