In my article The expanse: dimension, separation and explosion, I argued that the graphs of Wolfram Physics are going to have to be three-dimensional to be a true representation of our universe.

Take *this* graph:

*Is* this graph three-dimensional? It’s so convoluted that it’s difficult to tell. It looks like it might be *two*-dimensional. Or somewhere *between* two and three dimensions.

I’m going to make the question even *more* difficult.

Right now, we’re looking at this graph from the outside, from a God’s-eye view.

In reality, though, we’re not outside the graph. Remember, we’re hoping that the graphs of Wolfram Physics will prove to be a true representation of our universe, and we *can’t* be outside our *own* universe.

How could we tell whether a graph is two-dimensional, or three-dimensional, or even two-and-a-half-dimensional, from *inside* the graph?

How would we measure the dimensionality of our *own* universe?

## Crabs can walk in any direction

Imagine a two-dimensional crab in a two-dimensional universe:

If you find it *hard* to imagine a two-dimensional crab in a two-dimensional universe, think of it as a drawing of a crab on a flat sheet of paper. The drawing of the crab can move across the two-dimensional surface of the paper, but it can’t jump *off* the sheet of paper into a third dimension.

How would the two-dimensional crab *know* that it’s in a two-dimensional universe?

Well, here’s a way it could find out: it could measure how much space it covers moving different distances in every possible direction.

Let me break that down.

First, let’s decide the distance the crab’s going to move. There’s one obvious measure of distance in the crab’s universe, and that’s the length of the crab itself:

What happens if the crab moves that distance – one crab length – in every possible direction?

Well, it traces out what we, from our God’s-eye perspective, recognize as a circle:

Next, let’s decide how the crab, which doesn’t have the benefit of our God’s-eye perspective, is going to decide how much space it has just covered. Again, there’s one obvious measure of the amount of space the crab has covered in its universe, and that’s the amount of space taken up by the crab itself:

So how much space did the crab cover, by this measure?

Well, that’s a crab in the middle, with half a crab above it, half a crab below it, two more half-crabs either side, plus some small bits of a crab. That’s a total of three and a bit crabs’ worth of space.

I’m going to call it 3.14 crabs’ worth of space. I know, that’s weirdly precise, but this is a weirdly precise crab, when it comes to measuring space.

To summarize: by moving 1 crab length in every possible direction, our crab finds that it has covered 3.14 crabs’ worth of space.

If we call the distance the crab moves **r** and the amount of space it covers **V**:

r = 1 → V = 3.14

I’m using **r** for radius and **V** for volume. You’ll see why **r** for radius makes sense, seeing as, from our God’s-eye perspective, the crab is tracing out a circle. But you might be wondering why I’m not using **A** for area rather than **V** for volume. Well, here’s the reason. From our God’s-eye perspective, *we* can see that the crab’s universe is two-dimensional, and that the amount of space it covers is an *area*. But the *crab* doesn’t know that, yet. For all it knows, its universe might be three-dimensional, in which case **V** for volume would be more appropriate. Or its universe might be four-dimensional, in which case **h** for hypervolume would be better. I’m going to refrain from pre-empting the crab’s discovery that its universe is two-dimensional by always using **V** for an amount of space.

## Have crab legs, will travel

What happens if the crab move *two* crab lengths in every possible direction?

Well, it traces out a larger circle:

How large?

Well, that’s 9 whole crabs, plus a total of maybe 3½ bits of a crab. I’m going to call it a total of 12.57 crabs. Remember, this is a weirdly precise crab.

So by moving 2 crab lengths in every possible direction, our crab finds that it has covered 12.57 crabs’ worth of space.

r = 2 → V = 12.57

What happens if the crab move *three* crab lengths in every possible direction?

It covers 28.27 crabs’ worth of space.

r = 3 → V = 28.27

The crab can go on doing this until it reaches the edges of its universe:

r = 4 → V = 50.27

r = 5 → V = 78.54

## Crab power

As well as being weirdly precise, our crab is mathematically minded, so it comes up with a formula for how much space it covers moving a specific distance in every possible direction. You may have come across this formula before:

V = π r^{2}

Yep, it’s the formula for the area of a circle.

Our crab finds that this formula precisely matches the data it has collected about its universe:

r = 1 → V = π r^{2} = π × 1 × 1 = 3.14

r = 2 → V = π r^{2} = π × 2 × 2 = 12.57

r = 3 → V = π r^{2} = π × 3 × 3 = 28.27

r = 4 → V = π r^{2} = π × 4 × 4 = 50.27

r = 5 → V = π r^{2} = π × 5 × 5 = 78.54

From this formula, our mathematically-minded crab can deduce two things about its universe.

V = π r^{2}

That first number in the formula, π, tells the crab that its universe is flat. If, instead of a flat sheet of paper, the crab were living on a *curved* surface, that number would be more than or less than π.

V = π r^{2}

More important, for our current purposes, is the other number in the formula, 2.

The crab finds that the amount of space it covers increases with the distance it moves to the power of 2.

*That’s* how the crab can tell that its universe is *two*-dimensional.

## Some crabs can swim

Now imagine a *three*-dimensional crab in a *three*-dimensional universe.

This time, you can think of a crab in a rock pool, in which it can swim up and down as well as move forwards, backwards and sideways. And yes, though sideways is definitely a crab’s preferred direction, some crabs can, indeed, swim.

Just like our two-dimensional crab, this *three*-dimensional crab could measure how much space it covers moving different distances in every possible direction.

As before, it could use its own length as a measure of the distance it moves, and the amount of space it takes up itself as a measure of the amount of space it covers.

And as before, it could come up with a formula for how much space it covers moving a specific distance in every possible direction, a formula that precisely matches the data it has collected about its rock pool... I mean, its universe.

The three-dimensional crab’s formula would be *different* from the two-dimensional crab’s formula. Again, you may have come across this formula before:

V = 4/3 π r^{3}

Yep, it’s the formula for the volume of a sphere.

V = 4/3 π r^{3}

Again, the important number for our current purposes is the exponent, 3.

The *three*-dimensional crab finds that the amount of space it covers increases with the distance it moves to the power of 3.

That’s how *this* crab can tell that its universe is *three*-dimensional.

## Inside the universe

Thanks to these crabs, we have a way to measure the dimensionality of a universe, not from a God’s-eye view of the universe, but from *inside* the universe.

All we have to do is measure how much space we cover moving different distances in every possible direction, and determine whether that amount of space increases with the distance to the power of 2, or to the power of 3, or to some other power.

In my next article, I’m going to apply this method not to a continuous two- or three-dimensional space, like the crabs’ universes, but to a graph generated by Wolfram Physics.

At last I’ll be able to answer the question: how many dimensions are there in *this* universe?