On the Existence of Dragons

Here you go, Ruth. I'm actually writing about it.

A statue of two dragons in Varna.

In our world, it seems that dragons shouldn't be able to exist. Reptiles don't fly, don't breathe fire, and don't grow to the size of buildings.

Some of these could possibly have evolved. Dragons might have descended from the same common ancestor as bird and reptiles, and diverged at the same time; it wouldn't be impossible for them to have some features similar to modern birds and others similar to modern reptiles.

Organisms have also developed many kinds of chemicals and poisons for defense...why not something similar to triethylaluminum, which ignites upon contact with oxygen? It can be synthesized from hydrogen, aluminum, and methane, none of which is too hard to find. And maybe dragons have a way to extract more oxygen from the air in their respiratory system, so they breathe out almost pure carbon dioxide (which would prevent them setting themselves on fire).

But there's one big problem: the square-cube law.

Edit: I've made a new post about the square-cube law, look here.

The square-cube law states that, as you scale something up, the volume changes faster than the surface area. The surface area increases as the square of the length, and the volume as the cube.

If you look at a graph of the two, you can see that they keep getting further and further apart.

And this applies to biology, because strength is proportional to the cross-section of the muscle, which is an area, while mass is proportional to the volume of the animal. This is why ants can lift such heavy weights compared to their own, but we can't. If you scaled an ant up to the size of a person, its legs would snap under the weight of its own body.

This is even more of a problem for things that fly. If you scaled up an airplane to double its size, it would weigh 2³=8 times as much, but it would only get 2²=4 times the lift of the smaller plane. It wouldn't even be able to take off.

So let's imagine we have a very small, fire-breathing dragon, a bit smaller than a Komodo monitor. Its bones could be hollow, like a bird's, to make it lighter. It could presumably be able to fly--after all, we've seen birds approximately that size that can fly perfectly fine.

Our small dragon is about two meters long. We want the full-sized dragon to be, say, 64 meters long, 32 times longer. Its new surface area will be 1024 times its original, but its new volume will be 32,768 times larger!

So there's no way it could fly, although with enough muscle it might be able to support itself. Possibly.

Or are we missing something?

Now I'm going to deviate completely from reality, so you might want to read the previous two posts first.

Imagine that we have a fold in the world, but not a complete one like in the previous posts. Only a partial one. This dragon could essentially create a little "ledge" for itself, in which gravity would be reduced or even negated.

Now it would have no problems! There are animals larger than it on Earth which survive just fine--blue whales, because they live in the water. The buoyant force essentially weakens gravity, so they can move however they want in three dimensions.

Let's assume, however, that our dragon is purely three-dimensional. I don't even want to try analyzing the "cube-tesseract law."

TL;DR: It's perfectly possible that dragons exist, even though the square-cube law still holds. They could even fly and breathe fire.

Quick Follow-Up

A few more things about the Gravity and Portals post...

One of the What-If.xkcd.com answers recently concerned airflow through portals. According to Randall's answer, the flow rate through portals in Mexico City and Boston would be about 440 mph.

The same thing would happen with our world-folding portals, assuming that you moved the air along with everything else. Maybe there are transparent walls on the "front" and "back" of our world in the fourth dimension?

[Stupid diagrams! Upload!]

This seems like it would pose a problem, if we had a very (very very) tall room.

But remember, we would still have to fold it. That takes quite a bit of energy, which is proportional to the square of the height (remember, W_net = F Δx = m_world g Δx, and the mass of part of a uniformly-dense world would be proportional to the volume we lift).

We're already using a LOT of energy in our folding, so let's assume the mass-to-energy-converter in our portal gun can't quite lift enough of the world to make a significant difference in air pressure. Portals on the moon are right out.

Gravity and Portals

As you might be able to guess, I've been playing quite a bit of Portal 2 recently.

If you haven't played it, it's a physics-puzzle-based game where you have the ability to create portals on certain surfaces.

