Getting the square-cube law wrong

Time for something unrelated to politics…


The square-cube law explains the relationship between the height, surface area, and volume of a shape. As you can see in the diagram, when you increase the height of a 2D shape, its area doesn’t increase linearly—it increases by the change squared.


The same is true when you increase the height of a 3D shape—the volume of that shape increases by the change cubed.


The strength of a muscle is directly related to its cross-sectional area. So if you take a muscle from a shorter guy and scale it up proportionally to fit in a guy that is twice the height, it’ll have four times the cross section (and therefore four times the strength) of the smaller muscle—but eight times the volume (and therefore eight times the weight).


This also means that shorter men are much better at bodyweight movements. The square-cube law says that for every 10% you get taller, you get 21% stronger—and 33% heavier.


So as you grow taller, you get heavier much faster than you get stronger. It’s no coincidence that the tallest male gymnast to ever win gold was Alexei Nemov, standing a modest 5’8”. Past that height, the most impressive gymnastic feats become literally impossible.

The above passage illustrates a common misunderstanding regarding the cube-square law. The square in the cube-square law pertains to the strength of the bone and muscle as it pertains to resistance to shear stressors, not actual physical strength as in ability to apply force. Because weight grows cubically but the cross section grows quadratically, this means the added mass exerts more shear stress relative to cross-sectional bone area, and this can make weight-bearing bones more vulnerable to fracture. This explains why elephants have much larger legs relative to their bodies than, say, ants. But physical strength, as in work output, still grows cubically, not quadratically.

A human limb can be approximated by a cylinder, the volume of which is given by the formula π*r^2*h. If a human is re-scaled by a factor of 1.2, then volume grows by 72% (1.2^3)–that is 72% more total muscle, too, because muscle is growing not just in terms of cross section but also in terms of length, too. Strength is related to total muscle volume, not just the cross-sectional area.

To understand visually this common misunderstanding about the square law, imagine if a muscle were flattened into an infinitely flat disk. Such a person would be infinitely strong because the cross section about the x-y axis would be infinitely large. It would imply that even a mouse, if its muscles were flattened into a disk of microscopic thickness would be stronger than an elephant. That’s absurd. There must be a volumetric third dimensional component too.

So rather, because there is more total muscle both in terms of length and width, for every 10% you get taller, you get about 33% stronger, not 21%. But in some instances it’s even more than that. Due to the mechanical advantage of long limbs, an additional factor is added (depending on how the force is being applied), making the total gain not cubic but rather quartic. However, long limbs for certain exercises are unpropitious (such as for the squat and bench press), and the mechanical disadvantage results in a reduction of output work by a scaling factor, so a person who is 20% taller but otherwise equally proportioned would only benchpress 1.2^2 times the amount of weight as the shorter person. But being tall is advantageous for the dead lift because of the mechanic advantage of longer limbs, combined with the cubic increase of overall strength, so deadlifting ability grows by a whopping factor of 1.2^4. The quartic multiplier is applicable to pulling or bending (due to long arms acting as a lever), such as trying to bend a steel rod with one’s hands or dislodging something that is affixed (like King Arthur pulling the Excalibur out of the stone). A similar fourth-order advantage also exists for boxing, arm wrestling, and wresting, which is why weight classes are so important.

To be continued…