We know what it means when we say that our universe is three-dimensional: it means that we can move in three orthogonal directions: left-right; up-down; forwards-backwards.

But what would it mean to say that a universe is 2½-dimensional?

Or 3.37-dimensional?

Or 9-dimensional?

This is *not* an academic question. When I measured the dimensionality of *this* universe in my article Are Wolfram’s graphs three-dimensional?, I found it to be *at least* 3.37-dimensional:

If Stephen Wolfram is right, and graphs like this are a true representation of reality, then our universe might *not* be uniformly three-dimensional.

So maybe dimensionality isn’t quite what we think it is.

What, exactly, *are* dimensions?

## Don’t look too closely

The first thing we can say is that if graphs are as convoluted as the ones we’ve been looking at, then dimensions don’t make sense on a small scale.

Dimensions make sense for *this* well-ordered graph:

There are obviously two different directions here, left-right:

and up-down:

This graph is obviously *two*-dimensional.

Dimensions also make sense for *this* well-ordered graph:

There are obviously three different directions here, left-right:

up-down:

and forwards-backwards:

This graph is obviously *three*-dimensional.

But this *convoluted* graph?

There are no obvious directions. No left-right, no up-down, no forwards-backwards, just a convolution of connected nodes.

If our universe is as convoluted as this, then dimensions don’t make sense on such a small scale.

## Different neighbourhoods, different dimensions

There’s another thing we can say about the dimensionality of convoluted graphs.

Remember when I measured the dimensionality of this graph, I said it doesn’t matter which node we start from?

Starting from the node at the bottom, we found that the dimensionality reached 3.37:

Let’s start from the node at the top instead. This time, the dimensionality reaches 3.26:

Now, 3.26 is pretty close to 3.37. It’s not surprising that these numbers are a little different, given the randomness in this graph. Still, these numbers *are* different.

If we start in a different place, we get a different answer.

This is a small graph, and we’ve already seen that it’s hard to find the dimensionality of a graph that’s too small.

So let’s measure the dimensionality of a larger graph:

I generated this with the same rule of Wolfram Physics, but I let it run for longer.

Starting from a node bottom-left, we find that the dimensionality reaches 2, levels off and dips back down to 1.91, then reaches 3.57 before falling off:

Now let’s start instead from a node somewhere in the middle. This time, the dimensionality reaches 2.12, dips back down more precipitously to 1.3, but reaches 3.51 before falling off:

The numbers of dimensions we measure are similar, but not identical, for the two different starting points. There’s no getting away from it: if we start in a different place, we get a different answer.

For these convoluted graphs, dimensionality is *local*: different neighbourhoods of the universe can have different numbers of dimensions.

## Dimensions of a different kind

Confession time.

If you’re confused by my discussion of dimensions, it might be because I’ve been conflating two different kinds of dimensionality.

There’s the *directional* kind of dimensionality. This is the kind we’re most familiar with, the kind that allows us to say that our universe is three-dimensional because we can move in three orthogonal directions: left-right; up-down; forwards-backwards. If you want a mathematical name for it, these are called *topological* dimensions:

Then there’s the *metric* kind of dimensionality. This is the kind that allows a mathematically-minded crab to determine the dimensionality of its universe by measuring how much space it covers moving different distances in every possible direction. Again, if you want a mathematical name for it, these are called *Hausdorff* dimensions:

If a 3.37-dimensional universe seems unimaginable to you, that might be because you’re thinking of 3.37 *topological* dimensions – which, you’re right, is unimaginable – whereas what we’ve been measuring is 3.37 *Hausdorff* dimensions.

In a uniform graph like *this*, these two different kinds of dimension match up:

In this cubic grid, *Hausdorff* dimension is well-behaved: it’s the same in every neighbourhood and at every scale beyond the very smallest scale. It’s *three*.

In such a well-behaved universe, *topological* dimension makes perfect sense. In this universe, it makes sense to say that we can move in *three* orthogonal directions: left-right; up-down; forwards-backwards.

But what if our universe *isn’t* so simple?

