So recently I have been wondering what makes insects so strong. No really, they may not lift and pull the biggest weight in absolute terms. But those tiny beasts can sometimes carry a hundred and sometimes even a thousand times their body weight. Meaning insects have the most efficient body structure and use of muscle when it comes to doing work.

Take the **Onthophagus taurus*** – *a dung beetle if do not speak french(!) – the strongest males of this species can sometimes pull up to 1,140 times their body weight. To give you an idea of this feat, we can compare how this fares against the strongest males of our species. The dung beetle dragging around something that is a 1000 times its body weight is a more impressive feat than Brian Shaw (2015 winner of World’s Strongest man strongman competition) dragging around a blue whale.

That’s correct, a standard blue whale is only 740 times as heavy as Brian Shaw. So imagine a grown powerlifter dragging around a whale and not being impressed because you just saw a dung beetle drag around a large ball of, well, dung.

This is not an ability exclusive to beetles (although it is contender for an insect world record). Another insect that is all a serious lifter is a group of species of ants known as leaf cutter ants. These can carry more than 50 times its bodyweight in their jaws.

This is equivalent to a human lifting a truck with his teeth.

**What makes these feat possible?**

It lies at the heart of a mathematical/slash physics principle known as the** square cube law.**

If you bear with my primary-school diagrams, we’ll start with the basic principle.

Scaling length of each side of the cube by a factor of 10 makes it 10 times as high/long/wide. However, its surface area increases by a factor of 100, whilst its volume increases by a factor of 1000. We can then see that the volume and surface area are proportional to the cube and square to the length of the object.

The same is true for spheres. If you confer to amateur diagram number two, you’ll see this happening. It might not be as obvious because the sphere has weird number multiplying the cube of the length and the square of the length (which is equivalent to the radius), but we can just call that a constant of proportionality that makes the equation balance. The volume is still proportional to the cube of the length.

This can be generalised to more weird shapes, or rather shapes you are more likely to encounter in real life. The volume and surface area of most shapes follows the cube and square law, the only thing that is different is the value of the weird number that appears in front of the and .

So why am I dragging you back to tedium of geometry?

Unlike abstract 3D shapes, real physical objects have a weight that grows with volume. How heavy something is is directly proportional to volume, making it directly proportional to the cube of the length of the object.

Also of particular interest in this context is that for biological objects (my fancy way of saying, things that are alive) have muscles. The strength or force that these muscles exert or handle depends on the cross-sectional area of the muscle. A larger cross-sectional area contributes to a stronger muscle. Obviously there are lots of factors at play, such as leverage, the type of muscle finer and all sorts of other biological reasons that I do not know. But comparing two muscles with different areas, but otherwise identical, one will find that the muscle with larger area is stronger.

**To put it simply, if you enlarge an object whilst still maintaining the same shape that object will face a continual decrease in the surface area to volume ratio. **

The reason why such a relation between strength and area exists is beyond the scope of this post. Just treat it as a an experimental fact.

It may not look like it but we have gathered all the information we need to answer the question why insects can lift loads that far exceed their bodyweight.

We have all the experimental facts:

- The strength or force exerted by an organisms muscles is proportional the cross-sectional area, which is in turn proportional to the square of the length of the muscle. Making the strength of the muscle proportional square of the length
- The weight of an organism is proportional to its volume and therefore the cube of its length.

Okay, so now we do a common thing in physics and take away the the two forces from each other. This is called balancing the forces. It is a sort of comparison we can make using maths to see which one is larger.

So bear with me and let’s do some maths.

Where F is force and l is the length of the organism. So now let us compare the two forces, we can do this by finding the the ratio of the two forces. This gives an indication of the relative strength of the two forces.

This gives a mathematical function, which depends on the variable which is length of an organism.

What happens to the ratio as we increase the size and thus length of the organism. Fortunately, we are mathematicians so we do not have to actually go out and get ants, cats, dogs and lions and tell them to lift weights. We can plot the value of this function for every value of . What we get is this.

You can see that for small organisms (small ) the ratio has an incredibly large value, whereas for the larger organisms (big ) the ratio drops away getting smaller and smaller.

Of course this is a very simplified analysis and there many other factors one can include that will change how the function drops as increases. But the principle would remain the same. The force exerted by muscle compared to the bodyweight grows very large as organisms grow smaller and smaller.

This is the principle behind why insects can lift loads hundreds and thousand times their bodyweight. It also means if you were shrunk down you could jump distance several times larger than your height.

