If you read my article Where’s the computer that runs the universe? you’ll know that I have my doubts about the existence of a computer that’s whirring away, applying Wolfram’s rules to Wolfram’s graphs, performing the computations required to run our universe.

This computer, if it exists, is necessarily invisible to us, and as I warned in Beware invisible things, we should be wary of what we can’t see.

Still, I want to revisit this idea of a computer that runs the universe.

I want to come at it from a slightly different direction.

Rather than adopt the stance of the monkey with its hands over its eyes and insist that if I can’t see it, it’s not there, let’s suppose that there *is* a computer that runs the universe and ask a simple question:

How *big* would it have to be?

## The size of a space hopper

According to Wolfram Physics, the universe is a graph of nodes and edges.

It’s hard to estimate the *scale* of the graph.

A rough estimate I’ve heard from Stephen Wolfram is that the nodes of the graph are typically 10^{-100} metres apart, making the edges of the graph typically 10^{-100} metres long.

Actually, I’m getting things a bit backwards when I talk of an edge having a length in space, or of nodes being a particular distance apart in space. Space doesn’t exist independently of the nodes and edges: space *is* the nodes and edges. So I’m going to restate that rough estimate more precisely.

An object, such as a space hopper, that’s a metre across is about 10^{100} edges across. At the scale of a space hopper, space is approximately three-dimensional. So a space hopper contains roughly 10^{100} × 10^{100} × 10^{100} edges. That’s 10^{300} edges.

I’m assuming that the edges are laid out in something like a rectangular grid:

As we’ve seen, that’s not the case for most rules: the graphs they generate are more convoluted than that:

So there may be more edges than I’m counting here.

I’m also assuming that the number of edges scales linearly with the volume of space. Or, to be more precise, that the volume of space scales linearly with the number of edges. In other words, if we double the number of edges in a graph, we double the volume of space formed by the graph. Again, that might not be the case: there might be edges between distant regions of space. So again, there may be more edges than I’m counting here.

Still, I’m going to be conservative, and assume that there are roughly 10^{300} edges in a space hopper.

## Orders of magnitude of orders of magnitude

Now, I’m talking orders of magnitude of orders of magnitude here.

Usually, when we say *roughly*, we mean give or take 10%, or 20%, or, if we’re *really* being rough, maybe 50%. So when I say it’s roughly 20 miles to Cherryville from here, I mean it’s 20 miles plus or minus 10%, that is, plus or minus 2 miles. In other words, I mean that it’s between 18 and 22 miles. Or, if I’m *really* being rough, I mean that it’s 20 miles plus or minus maybe 50%, that is, plus or minus 10 miles. In other words, I mean that it’s between 10 and 30 miles. That’s *pretty* rough.

When we’re talking orders of magnitude, however, *roughly* means give or take an order of magnitude, or two, or, if we’re *really* being rough, maybe five.

By the way, if you’re not familiar with the term *order of magnitude*, it means *10 times*. So 1,000 is an order of magnitude more than 100, and 10^{7} is *two* orders of magnitude more than 10^{5.}

So when I say that the universe is roughly 10^{30} metres across, I *don’t* mean that it’s 10^{30} metres plus or minus 10%. In other words, I *don’t* mean that it’s between 0.9 × 10^{30} and 1.1 × 10^{30} metres across. Rather, I mean that it’s 10^{30} metres give or take an order of magnitude. In other words, I mean that it’s between 10^{29} and 10^{31} metres across. Or, if I’m being *really* rough, I mean that it’s 10^{30} metres give or take maybe five orders of magnitude. In other words, I mean that it’s between 10^{25} and 10^{35} metres across.

When it comes to the edges in a space hopper, however, the number is so enormous that we’re talking about a whole new level of roughness. When I say that there are *roughly* 10^{300} edges in a space hopper, I mean that the *order of magnitude* is 300 give or take an *order of magnitude*, in other words, that the number of edges in a space hopper is between 10^{30} and 10^{3000.}

At *this* level of roughness, it doesn’t really matter that the diameter of a space hopper is actually a little less than a metre, or that the volume of a space hopper is better approximated as 4/3 π times the cube of its radius. Such inaccuracies are miniscule when we’re talking *orders of magnitude* of *orders of magnitude*.

