# How would I move if I grew by a factor of 3 in each physical dimension?

Suppose, for the sake of this thought experiment, I am structurally identical to an average human, with the only difference being that my body is scaled in all directions by factor of 3. This would result in me having 27x more mass than a normal human.

In cinema, giants tend to move very sluggishly, and that seems to match our expectation of creatures of that size--after all, most people think of whales and elephants as being very large, slow-moving creatures.

However, I now also have 27x more muscle mass... Wouldn't my much larger muscles provide the necessary force to counteract the increase in inertia?

Barring a slight increase in air resistance, wouldn't my body movements (walking, running, moving my arms) be indistinguishable from that of a normal-sized human, and not sluggish as depicted in movies?

• Suggested book Jul 27 '17 at 9:42
• Also, read this. Jul 27 '17 at 9:49
• Hell no, you probably wouldn't even be able to walk. Your mass increases with volume, and strength with cross section of muscles, which is cubed vs squared, meaning the larger you are, relatively less strength you have in relation to your mass. Jul 27 '17 at 11:41
• Related Vsauce video. Basically it's related to square-cube law that has been mentioned in the answer. Might be worthy enough to watch the video from the beginning too (it's talking about the limit of human's size). Jul 27 '17 at 15:20

Assuming for a moment that your bones are proportionately stronger... (because you are asking about motion, not strength: but see for example this question about scaling in nature) That still leaves us with some physics that "doesn't scale well".

First, there is the issue of muscle mass: assuming your muscles are made of the same fibers, their strength (ability to exert a force) goes as the cross sectional area, while their power (force times velocity) scales with the total volume (if each element of muscle contracts by some amount, the total contraction of the muscle, and thus the velocity, depends on the length; and since the force depends on the cross section, the power scales with the volume; this also makes sense from an energy balance perspective, assuming that each cell expends a certain amount of energy per unit time, the total power will scale with the number of cells). So you are not actually "27 times stronger" when you are scaled up - your ability to accelerate yourself is less than if you were your normal size.

Second, there is the issue of inertia. Have you ever tried balancing a match stick on your finger? Hard, right? A fork is still quite hard, while a broomstick is easy. The reason for this is the moment of inertia. An simple rod has a moment of inertia $I=m\ell ^2$; if we scale all dimensions by 3x, the mass increases by 27x and the length by 3x, so the moment of inertia increases as the FIFTH power of scale. When you look at the effect of gravity on balance, the only thing that matters is the $\ell^2$ term - so the 3x bigger your will "tip over" much more slowly1.

This means that when you lift up a foot to take a step, it will take much longer for your body to start "falling forward" so you can actually take a step forward.

Of course once you are running, your superior strength will carry you further, faster - but when it comes to the "usual" maneuvering around, this extra size will be bothersome.

In nature, there is the additional complication that as things get bigger, they have to be built with stronger bones, etc. This is why a mouse seems to move so rapidly, and an elephant (giraffe) so slowly.

So yeah - the movies have it right. Giants are "lumbering". It's physics.

Incidentally, if I saw "giant you" running in the distance, it would look to me like gravity had been reduced: you would bounce "more slowly than I expected" because the time it would take you to land after jumping to "knee height" would be significantly longer (because your knee height is much higher than mine). That factor alone should mean that I would expect to have to speed up a movie of "giant you" by a factor $\sqrt{3}$ just so it would look normal. UPDATED: And the "slowing down" due to the moment of inertia thing has the same factor! This means that if we film "big" you, then speed up the movie by 1.7x, we should see something that "looks normal". And that's assuming you are strong enough to overcome the problem of muscle strength...

1: "...tip over much more slowly": the time constant of the motion goes as $\sqrt{\frac{2\ell}{3g}}$ as I derived in this answer about balancing a pencil on its tip. So when you are 3x taller, the time it takes to tip over will be about 1.7x slower.

• Thank you, this does clear things up a lot! I was hoping for the intuitive explanation you provided. I'll keep the question open for a day, then accept the best answer at that time. Jul 26 '17 at 21:35
• @maxathousand - I just realized a mistake in the last paragraph. I've now added a bit of math, and the conclusion is somewhat changed. Jul 26 '17 at 22:17
• @maxathousand -- Take a look at Walter Lewin's first or second lecture on YouTube, on dimensional analysis. He talks about this too. Jul 27 '17 at 6:04
• Is this answer trying to convince me that "elephant" is a different name for "giraffe"? ( ͡° ͜ʖ ͡°) Jul 27 '17 at 9:03
• @Floris: Sorry, I forgot about your comment until now. I first read about it in a book about Industrial Light and Magic's practical effects techniques that I don't remember the name of, but through Google Books I found another book talking about it: books.google.com/… (Zwerman and Okun, The VES Handbook of Visual Effects, 2014, pp. 362-364)
– user300
Aug 12 '17 at 21:29

Yes, you would have 27x the muscle mass. However, muscle mass is not actually the factor which defines the strength of a muscle. The strength of a muscle increases based on its cross sectional area, not its volume. For an intuitive image, consider a rubber band. How hard is it to stretch it apart? Now think of a much longer rubber band with the same cross sectional area. How hard is it to stretch? It's exactly the same.