**Does this mean it is better to just to be smaller? (Do not even go there 😉 )**

The square-cube law affects more than an animals capacity to lift heavy things.

The metabolic rate also falls victim to this law. I’ll show you how the rate depends on the size of the organism. Of course, with anything biological the factors defining and determining the metabolic rate are a bit complicated.

But we are physicists, so we can simplify it to mean the battle between the energy we expend and the energy we consume.

So consider a horse, which is standing very still. So still in fact that its only energy expenditure is the heat loss from its body.

What we will try and calculate is the minimum energy the horse needs just to maintain its body temperature, how much food it needs to metabolise to generate enough heat to keep a steady temperature.

Of course in this consideration we will be ignoring all the complicated details. These are, that their temperatures aren’t uniform across their body, plus, animals are generally hotter the deeper you go in their body. You know where the warm blood is.

So any heat that radiates away is going to be absorbed by the various layers of tissues on its way out.

But, what won’t change is the general trend (this is the basic physics law we will be using) is that the amount of heat an object radiates away depends on the surface area. The heat is proportional to the surface area of the object.

**This is the minimum energy the horse and any animal needs to consume just to stay alive. **

Now suppose we do an ‘experiment’ with the horse. We shrink our horse down, in such a way that its density remains the same. Meaning the shrinking horse will weigh less by the same fact we reduce its volume by.

The density can be written as

Where M is mass of the horse. Since has to equal some number that does not change, we can write the following.

So combining our equations for and we get the following,

If you did not follow the maths, what the result states is that the energy consumption increases with the mass (but in a slightly weird way as opposed to a direct proportionality).

But what is more interesting or more informative is a sort of relative consumption rate.

To find this, we divide by the mass of the animal.

Why?

Because doing so will show how much the animal needs to eat compared its bodyweight.

So I went right ahead and divided by . I know. Daredevil.

This is just a mathematical function is similar to the one we had above about the muscle strength. It blows up as M gets smaller and smaller. This shows that smaller animals needs to eat more and more just to stay alive due to their hyper fast metabolic rate.

And infact this is what we see with the shrew (amongst the smaller mammalian critters). Shrews have to eat nearly every few hours, and even eat beyond their bodyweight per day because the digest food so quickly.

So it isn’t always advantageous to be small. Larger animals like us have the luxury to not spend most of our time eating (though many have tried) to stay alive, leaving us open to pursue other activities.

You could jump very high but you would have to eat all the time. Doesn’t sound too bad actually.

As a last bit of deliberation, this will place a lower limit on how small something can get. After a certain point can’t reduce further; there just isn’t enough time in the day for gorging that could match the metabolic rates the mathematics suggests.

This is why you do not see mammals smaller than a shrew. It is physically impossible, to get any smaller and chemistry and digestion habits would have to change comprehensively changed to counteract the effect of energy consumption via heat loss.

Also as a final final deliberation. The calculation I presented to you was a super simplified version. A more comprehensive one resulted in reproducing what is knows as Kliebler’s Law. This is the experimental observation in 1932 that the metabolic rate is proportional to . Notice the different exponent on .

A couple of physicist and biologists in 1997 successfully derive a different law where is raised to the power of 3/4 rather than 2/3. They did this using the concept of fractals, basically considering all the complicated stuff I ignored. This included the fact that the tissues in large organisms have a supply problem. Some cells are too far away to receive the same level of oxygen from the bloodstream as other cells in the body. Kliebler’s law can derived by treating mammalian distribution networks to be “fractal like”. You can read more about it here : https://universe-review.ca/R10-35-metabolic.htm

**The power of a fundamental law of nature**

I hope I have shown you why insects are the strongest organism. But what I really hope to have shown you through this exercise is not much to do with insects or horses.

It is to appreciate the far reaching consequences of a mathematical truth. Most of the animals on Earth including us, are shaped and sized the way they are because of physical and mathematical limitations.

The square-cube law affects how small or big or bulky something can be. A tapeworm can be enormously long but is forced to be thin, because oxygen and food has to penetrate directly through its skin, owing to where it feeds from.

Some animals have a maximum size, it is impossible for that animal to grow any larger (ignoring marginal differences). This includes insects,if they were any larger they would be crushed under the weight of their exo skeleton.

Given this incredible variety of beasts with such complicated features and evolutionary histories, it is astounding that at its most fundamental level there are only a few mathematical and physics principles governing the amalgam.