Principles of geometry and physics predict the effect of large size differences on animal strength


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Super Bugs - Part 2

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In Association with
Why the Little Guys Can Do All the Pullups
     Are ants incredibly super strong? Would they be able to toss trucks around like volley balls if they were our size? What if we could shrink down to ant size? Say about 6.4 millimeters (1/4 of an inch for the metrically challenged) tall? How strong would we be compared to the mighty ant?

     And while we're at it, wasn't there always some little guy in PE class that could do 200 pushups and 30 pullups? The big guys, even the strong fast athletic big guys, just aren't usually the pull-up champions, are they? It's almost always the small guys.

     It's true for world class weight lifters too. The smaller people usually lift more with respect to their weight than the big people. It's not better muscles, or a more determined attitude, that helps the little guys, it's just geometry and the way muscles work.

     Like ants their feats are not "super" or even particularly impressive. If they couldn't lift more in proportion to their weight than bigger people, they would be real wimps.

     In cartoon land we created the "wild-eyed blue-legged cube creatures" to help visualize the geometry that explains why it's easy for the little creatures to be stronger for their size than the big creatures.
Jump down the next section below to learn their amazing story.
     Let me know what you think? Contact Me.
Click here to go back to Super Bugs - Part 1
Ponder the stuff below:
         Let's make up some facts about our cube creature.

     Look to the right at our first little Wild-Eyed Blue-Legged (WEBL for short?) cube creature. Its little cube body is exactly one cube tall and one cube wide and one cube deep.

      Let's say it weighs one pound. And let's say it can lift exactly ten pounds over its head. Pretty strong - ten times its weight over its head.
Surface Area

     The surface area is important also. Each flat surface on the outside of our cube creature is 1 flat square. So we can say the surface area of its face is one square.

      WEBL Size 1 Cube Creature:
        Height = 1; width = 1; thickness = 1
        Cross Sectional surface area of face = 1 square
        Weight = 1 pound

        Weight lifted over its head = 10 pounds
         It can lift 10 times its weight over its head.

What Happens When You Grow
         Look at the WEBL cube creature to the right. It has grown proportionately twice as big. It is now twice as tall, twice as wide and twice as deep.

     In other words it is two cubes tall, two cubes wide, and two cubes deep.
     But look closely. It has grown twice as tall, but its inside spaces have grown much more. Now it could fit 8 cubes of its former self inside itself. This measure of inside space is called volume. Our cube creature has grown twice as tall, but its volume has increased much more. Its volume is 8 times greater.

      If, as we said above, each cube section weighs 1 pound, then our cube creature now weighs 8 pounds. It is now 8 times heavier! Twice as tall but 8 times heavier!


     And don't forget the surface area. Look at our creature's face. Before, the surface area of its face was 1 flat square. Now its face has four flat squares. It also has four times the surface area through any cross-section, even though it only grew twice as tall.
WEBL Size 2 Cube Creature:

        Height = 2; width = 2; thickness = 2;
        Cross Sectional area of face = 4 squares
        Volume = 8 cubes
        Weight = 8 pounds
        Weight lifted over its head = 40 pounds
        It can now only lift 5 times its own weight over its head.
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     Ants and people are shaped and built very differently. Ants have six legs, big heads (with small brains), and huge jaws. We have two arms, two legs, small heads (with big brains), and two arms. Ants also have their skeletons on the outside and we have our skeletons inside.
     It would be very difficult (but not impossible) to compare ants and humans. Also, because of the way we are designed it would not really be possible to change our sizes as much as we are talking about on this page.

    We are "over simplifying"!

     Click on the above link to read about over-simplifying.

     "BUT WAIT!"
you say, "How do you know how much weight the Size 2 Cube Creature can lift?"

      How do I know that when the Wild-Eyed Blue-Legged Cube Creature gets 8 times heavier it can only lift 4 times as much? Now it can lift 5 times its weight. Before, when it was half as tall, it could lift 10 times its weight.
What changed?