This is known as the square-cube law. Many properties governing strength, such as muscle strength and bone strength, vary by the square of your scale. Others, such as mass, vary by the cube of your scale. As you get larger, many factors such as how you move and how you get rid of heat do not scale. You need different solutions on the larger scales.

If you assume the strength of your bones grows proportionately to your mass, then look to Floris' answer, which covers the physical challenges when strength is not a limit.

• "Now think of a much longer rubber band [...] How hard is it to stretch? It's exactly the same." Actually it's not, if we're speaking about stretching by a fixed distance (that is not relative to the length of the band). It's "easier" (in terms of work done) to stretch a longer rubber band by a fixed distance. Also note that the linear density of the longer rubber band will be larger (after being stretched) than the one of the shorter rubber band (being stretched by the same distance). Jul 27 '17 at 8:44
• Square-cube law is the central point here, but @a_guest is right: if you make it longer it will be easier to stretch (just calculate the elastic constant of two spring in series with identical elastic constant $k$: you will find $k/2$). So correct explanation but bad example... Jul 27 '17 at 9:48
• It's constant if you stretch by constant factor - i.e. to twice it's length. Jul 27 '17 at 10:07
• @Baldrickk That's true for the restoring force after the rubber bands (or springs) have been stretched but not for the work which was necessary in order to stretch them. A precise definition of what "hard" shall mean in this context would prevent any such ambiguities. Jul 27 '17 at 14:54
• @a_guest I should update the wording. The part I wanted to convey was that a rubber band that's 5x longer does not take 5x as much force. I'll see if I can modify the wording to capture that better. Jul 27 '17 at 14:59

Let's take a good look at this question. Taking it straight back to physics, we know that gravity is actually the weakest force, though it seems to be the strongest to us. This is of course because of size - because as mass grows, it's relationship to the outside world often does not grow proportionally.

Take for example, a flea. It can jump roughly 100 times its hight and land fine.

If you were shrunk to a size, likely about that of a rat (this is an estimate, I do not have a citing to back it up) you would be able to jump one if not several times higher than you are tall (rats not being able to because of their build).

Taking it back to bugs, if a bug was made the size of a human, as one might imagine, it's legs would almost certainly snap, not having enough support.

Though of course, no one has ever grown like you are suggesting (or bugs, hopefully) one can predict that there would be quiet a bit more strain on your body. It is unlikely that your bones would just snap, but you would be sluggish, and have a lot more trouble doing stuff, in general (without even getting into the digestive, endocrine, circulatory, etc. systems).

Just to add some factors from thermodynamics and chemistry:

The loading from body weight on joints would be $27\times$ greater; the surface area of the joint would be $9\times$ greater;

You would need to get $27\times$ as much air per second down a throat with only $9\times$ the cross-section;

You would need to get $27\times$ the food/day down the same throat;

You would need to absorb $27\times$ as much oxygen and exhale $27\times$ as much carbon dioxide through $9\times$ the lung surface area;

You would need to dispose of $27\times$ the heat energy through only $9\times$ the skin surface area;

Urine and feces are left as an exercise for the reader

• "9× the lung surface area" -- well this is an interesting point. If we assume that the lung has the same number of alveoli, each 27 times the volume, then we only have 9 times the surface area. A larger number of smaller alveoli would be more. But if we assume that the larger body has the same number of cells, each 27 times the volume, then I strongly suspect the cell membrane chemistry no longer works and we rapidly die of hypoxia no matter how much air we can breathe. So "structurally identical" is going to need some hand-waving to sort out the details down at smaller scales :-) Jul 28 '17 at 9:25
• Also, if the lung surface area is considered fractal, its dimension might be $>2$ and the scale factor would be different :) Aug 2 '17 at 9:12

The square-cube law described by Cort Ammon is the essential thing. Muscle power and bone strength grow like the square of linear dimension. Mass and inertia like the cube.

A vivid example is that a mouse, despite its flimsy skeleton, can fall off a high cliff and suffer no harm, but an elephant would break all the massive bones in its body because stress = force/area = mass x acceleration/area. For the same acceleration, stress is proportional to size.