What *matters* is that number of edges in a space hopper probably isn’t as low as 10^{30} and probably isn’t as high as 10^{3000.} It’s probably somewhere in between. Let’s call it 10^{300.}

## How big is the computer that *simulates* the universe

As I’ve mentioned before, I run my simulations of the universe on this low-powered laptop:

It’s not as big as a space hopper. It’s about 40 cm × 30 cm × 1 cm, which is about 10^{-3} cubic metres. According to my rough estimate, there are about 10^{300} edges in a cubic metre. So let’s say that there are 10^{297} edges in my low-powered laptop.

On *this* computer, I can simulate a universe of about 1,000 edges. Beyond that, it slows to a crawl.

I should say that one of the reasons for this slowing down is that the more nodes there are in a graph, the longer it takes my software to decide where to *position* each of these nodes on the screen. Those decisions aren’t part of *simulating* the universe, they’re part of *drawing* the universe so that I can show it to you in my articles.

I should also say that *how* I apply Wolfram’s rules has a significant effect on how dramatically the simulation slows down. As I explained in Where to apply Wolfram’s rules?, every time I apply a rule to a graph, I find *all* the sets of edges that match the rule, then select one at random. That takes time.

If I did as Stephen Wolfram suggests, and apply the rule to *all* the sets of edges, rather than just *one* selected at random, it’d take even more time. *Much*, *much*, *much* more time. The multiway graph generated by this approach branches into a monstrous tangle after only a few iterations.

These quibbles aside, my low-powered laptop, which, in *our* universe, contains 10^{297} edges, can simulate a universe of about 1,000 edges, in other words, 10^{3} edges. Which means that *this* computer can simulate a universe with 10^{294} times fewer edges than it contains itself.

Obviously, there are computers more powerful than my laptop. In the future, there’ll no doubt be computers that are much, much more powerful than my laptop. There might be processors specifically engineered to apply rules to graphs, performing computations in parallel, rather than one at a time, like *my* pedestrian processor.

The trouble is, the computations of Wolfram Physics *explode* in complexity as the number of edges increases. To simulate a universe with much more than 1,000 edges, you’d need a *much*, *much*, *much* more powerful computer.

And if you wanted to generate the multiway graph, well, I don’t think I could repeat the word *much* enough times to communicate how powerful the computer would have to be.

Since we’re talking orders of magnitude of orders of magnitude here, even a *much*, *much*, *much* more powerful computer seems unlikely to make a significant dent in the numbers.

Let’s be optimistic. *Ridiculously* optimistic. Let’s say we had a computer the size of my laptop that could simulate a universe of 10^{20} edges. I find it difficult to imagine that humans could ever build such a powerful computer. No matter. As I say, let’s be *ridiculously* optimistic here.

This impossibly powerful computer could simulate a universe with 10^{277} times fewer edges than it contains itself (because this computer, like my laptop, would contain 10^{297} edges, and it could simulate a universe of 10^{20} edges, which is 10^{297} ÷ 10^{20} = 10^{277} times fewer edges).

## Flipping enormous

Flipping this calculation, you can see that if there’s a computer that runs *our* universe, it likely contains 10^{277} times *more* edges than the universe.

Now our universe is pretty big. It’s roughly 10^{30} metres across. (Did I already mention that?) So if there are roughly 10^{300} edges in a space hopper, then there are roughly 10^{390} edges in the universe (because a space hopper is about 1 cubic metre in volume, and the universe is roughly 10^{30} × 10^{30} × 10^{30} = 10^{90} metres in volume, so the number of edges in the universe is 10^{300} × 10^{90} = 10^{390}).

So if there’s a computer that runs *our* universe, and it exists in its own universe, and that universe evolves according to the same laws of physics as *our* universe, then the computer that runs *our* universe likely contains 10^{390} × 10^{277} = 10^{667} edges.

Also, *our* universe is roughly 10^{93} times the volume of my laptop (because my laptop is about 10^{-3} cubic metres in volume and the universe, again, is roughly 10^{90} metres in volume). Arriving at the same number in a different way, there are roughly 10^{93} times the number of edges in our universe than in my laptop (because there are roughly 10^{297} edges in my laptop and roughly 10^{390} edges in the universe).

So if the computer that runs *our* universe exists in its own universe, and the volume of that universe is as many times the volume of the computer that runs *our* universe as the volume of *our* universe is of the volume of my laptop, or, to arrive at the same number in a different way, if the number of edges in that universe is as many times the number of edges in the computer that runs *our* universe as the number of edges in *our* universe is the number of edges in my laptop, then that universe likely contains 10^{93} × 10^{667} = 10^{760} edges.

I’ve just bombarded you with a *lot* of assumptions and a *lot* of numbers.

