Tag Archives: square-cube law

The Problem of Scale

Veritas Readers,

 

                A long time back ( 13 Feb 2000), I had written a VERITAS article “Size Does Matter” http://unvarnished-veritas.posterous.com/size-does-matter. I read it again and felt that it was badly written and did not really convey what should have been conveyed. So in this article, I will try to do a better job.

Ken: What are you reading son?

Sid: “Amazing Animals”, papa. Aren’t the pictures pretty. [ He shows Ken a page which has a number of animals drawn on the page]

Ken: Isn’t it interesting that the mouse and the man seem to be of the same size. Also the whale seems to be smaller than the elephant.

Sid: Yes. It is just a drawing. But wouldn’t it be interesting if the relative sizes of animals were changed?

Ken: You mean make a mouse 10 times bigger and an elephant 10 times smaller, or something like that?

Sid: Yes. Maybe in a different world this is how it happens.

Ken: It is an interesting thought- take a picture of animal/thing from a book and scale it to whatever dimension. But do you think there would be no other problems?

Sid: Yes there will be differences. So if all our dimensions were made 10X then we will have 10 times larger houses and would eat 10 times more food. But what is the big deal. We will just get used to the new scale soon and then that scale would be our “natural” scale. In fact, I think that if magically the scale of everything around us were to be made 10X larger or smaller, we would not even be able to notice.

Ken: I think there is a flaw in your thinking. Galileo pointed this out in his book “Two New Sciences”.  Haldane had written a very interesting essay about this topic- “On being the right size”. The principle is known as the square-cube law.

Sid: Square-cube law?! What is that?

Ken: Let me explain using an example: Take a cube of side 1 cm, Now magnify each side by 10 times. What is the new side:

Sid: 10 cm.

Ken: What is the new area compared to old area:

Sid: [ thinks for a while] 100 times.

Ken: And the new volume?

Sid: 1000 times!

Ken: So if you increase the linear dimension by D  the area increases by  D squared and volume increases by D cubed.

Sid: That is correct.  But how does that relate to size of animals?

Ken: Imagine that you become 10 times larger in all dimensions. Your bones will become longer by 10 X. Your arms, legs etc – all longer by 10 X. Tell me what the cross section of the bones will be.

Sid: Thinks for a while….. umm. That is area so it will become 100X. But that is okay.

Ken: That is okay and probably expected but now tell me your new weight considering that the volume becomes 1000X.

Sid: 1000 times!

Ken: Don’t you see a problem now: a 10 times increase in size has caused 100 times area increase and a 1000 times weight increase because weight is proportional to volume.

Sid: So my 10 times thicker bones, with 100 times more area have to bear 1000 times more weight! EEKS! They will break!

Ken: Yes. That is correct. So the general principle is that if the linear dimensions increase by X the volume and thus the weight increases by X cubed. Therefore the shape cannot be scaled to any dimension. There has to be a change of shape to be able to support the many times extra weight.

Sid: And so every animal has a right  or proper size.

Ken: Yes. And that is why the very small people and very large people described in Gulliver’s travels cannot exist!

Sid: Interesting.

Ken: Let me give you some more examples: muscle strength is a factor of muscle cross section(area) and bone strength is a function of bone area and these vary as square of the dimension of the animal. However as we have seen, the weight is a function of the volume or the cube of the dimension. So the relative bone and muscle strength is much more in small animals than in large animals. You have seen cats jump from great heights and land softly. Try that with a horse! Or another example: an ant can carry 50 times its own weight, a man can carry about its own weight and an elephant can carry only a fourth of its own weight.

Sid: Wow. So you cannot just scale a shape.

Ken: Yes. Lets apply this principle to flying.  Flying is easy for small creatures. And the limit of size reaches very fast. A 2 times larger bird does not require 2 times more power – it requires much more than that.

Sid: Cool.

Ken: Now take the case of a worm. It breathes through its skin. And it has a straight gut to absorb its food. Now increase its dimension 10  times and its weight becomes 1000 times. So it will require 1000 times more food, 1000 times more oxygen and will excrete 1000 times more waste. But with 10 times increase in length how much is the increase in the area of skin and area of gut?

Sid: 100 times.

Ken: So that is not enough. So as an animal becomes larger it cannot breathe through its skin and cannot have a simple straight gut. A man has lungs because we need the 70 sq m of  surface to take in the oxygen that we need.  And our size is the reason why we have more than 25 feet of gut and all that is coiled up. Imagine a man with 70 sq metre of skin –if he has to breathe through the skin. And if our gut has to be straight- it has to be 25 feet long.

Sid: So it is because of our size that we need complex structures like lungs and long coiled intestines.

Ken: The human lung has 2400 Km of airways and 750 sq feet of surface area( equal to about a tennis court) in a small compact structure.

Sid: Wow!

Ken: Yes. As Haldane said: “The higher animals are not larger than the lower because they are more complicated. They are more complicated because they are larger.”

Sid: So it seems that larger animals have to do a lot of complex things whereas smaller animals can more easily adapt to their surroundings.

Ken: There are situations where being large is advantageous. Let’s take the case of heat dissipation from the skin. All warm blooded animals lose heat from their skins. Larger animals have less surface area with respect to their weight as compared to smaller animals. So large animals lose heat more slowly compared to small animals. So we need to eat less for our weight. A mouse has to eat about a quarter of its weight every day- most of it just to keep warm. And that is why there are no small mammals in cold countries.

Sid: Interesting.

Ken: And that is why babies of warm blooded animals are much larger compared to the adult – they have to maintain the temperature and large size gives that advantage. A cold blooded crocodile can have tiny babies less than an inch but look at our babies- much much larger. The smallest mammals like bats have babies about one fourth of the adult weight.

Sid: So size does decide a lot of stuff!

Ken: Similarly the eye becomes an efficient organ only if it is large. Read the details in Haldane’s essay. This is related to the number rods and cones and how things come to focus.

Sid: So life is very different at different sizes.

Ken: Yes. Even the forces that are important are different. Our lives are dominated by gravity. For a man, falling from even 10 feet can be “bad”. Anything more can be fatal. An ant could fall from a 50 story building and nothing happens.  However for small animals the force of surface tension becomes very important. An ant that falls into a drop of water may never be able to come out. That is why insects have a proboscis  to drink water.

Sid: Interesting.

Ken: So the next time you look at a creature, realize that it has that shape and size for a reason. And that size is most optimal for its body and its place in the natural world.

[ Just then the door bell rings. It is the pizza delivery man. Ken and Sid open the box and are delighted to see that the Pepperoni Pizza is just the right size – Large]

 

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Go wondrous creature, mount where science guides

go measure earth, weigh air, state the tides,

instruct the planets in what orbs to run

correct old time, regulate the sun

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