If aliens ever visit the earth, might they take the particularly schlocky form of giant ants? Should we be worried about the ape king of Skull Island clambering up the Empire State Building, Hollywood starlet in one hand and a military helicopter in the other? And can we look forward to the exciting prospect of enormous radioactive monsters like Godzilla and Mothra duking it out in the Japanese sea?

For better or worse (depending on how you feel about giant monsters) the answer is no. All of these creatures are essentially normal terrestrial animals scaled up to a ridiculous size, and when you scale a normal animal up to a ridiculous size, it simply collapses under its own weight. We don’t need to worry about getting invaded by giant ants from outer space, or having our landmark buildings ruined by oversized primates. It’s not going to happen.

To see why a giant ant or lizard or gorilla collapses under its own weight when you make it big enough, we’ll need to understand the maths and physics of scaling, in other words, how things change with size. And in fact, we’re going to do a lot more than simply rule out the existence of giant monsters. We’ll turn our understanding of scaling into a predictive tool, something you can use to learn positive facts about the world, and not just negative, sort of boring facts, like “King Kong vs. Godzilla is physically implausible”. I mean, we already knew that. Instead, we can do cool things like estimate the weight of the largest tree in the world.

But here’s the really cool thing. If we want to explain something as simple as a heartbeat, or the diet of a chipmunk, we’ll actually be led to the startling conclusion that animals are biological fractals: they are built out of structures that look the same when you zoom in. Maintaining a living organism comprises a bunch of interesting design problems, and nature evolved fractals in order to solve them. This is pretty damn cool.

The jelly cube monster

To illustrate the main ideas, let’s imagine a strange organism which looks like a cube, which we’ll call the cube monster. The scale of the cube monster is anything we can conveniently measure with a ruler, say the side length, which I’ll call $L$. To change the scale of the cube monster (to “scale up” or “scale down”) means to change its side length but keep the shape and proportions fixed; the cube remains a cube.

Different properties of the cube monster will vary in different ways as we vary uts size. For instance, the volume is the product of the length, the width and the height, just like any box. But since length, width and height are all equal for a cube, the volume is just

\[L \times L \times L = L^3.\]

If we scale up the cube by a factor of two, in other words, we double the side length, then the volume increases, not by a factor of $2$, but by three factors of two, one for each factor of $L$:

\[L \times L \times L \to 2L \times 2L\times 2L = (2 \times 2 \times 2)L^3 = 8 L^3.\]

The volume increases by an overall factor of $8$. More generally, if we change the scale by a factor of $f$, the volume will change by a factor

\[f \times f \times f = f^3.\]

Now, the cube monster is a material being, but instead of being constituted of flesh and bone like a human, we’ll imagine that it’s made out of jelly. And just like humans, who big or small are made of the same stuff, bigger and smaller cube monsters will be made of the same type of jelly, with the same material properties. One important property of a material is its density: how much weight it has for a given volume. If the density of jelly is the same for every cube monster, in the same way that bone and flesh have the same density for people of different sizes, then what determines the weight of a cube monster is its volume:

\[\text{weight} = \text{jelly density} \times \text{volume}.\]

This may all seem kind of obvious, but it leads to an important observation: weight scales the same way as volume. If we change scale (in other words, multiply the side length) by a factor $f$, the volume will change by a factor $f^3$, and since the density doesn’t change, the weight also changes by a factor $f^3$.

Suppose our jelly cube monster shuffles around on the land. It has to prop itself against the pull of gravity, and to understand the physics of this propping up, we can imagine a thin column of jelly sitting on the ground. As we make the column taller and taller and taller, the jelly at the bottom is going to get more and more pressurized by the weight of the jelly on top. Eventually,