These portals act quite a bit like the ones I mentioned in my earlier post, where you could "fold" the two-dimensional world.

But the problem with the portals in the game is that they break the Laws of Thermodynamics. These are a set of rules about how energy works, which can be reduced to this:

1. You can't get out more than you put in--there's no way to create or destroy energy.
2. You can't even get back just what you put in--a little bit spreads out into the universe. More subtly, some things are irreversible. If a perpetual motion machine fails because friction keeps turning some energy into heat, you can't put it in a box and extract that heat again in a useful way.
3. You can't break either of the first two rules. Ever. Under any circumstances. The only way you could do it would be to lower a perfect crystal to absolute zero, and this law says that's impossible*.

*Technically it's possible, but it requires that you do something an infinite number of times, which is rather difficult.

So what happens if you have portals to work with?

You can create energy by teleporting up to a ledge, which gives you more gravitational potential energy. Normally you would have to convert kinetic (motion) energy in order to get up there, which makes up for it. But if you can teleport, then that restriction is gone!

And you can also destroy energy, which should also be impossible. Say you step on a Faith Plate, a catapult-like thing that launches you upwards. The catapult converts potential energy from its battery or power source into kinetic energy, which it then imparts on you. That all works. Let's say you land on a ledge at the top. When you hit the ledge, your energy is transferred to it. Still following the laws.

But what if you place a portal underneath you at the top? Now you're at the bottom again...but where did all the potential energy go? It's just gone!

[Images Pending, Sorry]

But wait a minute.

The third law says that there's no way to break either of the first two. Did we just get around that? Is our test subject freezing to death in a room colder than outer space?

It makes more sense if you look at a two-dimensional Flatlander doing the same thing. Let's turn Flatland on its edge, so that we can see the effects of gravity.

Now, when we make our first portal up to the ledge, what happens?

There's our answer! We didn't actually gain potential energy at all--because when the portal is open, the two places are at exactly the same height!

When we created the portal, we actually lifted up the lower part of the universe, and that's where the energy came from.

And, if our Flatlander jumps off the ledge to use some of that energy, then gravity starts acting strangely...when they're in the "folded" part, there seems to be no gravity at all. That's not what we see in Portal 2, although it might be interesting if we did.

Now, when we close the portal, we open out the universe again, and everything goes back to normal.

So what about the second example? Can we still destroy energy?

As it turns out, no. Let's say we open the portal when our little test subject is at the peak of their trajectory, about to land.

[Images Pending, Sorry]

Since we lifted up half of their world, they didn't lose any energy. Hitting the ground below was the same as if they hadn't used a portal at all.

But when we CLOSE the portal, and set everything back to normal, then there's a release of energy. And since we set down a little bit more than we lifted up (since the Flatlander wasn't in the area we folded the first time), that's where the extra energy goes.

So that makes sense now, although gravity still behaves strangely around portals.

If the test subject jumps off a high platform, hopefully above something soft, then what happens when we open a portal?

Left: what we 3-dimensional beings can see. Right: What it looks like to a test subject.

To us it looks like they fall straight down, but to them gravity has suddenly broken. I've used a bit of hacking to show what it would look like in the video game itself. (This also shows the examples I was talking about.)

Strange, no? I can see why Valve didn't build their game this way. Maybe that portal gun contains a nuclear battery or some other small power source, which can not only provide enough power to fold the universe but which can change the effects of gravity (look at the effect when you hold a cube).

One-Point Perspective

People seemed to like my first post about the fourth dimension, so I'm going to attempt to write another. This one gave me a migraine when I first started thinking about it, so be warned.

As you might remember from geometry classes, there are five "Platonic Solids" in three dimensions. These are objects that are made up of a single two-dimensional polygon, repeated and connected. In math terms, they're the tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron. In geek terms, they're the d4, the d6, the d8, the d12, and the d20. They're named after Plato, who believed that the four elements of the universe were made up of little pieces, shaped like these solids. Water was the icosahedron, which is almost spherical and can roll around. Fire was the tetrahedron, which has sharp corners and is only stable with a point facing upwards. Earth was the cube, which can be packed together into bigger cubes without space between them. And air was the octahedron, for no real reason. (The dodecahedron was left out of this theory. It's now best known as the D&D die that nobody ever uses, the d12.)

