The “Square Cube Law”
In: Mathematics
The square-cube law is the mathematical relationship between “area” (or “surface area”) and “volume” and is used to explain certain phenomena we observe.
Consider a living cell as a sphere with a radius of r = 1. The surface area of this sphere is C * r^2 = C, where “C” is some constant number, and the volume of this sphere is D * r^3 = D. where “D” is a different constant number. If you increase the radius to 2, the surface area becomes 4C, and the volume becomes 8D. At r = 3, we have 9C and 27D.
The point here is that, since volume is the radius *cubed*, but surface area is the radius *squared*, volume will grow much faster than surface area as the radius increases.
Why does this matter? Let’s go back to our cell. The cell has stuff inside it that it needs to fuel, and it had stuff outside it that it needs to gather for that fuel. But the only place that things can move from “outside” to “inside” is on the surface.
Let’s suppose the cell eats bugs. With a radius of 1, it has a volume of ~1, so it needs 1 bug per hour to fuel itself. It has a surface area of ~1, so it can move 1 bug per hour through its surface. One bug in, one bug used, no problem.
At r = 2, we can move 4 bugs per hour through the surface, but we need 9 bugs per hour to fuel the inside! At r = 3, the cell can move 9 bugs per hour but needs to eat 27.
The punchline here is that this law makes it harder for living things, or really any similar systems, from getting too big.
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