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Big Friendly Giant and Square to Cube Ratio

Before you continue reading, I would like you to participate in a small poll. Ready? Imagine a giant grasshopper which is otherwise identical to a normal insect grasshopper, but is 50 cm (20 inches) of height. It is like a 20-fold zoomed-in version of the grasshopper, all body parts increased proportionally. Regular tiny grasshoppers can jump up to twenty times times their own body height. The question is:

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Big Friendly Giant (2016)

In this post I will explain which is the right answer to the poll and why the giants in the giant country in the movie Big Friendly Giant violate the laws of physics. My aim is not to critisise the movie, but to point out that physics in it is flawed (which is not necessarily a downside artistically speaking). I will also try to give an impression of what the physics of the giant folks should look like which would give the reader an intuition why there are no giants, why there are no giant grasshoppers and why elephants are different in their movement patterns than scaled up insects. Perhaps also why elephants are among the biggest land animals that have ever lived. . After the 1989 animated film, the recent movie Big Friendly Giant is the second popular screening of the cute fairy tale originally published as a book by Road Dahl where a little girl is stolen by a giant… Don’t worry, I am not gonna spoil more!… Below you can see illustrations and snapshots from the book and the movies of the scene where the giant is taking the girl with him.

It might not be so evident from the snapshots, but if you watch the trailers, or if you’ve seen one of the movies, you see that the giant is walking effortlessly in the same way as a human sized

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The original book, one of the first illustrations.

person would. He (and the other, even bigger giants in the movie) are moving around, running and walking in exactly the same way as if they were 1,82 metres (6 feet) tall.

Thus, if there were a giant grasshopper in the giant country, it could indeed jump in proportion to a tiny one, i.e. 20 times its height. If you answered “yes” to the poll above, then there is nothing weird in the giant’s behaviour in your opinion. However, that answer is wrong.

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Animation 1989

The problem is that even if you scale an animal in precise proportions, as if it were drawn on a computer screen and you “resize” it (this is how the giants in the giant country were made, I suppose), not all physical properties will scale up proportionally. This can be easily illustrated already with a simple geometric example: suppose you take a cube and “resize” it so that each side becomes twice as large:

cubes

When the size of a cube is doubled, its surface grows by a factor of 4 and the volume by a factor of 8.

Then the resulting cube will be 2x2x2=8 times in volume. In general, stretching the side of the cube n-fold will result in n³-fold increase in volume. By thinking of a grasshopper or a human as composed of tiny cubes, you can understand why this reasoning applies also to them. If you double the height of a living being, its weight is multiplied by eight. So, if you are 1,8 metres (~6 ft) tall and your weight is say 75 kg (~175 lb), then when your height is doubled and every body part increased proportionately (the length and circumference of the arms, waistline; basically everything that can be measured in length is doubled), your weight would be 8-fold and become 600 kg (1323 lb = 94 st). If your size is ten-folded, your weight is 1000-folded and you would weight 75000 kg (175000 lb). In the movie, BFG is around 4 times taller than the girl and around three times taller than an adult human, so he, being quite skinny would (given normal physics) weight around

3³ x 60 kg = 1620 kg (~265 st).

What about the grasshopper? A regular grasshopper weights around 3 grams (0.1 ounce) so if it is stretched by the factor of 20, its weight will become

20³ x 3 g =  24000 g = 24 kg (~3.75 st)

So what? These numbers don’t tell us whether the giant grasshopper can jump 20 meters high or why can’t BFG walk and run around in the same manner as a human does. The problem is that muscle strength increases proportionately to the cross-sectional area of the muscle, hence proportionate to the square (and not the cube) of the length. There are many intuitive ways to understand why this is so. A muscle is composed of muscle fibers. Here is a picture which I stole from Wikipedia:

The strength of a muscle is proportionate to the number of muscle fiber in it which in turn is proportionate to the cross-section area of the muscle.

The strength of a muscle is proportionate to the number of muscle fiber in it which in turn is proportionate to the cross-section area of the muscle.

The number of muscle fibers is directly proportionate to the cross-sectional area of the muscle and the number of these fibers determines the strength of the muscle. (Yes, also insect muscles are composed of muscle fibers.) Another way of looking at it is that force is always exerted in one direction by a three dimensional muscle which leaves two dimensions as carrying “force-exerting” potential. Area scales as a square of the length: if the side of a square is doubled, its area is quadrupled and if it is increased n-fold, the area is only increased n²-fold:

The area of a square is quadrupled when the side length is doubled.

The area of a square is quadrupled when the side length is doubled.

Getting back to our examples above: Suppose your mass is m = 75 kg. If you can jump to h = 50 cm into the air after exerting force (F) onto the ground for one second, you achieve the average acceleration of a = 3,13 m/s² (your potential energy at 50 cm is mgh = 75 x 9,81 x 0,5 Joules and this is equal to the kinetic energy mv²/2 in the beginning of the jump from which v and the average acceleration is therefore from 0 to v, i.e. a = v/s). Since

 F = ma,

increasing the force by a factor of 4 but the mass by a factor of 8 decreases the resulting average acceleration to half of the original one which means that after exerting the 4-fold force for one second, the double-sized you would achieve the speed of v/2, which means that the kinetic energy is twice as large (the m in the formula of kinetic energy mv²/2 is 8-fold while is one quarter of the original). Going back to the formula for potential energy at the peak of the jump, we get that the jump results in 1/4 of the original 50 cm, thus only 12,50 cm: If m’ is your new mass and v’ and h’ your new speed and jumping height, we have that the new potential energy m’gh’ is twice the old one mgh, so we have the equation

m’gh’ = 2mgh

 Where m’ = 8m and h = 0.5 m,  so we get

8mh’ = m   ->  h’=1/8 = 0.125 meters.

