The Last Theory
The Last Theory
The Last Theory
13 October 2022

# How big is the computerthat runs the universe?

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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 10100 edges across. At the scale of a space hopper, space is approximately three-dimensional. So a space hopper contains roughly 10100 × 10100 × 10100 edges. That’s 10300 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 10300 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 107 is two orders of magnitude more than 105.

So when I say that the universe is roughly 1030 metres across, I don’t mean that it’s 1030 metres plus or minus 10%. In other words, I don’t mean that it’s between 0.9 × 1030 and 1.1 × 1030 metres across. Rather, I mean that it’s 1030 metres give or take an order of magnitude. In other words, I mean that it’s between 1029 and 1031 metres across. Or, if I’m being really rough, I mean that it’s 1030 metres give or take maybe five orders of magnitude. In other words, I mean that it’s between 1025 and 1035 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 10300 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 1030 and 103000.

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 1030 and probably isn’t as high as 103000. It’s probably somewhere in between. Let’s call it 10300.

## 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 10300 edges in a cubic metre. So let’s say that there are 10297 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 10297 edges, can simulate a universe of about 1,000 edges, in other words, 103 edges. Which means that this computer can simulate a universe with 10294 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 1020 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 10277 times fewer edges than it contains itself (because this computer, like my laptop, would contain 10297 edges, and it could simulate a universe of 1020 edges, which is 10297 ÷ 1020 = 10277 times fewer edges).

## Flipping enormous

Flipping this calculation, you can see that if there’s a computer that runs our universe, it likely contains 10277 times more edges than the universe.

Now our universe is pretty big. It’s roughly 1030 metres across. (Did I already mention that?) So if there are roughly 10300 edges in a space hopper, then there are roughly 10390 edges in the universe (because a space hopper is about 1 cubic metre in volume, and the universe is roughly 1030 × 1030 × 1030 = 1090 metres in volume, so the number of edges in the universe is 10300 × 1090 = 10390).

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 10390 × 10277 = 10667 edges.

Also, our universe is roughly 1093 times the volume of my laptop (because my laptop is about 10-3 cubic metres in volume and the universe, again, is roughly 1090 metres in volume). Arriving at the same number in a different way, there are roughly 1093 times the number of edges in our universe than in my laptop (because there are roughly 10297 edges in my laptop and roughly 10390 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 1093 × 10667 = 10760 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 10760 edges.

## How big is big?

Do you have any idea how large a number 10760 is?

I know I don’t.

It’s literally unimaginably large.

Remember, our universe is roughly 1030 metres across. Can you imagine how big our universe is? Probably not. Try to hold it in your mind, a universe 1030 times the diameter of a space hopper.

Can’t do it? You’re not the only one.

And that’s just 1030. If you think you can imagine 1030 space hoppers, try to imagine 1090 space hoppers, the number that would fit into our universe, if our universe consisted entirely of space hoppers.

Remember, 1090 is not three times higher than 1030, it’s 1060 times higher.

If you think you can truly conceive of numbers as large as 1090, 1060 or even 1030, I think you’re fooling yourself.

And if you can? Well, that’s just 1090. It’s a long, long way from there to 10760.

I’m not saying that it’s not possible that there’s some other universe that’s 10370 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 10370 times the size of our own.

Where’s the computer that runs that universe?

Is it in another universe that’s 10370 times the size of this unimaginably immense universe, a universe that’s 101130 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 10370 times the size of this even more unimaginably immense universe, a universe that’s 101500 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 = 10100 /m

Edges/m3 = (10100)3 = 10300 /m3

Volume of a space hopper = 1 m3

Edges in a space hopper = 10300 /m3 × 1 m3 = 10300

Volume of my laptop = 0.4 m × 0.3 m × 0.01 m = 10-3 m3

Edges in my laptop = 10300 /m3 × 10-3 m3 = 10297

Max edges in a universe simulated by my laptop = 103

Ratio of edges in my laptop to edges simulated by my laptop = 10297 ÷ 103 = 10294

Max edges in a universe simulated by an impossibly powerful computer the same size as my laptop = 1020

Ratio of edges in this computer to edges simulated by this computer = 10297 ÷ 1020 = 10277

Volume of our universe = 1030 m × 1030 m × 1030 m = 1090 m3

Edges in our universe = 10300 /m3 × 1090 m3 = 10390

Edges in the computer that runs our universe = 10390 × 10277 = 10667

Ratio of volume of our universe to volume of my laptop = 1090 m3 ÷ 10-3 m3 = 1093

Ratio of edges in our universe to edges in my laptop = 10390 ÷ 10297 = 1093

Edges in the universe that contains the computer that runs our universe = 1093 × 10667 = 10760

Ratio of edges in the universe that contains the computer that runs our universe to edges in our universe = 10760 ÷ 10390 = 10370

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