If you think about it, these could be considered the 3-dimensional equivalents to regular polygons--shapes where every edge is the same length, such as a square or an equilateral triangle. So could there be four-dimensional equivalents, built out of 3-dimensional solids?

The easiest one to think about is the cube, so I'll just focus on that one for now.

The cube can be thought of as part of a series, going up through dimensions: in one dimension there's the line segment, in two dimensions the square, in three the cube, and...so on? The four-dimensional one is sometimes called the 4-hypercube, or tesseract.

Now, if you start from the one-dimensional line segment, you can form a square by duplicating the segment, then connecting them. So if we start with a one-inch line segment pointing forward-backward, we can place another segment an inch to the right, and connect the endpoints.

That gets a square. And we can repeat the process to make a cube: place another square an inch above the first, and connect the endpoints.

To make a tesseract, then, it would seem that you could just place another cube one inch along the fourth dimension, and connect the lines. That makes theoretical sense...but what would it look like, if you actually created one? To figure that out, we can turn to perspective.

Generally perspective is used to portray a three-dimensional object in two dimensions, by stretching and skewing parts of it.

If we took a folded paper cube, made of six squares, we could show it to a Flatland being by placing the six squares next to each other--"unfolding" it into a cross shape.

I didn't make this image, I'm bad at 3d animation.

And we can do the same for the eight cubes of the tesseract.

Nor this one, sorry.

Salvador Dali once made a painting based on this net.

But this picture doesn't help us very much. I, for one, can't really tell anything about what the finished tesseract would look like from seeing the arrangement of cubes there, even though I can see what shapes comprise it.

So the other way to show it to the Flatlanders would be to use perspective.

If you look at the picture, you can imagine the cubes being there, although of course it's two-dimensional. But if you tell the Flatlanders that each cube made of six squares, they will probably be very confused. Looking at the picture, one can see six rhombuses, but only two actual squares (the front and back). The other four are distorted by the perspective. And if you somehow animated your drawing to make the cube rotate, it would blow their minds. Somehow these squares are becoming rhombuses, and passing through each other, and even changing in size! But somehow this is supposed to be a solid object?

If it doesn't rotate, click it. Note: I didn't make this!

It might be easiest to draw the cube with perspective using one-point perspective, where every line leads to a single vanishing point. [The cube on the right.]

You can still see that this is a cube, although it's less clear than some other ways of drawing it.

I was thinking a few days ago that this form of perspective could also possibly be applied to a four-dimensional object. One-point perspective is probably the easiest to make. But somehow it will need to be applied to a three-dimensional object...

Looking at the cube, it looks like a smaller square inside a bigger square, with diagonal lines representing the third dimension.

So, why not try that with our tesseract?

Unfortunately, I can only upload two-dimensional pictures onto this blog, but here's a photo of my model.

Here's my simple model. The vanishing point is at the very center.

We could also put the vanishing point off-center, so that the squares or cubes aren't inside each other. That also looks weird, but different.

The lines go through the cube for clarity. The vanishing point is off to the right.

So I don't know if there really is a "best way" to show four-dimensional shapes. I personally like the perspective models better.

Animation can help, also, by letting us see how it looks as it rotates. Sadly, I don't have a way to project animation into three dimensions, so it will have to be on a screen or a page. It's good that our brains are tuned to see two-dimensional pictures as three-dimensional.

So here's our tesseract, rotating.

Made by Jason Hise, released into the public domain.

Now you can tell how those poor Flatlanders felt when they saw the rotating cube turn itself inside out.

And, just because I haven't screwed with your minds enough yet, here are the four-dimensional versions of the D&D dice. (You can also see the view from inside the "4d120".)

Enjoy! >: D