     Well, his weight is simply increasing faster than his muscle strength is growing. Bio-physicists have told me in various papers and books, that muscle strength changes approximately with the change in muscle cross-sectional area. So that's all that has happened to our cube creature. The cross-sectional areas have increased by a factor of four, while the weight increased by a factor of eight.
      So even though a big guy can be much stronger than a little guy, if he is growing proportionately, his strength can't increase as fast as his weight is increasing.


     When muscles get bigger or smaller proportionately, the force they can exert changes mainly with the cross sectional area, not with the weight or length or width.

    This is true for all animals, including ants and humans.
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         Now look to the right at our Wild-Eyed Blue-Legged Cube Creature. It has grown ten times taller proportionately. When we say something has grown proportionately, we mean all the dimensions have increased by the same amount. It might be clearer to say all the dimensions have grown in the same proportion or by the same amount. So the WEBL to the right has grown 10 times taller, 10 times wider, and 10 times deeper.

     The same is the case with its muscles. They have grown 10 times longer, 10 times thicker, and 10 times wider.

     But look closely at the cube shaped guy. Now his surfaces are 100 times bigger (so the cross sectional area of his muscles are 100 times bigger) and his inside volume is 1000 cubes (10 times 10 times 10). He is 1000 times heavier!

    Look! Our cube creature has 1000 cubes inside itself! It got 10 times taller, but 1000 times heavier. Weight changes a lot faster than height. So it is with all animals if all dimensions change in the same proportion.

        Those of you that know algebra will recognize that our cube creature's (and any other animal's) weight increases or decreases with the cube of the change in length.

     So here is the bottom line. Our strange guy is ten times taller, 100 times stronger, and 1000 times heavier
    See what's happening. Our creation is getting stronger, but his strength is not increasing as fast as his weight. There is nothing to be done about it. It is a simple fact of geometry that can't be changed.

    Even though his muscles are, pound for pound, every bit as strong as they ever were, the WEBL went from being able to lift 10 times his weight, to being just barely able to lift one times his own weight.

     As you grow larger, your muscle strength grows, but not as fast as your weight.

(And actually, for reasons partially descibed in the over simplifying appendix below, we suspect your strength would actually increase less than is estimated here).

If you think you understand the way I am doing my figuring, try doing some of your own estimates. Let me know if I am doing this right. E-mail Me.
WEBL Size 10 Cube Creature:

        Height = 10; Width = 10; Thickness = 10;
        Cross Sectional area = 100 squares
        Volume = 1000 cubes
        Weight = 1000 pounds
        Weight lifted over its head = 1000 pounds
        Now it can only lift 1 times its weight.
(OK, the drawing doesn't look quite right. I should have drawn the barbell and weights bigger. They don't look like they weigh the same as the Cube Creature, do they?)

Let's do it for an ant.
     Let's say we have an ant that is 6 millimeters long and weighs about 3 milligrams (or 0.003 grams). This, according to one of my references, is about the size of a leafcutter ant. Pictures of leafcutter ants carrying leaf sections much bigger than themselves are common and they are often touted as being super strong.

     Well we found lots of claims about ant strength. The weight lifting claims ranged from 10 to 50 times their own weight. Every one of the sources claimed this was the equivalent of a man lifting 10 to 50 times his weight.
    Let's use 50 which I suspect is on the high side for most ants. That means our ant can lift 150 mg, or 0.15 grams.

     Now let's grow our ant to 6 feet (1829 millimeters) in length. An increase of 304.8 times the original length of 6mm. If she (most ants are females) grows proportionally, she will weigh about 187 pounds when she gets to 6 feet in length. Continuing to use the relationships described above, we can calculate that this frighteningly large ant will now be able to lift 13,935,456 milligrams. That sounds like a lot more. But alas, it is really only 13,935 grams or just about 30.7 pounds. (see the stuff below on over simplifying).

     Now our giant ant cannot quite lift one sixth of her body weight! No one would brag about that. And I suspect, though won't try to prove it here, that she will not even be able to lift that. She'll be lucky if she can lift her head.