In a convoluted graph like *this*, the two different kinds of dimension *don’t* match up:

In this graph, *Hausdorff* dimension is *not* so well-behaved: it’s different in different neighbourhoods and it varies wildly at smaller scales. It’s 2, or 1.91, or 3.57, or 2.12, or 1.3, or 3.51, depending on where you start and how far you go.

In such a badly-behaved universe, *topological* dimension makes no sense. In this universe, it simply doesn’t make any sense to say that we can move in *two* or *three* or any other number of orthogonal directions.

At the tiniest scale, there *is* no left-right, no up-down, no forwards-backwards.

It’s more complicated than that.

You just can’t simplify a graph as complex as *this*:

into a three-dimensional space as simple as *that*:

At the tiniest scale, all there is is the graph, in all its complexity.

## What are dimensions?

So what *are* dimensions?

They’re an attempt by large-scale creatures like ourselves to make sense of space.

They’re *our* simplification of an unimaginably complex graph.

That’s not to say that our simplification isn’t correct, or at least *close* to correct at our own scale.

Indeed, I’m quite sure that our simplification of the graph into three-dimensional space *is* correct. As I’ve said before, if Wolfram’s graphs are going to be a true representation of our universe, they’re going to have to be three-dimensional, or at least *approximately* three-dimensional at a *large* scale.

But that doesn’t mean that Wolfram’s graphs need be *precisely* three-dimensional: it’s possible that there are some neighbourhoods of our universe where space is 2.99999-dimensional, and other neighbourhoods where it’s 3.00001-dimensional.

Nor does it mean that Wolfram’s graphs need be three-dimensional on a *small* scale: our universe might be 2½-dimensional or 3.37-dimensional or 9-dimensional on the tiniest scales. All that matters is that Wolfram’s graphs *approximate* to three dimensions at a *large* scale.

## Which is real?

To conclude this trilogy of articles about dimensionality, I’m going to flip everything I’ve written so far on its head.

I’ve been starting from the assumption that our universe is three-dimensional, and asking whether the graphs generated by Wolfram’s rules might be a true representation of this three-dimensional universe.

But what we’ve found is that Wolfram’s graphs *aren’t* three-dimensional on the scale of nodes and edges. Indeed, on this tiniest of scales, the notion of *topological* dimension doesn’t even make sense.

So which is *real*? Wolfram’s graphs, with wildly varying *Hausdorff* dimensions and without *topological* dimension? Or the uniform, continuous, three-dimensional space in which we live?

It’s natural to start from the assumption that uniform, continuous, three-dimensional space is real and that it’s Wolfram’s graph that’s the fabrication.

We’re going to have to flip our thinking.

If Wolfram is right, then it’s the graph that’s real, and it’s our concept of uniform, continuous, three-dimensional space that’s the fabrication.

Don’t get me wrong: it’s a *fine* fabrication, a close approximation to the structure of the graph at a large scale.

But as long as we keep in mind that *it ain’t necessarily so*, we’ll leave our minds open, not only to a deeper understanding of Wolfram’s concepts, but also to potential differences between those concepts and our current simplifications.

Suppose, for example, we measured the dimensionality of our own universe, and found that it is, indeed, 2.99999-dimensional in some neighbourhoods, and 3.00001-dimensional in others. If such fluctuations could be precisely predicted by Wolfram’s rules, it would be a way to verify the reality of Wolfram’s graphs.

## Mindset shift

This is going to be a theme throughout future articles.

If we’re going to wrap our minds around Wolfram Physics, we’re going to have to flip our thinking.

We’re going to have to stop thinking of our current concepts of physics – three-dimensional space, the curvature of four-dimensional space-time, and so on – as *reality*. And we’re going to have to stop thinking that Wolfram’s graphs need to conform to those concepts.

Instead, we’re going to have to start thinking of *Wolfram’s graphs* as reality, and our current concepts of physics as mere approximations to that reality.

I don’t know whether it’ll work out, but I *do* know that it’ll take a mindset shift to find out.