Here’s the bottom line: the computer that runs *our* universe exists in a universe of maybe 10^{760} edges.

## How big is big?

Do you have any idea how large a number 10^{760} is?

I know I don’t.

It’s literally unimaginably large.

Remember, our universe is roughly 10^{30} metres across. Can you imagine how big our universe is? Probably not. Try to hold it in your mind, a universe 10^{30} times the diameter of a space hopper.

Can’t do it? You’re not the only one.

And that’s just 10^{30.} If you think you can imagine 10^{30} space hoppers, try to imagine 10^{90} space hoppers, the number that would fit into our universe, if our universe consisted entirely of space hoppers.

Remember, 10^{90} is not *three* times higher than 10^{30}, it’s 10^{60} times higher.

If you think you can truly conceive of numbers as large as 10^{90}, 10^{60} or even 10^{30}, I think you’re fooling yourself.

And if you *can*? Well, that’s just 10^{90.} It’s a long, long way from there to 10^{760.}

I’m not saying that it’s not possible that there’s some other universe that’s 10^{370} times the size of our own, in which there’s a computer that’s running *our* universe.

I’m just saying that it’s a stretch.

## Turtles all the way down

Let’s get deeper into trouble.

That universe that’s 10^{370} times the size of our own.

Where’s the computer that runs *that* universe?

Is it in another universe that’s 10^{370} times the size of this unimaginably immense universe, a universe that’s 10^{1130} times the size of our own?

And you know what I’m going to ask next.

Where’s the computer that runs *that* universe?

Is it in another universe that’s 10^{370} times the size of this even more unimaginably immense universe, a universe that’s 10^{1500} times the size of our own?

This is getting silly, I know, but it’s important to take this argument to its limit, to show that, in fact, it has no limit.

It’s an age-old problem.

When the ancients asked the question: *what supports the world?*

and answered: *a turtle supports the world*

it begged the question: *OK, so... what supports the turtle?*

Maybe *another* turtle supports the turtle.

And maybe *yet another* turtle supports *that* turtle.

Maybe it’s turtles all the way down.

Or *maybe* the idea that there’s a turtle that supports the world is just the wrong way to think about it.

## How big is the computer that runs the universe?

How big is the computer that runs the universe?

It’s big. I mean *really*, *really*, *really* big. You can’t imagine how big. Literally, you can’t imagine.

And how big is the computer that runs the universe that contains the computer that runs *our* universe? Well, it’s even bigger. *Really*, *really*, *really*, *really*, *really*, *really* big.

And how big is the computer that runs the universe that contains the computer that runs the universe that contains the computer that runs *our* universe?

OK, I’m going to stop there.

Maybe it’s computers all the way up.

But *maybe*, just *maybe*, the idea that there’s a computer that runs the universe is just the wrong way to think about it.

—

For reference, here are the *very*, *very*, *very* rough numbers I’ve used in this article:

Edges/m = 10^{100} /m

Edges/m^{3} = (10^{100})^{3} = 10^{300} /m^{3}

Volume of a space hopper = 1 m^{3}

Edges in a space hopper = 10^{300} /m^{3} × 1 m^{3} = 10^{300}

Volume of my laptop = 0.4 m × 0.3 m × 0.01 m = 10^{-3} m^{3}

Edges in my laptop = 10^{300} /m^{3} × 10^{-3} m^{3} = 10^{297}

Max edges in a universe simulated by my laptop = 10^{3}

Ratio of edges in my laptop to edges simulated by my laptop = 10^{297} ÷ 10^{3} = 10^{294}

Max edges in a universe simulated by an impossibly powerful computer the same size as my laptop = 10^{20}

Ratio of edges in this computer to edges simulated by this computer = 10^{297} ÷ 10^{20} = 10^{277}

Volume of our universe = 10^{30} m × 10^{30} m × 10^{30} m = 10^{90} m^{3}

Edges in our universe = 10^{300} /m^{3} × 10^{90} m^{3} = 10^{390}

Edges in the computer that runs our universe = 10^{390} × 10^{277} = 10^{667}

Ratio of volume of our universe to volume of my laptop = 10^{90} m^{3} ÷ 10^{-3} m^{3} = 10^{93}

Ratio of edges in our universe to edges in my laptop = 10^{390} ÷ 10^{297} = 10^{93}

Edges in the universe that contains the computer that runs our universe = 10^{93} × 10^{667} = 10^{760}

Ratio of edges in the universe that contains the computer that runs our universe to edges in our universe = 10^{760} ÷ 10^{390} = 10^{370}