In 1638, Galileo Galilei described a simple law:

The ratio of two volumes is greater than the ratio of their surfaces.

To understand what Galileo was talking about, just imagine a cube. If you double the size of a cube, the surface area increases by the square of the length (2len^{2}) and volume increases by the cube of the length (2len^{3}). The surface area will be 4 times larger (2^{2} = 4) and the volume will be 8 times larger (2^{3} = 8). The ratio of volumes is greater than the ratio of surfaces.

The square-cube law says that as a shape grows in size, its volume grows faster than its surface area. This turns out to be a powerful idea for many scientific fields. For example, in mechanical engineering, it explains why a scale model engine won’t account for the heat loss of a full-scale engine, or why an airplane’s wings need to scale faster than the planes fuselage, or why building taller and taller skyscrapers is increasingly difficult.

Living things are also shaped and constrained by this simple mathematical principle. In this post, we will explore the biomechanics of the square-cube law as they relate to a human. Of course this could be generalized to any organism.

### Biomechanics

Consider a simple model of a human that contains four variables: weight, muscle strength, bone load-barring capacity, and skin heat-dissipating capacity.

Weight depends on the volume of the human. Think of weight as a function of mass and gravity, mass as the amount of matter in an object, and volume as a good proxy for that.

Muscle strength depends on the cross-sectional area of a muscle. Think of this as the muscle volume divided by the fiber length of that muscle. Dividing volume by a length, we get an area.

Similarly, bone load-barring capacity depends on the cross-sectional area of a bone. Think of this as the width of the bone (why elephants have short wide legs).

Skin heat-dissipating capacity depends on surface area. Think of how a small ice-cube melts faster than a large ice-cube.

### Giants

Let’s make our human 10 times larger and see what happens. He now weighs 1000 times more (weight^{3}) but he is only 100 times stronger (strength^{2}). His bones will be subject to 10 times as much stress as before (weight^{3} / load-barring^{2}) and his heat-dissipating capacity will be 10 times less (weight^{3} / heat-dissipating^{2}).

Our giant wouldn’t be able to lift his arms, his bones would shatter, and he would overheat.

### Ant-Man

Shrinking brings another set of problems. He would freeze since his heat-dissipating capacity is more than his metabolism ability to create heat. He would suffocate since he has 8 times less lung volume to process oxygen. He would be blind since his iris opening wont be able to process a fixed wavelength of light.

The point is that size and scale matter.

### Breakpoints

At a certain scale, all systems reach a critical mass that either breaks them or transforms them. In our human example, small changes in size are manageable, but at a certain point the system breaks down. In chemistry, small additions of energy are manageable, but at a certain point the system transforms.

This model also translates to business. When a business is young, increasing scale should result in better returns (lower average cost and higher sales). But there is a point where the business can get too big. This is called diminishing returns to scale. In the 1970s, we saw a bunch of giant conglomerates go bust. As they got larger, they lost focus and the ability to execute nimbly. Scale matters.

### Conclusion

The square-cube law explains how volume and area vary as the size of shape changes. Volume varies faster than area. This is why a doubling in size doesn’t always lead to a doubling in performance. This is true for physical systems, biological systems, and business systems.

References:

https://en.wikipedia.org/wiki/Square-cube_law

https://en.wikipedia.org/wiki/Biomechanics

https://en.wikipedia.org/wiki/Physiological_cross-sectional_area

http://amzn.to/2dD5JIF