Here is the familiar image of King Kong atop the Empire State Building. There’s something profoundly incorrect in how he is depicted, a problem that is shared with almost all giant creatures in the movies, be they flies, dragons, or humans. The problem involves some math, which is why I’m discussing it here.

Before I can tell you what the problem is, we need to understand these facts:

1. If we enlarge any shape, its area increases by the square of the enlargement factor. For example, a rectangle of length L and width W has an area of LW. If we enlarge it 3 times, the area becomes (3L)(3W) = 3²LW = 9LW. This holds true for any shape, not just rectangles, and it is also true for the surface area of 3-dimensional shapes.
2. If we enlarge any 3-dimensional shape, its volume increases as the cube of the enlargement factor. As an example, a cube with edge length L has volume L³. Enlarge it by a factor of 3, and the volume becomes (3L)³ = 3³L³ = 27L³. Again, this holds for any shape.
   Note that this means that the weight also increases as the cube of size (provided the material it’s made of doesn’t change).
3. Consider a shape, say a cylinder, that is being compressed or stretched by opposing forces on its ends. For instance, it might be a concrete column supporting a roof. If the force along its axis is large enough, it will shatter. We can quantify this by computing the stress, namely (total force) / (area). If the stress exceeds concrete’s stress limit, the column shatters.
   As a corollary to this, the total allowable force for the column increases as the area of its cross section (provided the material it’s made of stays the same).

Now, think of some object, say your average gorilla. If we increase that gorilla’s size by a factor of S, keeping the shape the same, then his new weight becomes w S³ (property 2), where w is the initial weight. The weight bearing strength of his legs (which are columns), will increase by p S² (property 3), where p is the initial strength. So, the stress on those legs is (w S³) / (p S²), which is a constant times S. Do you see the problem? If you make the gorilla 5 times as big, the stress on his body caused by his weight will increase by a factor of 5 also. The poor guy’s bones will be crushed by his weight.

This simple phenomenon is the reason elephants have such stocky legs, and mouse legs can be very spindly.

Now let’s look at another aspect of scaling that is related to the ability of creatures to tolerate heat or cold.

First, consider that each cell gives off some heat as a result of metabolism. The amount of heat produced is therefore (roughly) proportional to weight. From property 2, then, this internally produced heat goes up as the cube of size.

Heat escapes at a rate (roughly) proportional to surface area, which is the square of the size.

Heat loss per unit of body weight would then be proportional to (k₁ S²) / (k₂ S³) = k / S. So, as the animal becomes larger, a smaller fraction of its internally generated heat escapes through the surface. Because of this, we would expect to see fewer very small animals in cold climates – it’s just too hard to retain body heat if the animal is small.

Finally, note that these considerations apply to lots of other things besides animals. An example is buildings; we can’t make a building 3 times as big without changing its proportions, or by making the material it’s made of stronger. If fact, it wasn’t possible to build skyscrapers until builders switched to steel for the support structure, because steel can withstand much higher stress than wood or masonry.

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This entry was posted on Thursday, October 1st, 2009 at 10:21 pm and is filed under Thrilling Math. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.