Thus with the same type of movement you would only be able to jump to 1/4 of the original height. But in fact there is more to it. You cannot even perform the “same kind of movement”. As we noted, the force results in half the acceleration of your body, so if your jump is preceded by an upward squat, the squat will accelerate with half the acceleration of your normal squat and so in the same time (one second) you will have accelerated to v/2 instead of v and your average speed during the squat will be v/4 instead of v/2, so during this second you move half of the normal squat, and since you are twice as tall, it will only be one quarter of the squat in proportion to your height. If you perform the full squat, however, it will take 4 times longer, but you will jump 1 meter high which means that your jump scales proportionately! In this case each of your muscle fibers will work four times as much and since there is 4 times more of them, you end up spending 16 times more energy (which is exactly how much is required to lift your new 8-fold body-weight twice as high). Or, you can compromise and perform half of the proportional squat in which case you will be propelled to the same height as the normal you — 50 cm. In a nutshell, the double-sized you has the following options (for mathematical details see below):

(1) Jumping 1/4 of the same height which in proportion to your body height is now only 1/8 of the jump. In this case your pre-jump movement looks like one-eighth of the movement you would normally do in the same time. The pre-jump movement takes the same amount of time as normally.

(2) Jumping twice as high as your original self would, i.e. the same amount proportionate to your body height, but the pre-jump movement takes four times more time and takes sixteen times more energy.

(3) A compromise: jumping to the same height as you normally would which is half in proportionate to the body height, in which case your pre-jump squat takes only twice as long as normally, but you still spend eight times more energy.

So we see that there is no way in which your jump would look anything close to normal when you are double sized. Your movements are generally four times slower, your jump is either quarter of the proportionate version of it, or you exhaust yourself with a 16-fold energy outburst. The general math including for the 20-times increased grasshopper and BFG can be found below.

But what happens if you are not scaled up proportionately? What if the muscles are thicker and stronger, can you then seem to move around normally? Having all your muscles 1.41 (the square root of two) times thicker in diameter will make their cross sectional area double in which case they would be strong enough to exert the 8-fold force.  This is definitely not the case with BFG, but might be the case with some other giants in the movie who are quite pumped-up. In this case you are not proportionately scaled up, and in fact you still move in a weird way: now half of the proportionate squat will propel you proportionately high while a proportionate squat (in the amount of lifting, but not in the amount of time) will propel you even higher. Problem is that these variables never scale proportionately to each other, so the movements of a giant will never look natural scaled versions of a normal human.

What about the grasshopper? For the sake of generality assume that it is scaled to be n  bigger (in the case of the above questionnaire, n = 20) and that its original height was (= 2,5 cm = 1 inch) and the scaled-up height is h’=nh. Denote the mass of the original grasshopper by whence the mass of the scaled-up one is m’ = n³m. The energy needed to propel the n³-weighting grasshopper 20 times its body height is

20m’gh’ = 20n³mgnh = 20n⁴mgh

which is 20n⁴ times the energy needed to propel the small grasshopper 20 times its bodyheight. The force that can be exerted by the legs of the giant grasshopper is F’n²F where F is the force that can be exerted by the small grasshopper. This force needs to be exerted for the distance of 20times as much as in the small grasshopper in order to achieve the same energy, because energy is force times distance (compare to the length of the squat of the dobule-sized you above). If we, however, insist that the force must be exerted only n times as much as in the original grasshopper, then that would require F’ to be equal to 20F. The former could only be achieved if the legs of the grasshopper are 20times longer than the already scaled-up legs and the latter can be only possible if the legs have the diameter equal to the square root of 20n³ times the original diameter which is the square root of 20times thicker than the scaled-up legs. Converted to concrete numbers in our specs with n=20 and 50 cm high giant grasshopper this means:

either 400 times longer legs (200 meter legs)

or

20 times thicker legs (probably around 80 cm in diameter)

Neither of these is clearly a proportionate scale-up. What about the proportionate situation? The kinetic energy it obtains by exerting the n²F force by his legs that stretch n times the original legs is n³ of the kinetic energy of the tiny grasshopper. If the original grasshopper jumps 20 times its body height, it is exerting the energy of 20mgh. Suppose the giant grasshopper jumps x times its body height in which case the energy is xm’gh’ = xn⁴mgh = n³20mgh, so we have that

x = 20/n.

In case of n = 20, the giant grasshopper can jump exactly its own body height. Note that the jump would look rather cumbersome as it would take 20 times more time to execute (given that the grasshopper normally jumps extremely fast, this would probably look like a regular kangaroo jump). Note that our choice n = 20 is only coincidentally the same as the twenty-times its own body height. If we choose n=40, the grasshopper can only jump half its body-height etc. Thus, in fact the third answer in the poll is correct; before the calculations I thought that the last answer (that it can’t jump at all) is the right one.

Applied to the giant, the formulas above tell us if a normal human being can jump 1/3 of her height, then BFG being 3 times taller (n=3) would be able do jump 1/9 times his body height after performing a triple long motion. Hopefully you have now better intuition why elephants move in quite a different way than insects and why there are no gigantic insects nor tiny elephants.

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3 replies
  1. Sam Hardwick
    Sam Hardwick says:

    I still think the last answer is correct, because the scaled-up grasshopper’s body wouldn’t be able to withstand the forces. It doesn’t have bones, and I think its legs would buckle somewhere.

    Reply
    • kulikov
      kulikov says:

      It has an exosceleton, and there are reasons (also related to square-to-cube ratio) why big animals don’t have exosceletons, so you might be right. The whole grasshopper might just explode immediately upon coming into existence. Poor guy.

      Reply

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