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     If you shrank to 6 mm tall would you be able to toss ants around like juggling pins?
     The examples to the right and above would seem to indicate that we are stronger than ants. I think we are. Even stronger, maybe, than these estimates suggest.
     I still don't think l'd want to mess with them. Ants have those giant sharp pincher jaws and a nasty stinger on the other end, not to mention thousands of nest mates that will eagerly come running to help cut you up into bite sized morsels.
      As we said in
Super Bugs - Part 1
, the real strength of ants comes from their non-stop determination and huge organized numbers. They're everywhere.

     Pretend you are 6 feet tall and weigh 200 pounds. You're a pretty strong guy or girl and can push 100 pounds worth of barbell and weights over your head. Not exactly olympic material, but not bad. That's one half your body weight.

      Seems pretty pathetic next to our leafcutter ant that can lift and carry 50 times its weight (though I have never actually verified this. Have you?).      
(Remember we are over simplifying)

     But what happens if we shrink you?
     We zap you with the "Honey I Shrunk the Kids" ray and you shrink all the way down to about 6 mm or 1/4 of an inch.
    According to my estimation using the relationships described above, you now weigh 3.2 milligrams. Based on the change in cross-sectional area of your muscles you can now lift 488 milligrams.
     That's almost 153 times your weight!
     Now you are 3 times stronger than the whimpy ant!
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    Changing size proportionately - What we really mean when we say this, is that all the length dimensions of an object or animal's body, are changing in the same proportion, or by the same amount.
     Let's say you are six feet tall and you grow proportionately ten times taller. That means you are now 60 feet tall. According to our definition of proportionate growth all your other length dimensions have also increased by a factor of 10. If your leg inseam was 36 inches, now it is 360 inches. If your leg was 7 inches thick, it is now 70 inches thick. If your chest was 40 inches around, it is now 400 inches around. All your length dimensions, heights, thicknesses, widths, circumferences, increase by ten.

Simplifying -
Something engineers and scientists (and politicians) do all the time.
     It can be very useful and helpful, but also very dangerous, if you don't understand what you are doing.

     In the explanations above we are purposely ignoring many important factors that affect the weight lifting abilities of animals. We are over simplifying in order to show more clearly the effect of size on animal strength and weight.

     For example, we created the cube creature in order to more easily visualize how volume changes with size. It is easy to see in our cube creature how cross-sectional surface area increases with the square of length and how volume increases with the cube of length. These relationships hold true for complicated insect and human shapes too, but would be very difficult to show on a website, and imagine trying to calculate the volume of an ant by measuring its outside dimensions.

     Also, there are many factors, in addition to size, which affect an animal's strength. For example, a large kangaroo that is the same weight as a small human would certainly be able to jump higher and farther. Its legs are better designed for jumping and the muscles used for jumping make up a much higher percentage of the kangaroo's total weight than the jumping muscles of a human.

      Then, of course, when it comes to ants and humans there is the not-so-little matter of extremely different body types. Not just the number of legs and body shape, and the fact that ants lift with their heads, necks, legs, and jaws and we use hands, arms, backs and legs; but there is the fundamental difference of exoskeleton versus endoskeleton. The ways that muscles are attached to the skeleton has to be different and therefore there must be a difference in the leverages and efficiencies of the muscles. Also the rigidity of the exoskeleton must limit how big certain muscles can be.

     Finally, it is not so easy to grow bigger or smaller. Lots of things can't really be scaled up in proportion as we have assumed here. Bones for example. As we have shown here, increasing size means weight is increasing 3 times as fast as height. Bones have to get a whole lot stronger which means they have to be a lot fatter and take up more room (eventually they would have to get so fat there wouldn't be room for muscles and organs). More and more of a large animal's strength is used to just lift his bones and mass. If you grew taller and taller keeping all the same proportions you have now, your bones would eventually collapse under your own weight. Yuk!

     Remember the Amazing Colossal Man? A movie about a 50 foot tall man. It wouldn't work. His bones wouldn't have supported him.
      And I suspect an insect the size of a human would not be able to even lift its own weight. Seen any really big insects lately?


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