Saturday, July 27, 2013

The square-cube law


Image by Jonathunder, under CC-BY-SA 3.0 licence


Here be dragons giants

The myths, folklore and literature of humanity account for many giants:

  • Giants of the Greek mythology who were the children of Gaia (Earth)
  • Christianity's Nephilims who were the children of angels/demons and women
  • Norse J├Âtnar, who were a mythological race of big fellows
  • Giants in Gulliver's travels
  • Giants in Francois Rabelais's novels of Gargantua and Pantagruel


Have you ever wondered whether giants are even possible? The Guinness book of records surely accounts for tall people. But my question is not on the actual existence of giants, but on the possibility of the existence of giants. So think about it for a few seconds! Could giants exist at all?

Care to take a few more seconds?

OK. I'll assume you're done thinking and I'll move to the answer. The answer is "No. They could not exist and their existence is impossible". Sorry! This article is intended to explain why.


So what about that square-cube law?

It is a description of how some body characteristics scale with the size of creatures. Imagine a standard person or object! Scale it up by a factor of 2! You get double the height, double the width, and double the depth (or thickness, however you call it). These measures scale linearly. The surface on the ground, or the section is multiplied by 4, because it is the product of 2 dimensions (let's say width and depth). Had we multiplied the scale by 3, the section would have been multiplied by 9. This is a square scale. As for the body's volume, you understand it increases as a cubic function since it scales with all 3 dimensions.

All things being equal in our object, weight scales like volume, as a cube function.

Before putting this data together, we also need a quick reminder of physics class:
Pressure = Weight / Surface

If you need an illustration to remind of this, think of snowshoes! they distribute your weight over a greater surface and thus reduce the pressure on snow and prevent you from getting knee-deep in the snow.


Why this matters is because weight scales as a cube function. Our skeleton, our bones, is what keeps us standing. Bone density being more or less constant, what matters is the section of bone on which the pressure of our weight will apply. More precisely, what matters are the bones in our legs since they are at the bottom of our body and support the greater weight. The section of our bones scales as a square of our height while our weight scales as a cube. So when we scale up in body size, the pressure endured by bones will scale as a cube divided by a square... which makes the pressure scale linearly.

If we double the size, the pressure endured by bone will be multiplied by (2³ /2²) = 2. As you can imagine, bone will break if too much pressure is applied on it. Due to evolutionary considerations (economy of calcium within the body)  we cannot consider denser bones. Instead, it is observed that the limbs of taller animals (e.g. elephants) have larger bones proportionally to the size of their legs in order to support their gigantic weight. But the downside of this is that it leaves less room for muscles.

And that's why giants are impossible. Because scaling with their size, the square-cube law explains that the pressure on their bones would force them to have deformities compared to a human body, like massive legs. And even these massive legs would account for extra weight, thereby creating a limit to what is possible.

Conclusion


  • You learned a cool scientific law.
  • You learned that giants are impossible.
  • Movies and myths featuring giants anatomically similar to humans are wrong.
  • Same rule applies to robots, unless more resistant materials are available.

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