The Last Theory
The Last Theory
The Last Theory
3 February 2022

Nodes, edges, graphs & rules:
the basic concepts of Wolfram Physics

Are you ready?

Today, I’m going to dive right into Wolfram Physics.

If you’ve never heard of Stephen Wolfram or his team’s project to find the fundamental theory of physics, don’t worry.

Think of it like this: I’m going to dive right into the fundamental structure of the universe.

And, well, you might not believe that the words “simple” and “physics” can go together, but I’m going to keep it simple.

Welcome to the universe

Without further ado, here’s the universe:

Well, I told you I was going to keep it simple, didn’t I?

Actually, this isn’t the universe so much as one possible universe.

And it’s that universe right back at the beginning of time, before it explodes into... well, we’ll soon see what it explodes into.

This universe consists of three nodes connected by two edges.

The nodes are just points. There’s one node on the left, and two on the right.

The edges are just lines between the nodes. One edge goes from the node on the left to the upper-right node. The other edge goes from the node on the left to the lower-right node.

Each edge has a direction, indicated by the arrowhead.

A set of nodes and edges like this is called a graph.

It’s a confusing term.

When I talk about a graph, I’m not talking about a plot of y against ×:

I created this graph at FooPlot, my go-to online graph generator

Rather, when I talk about a graph, I’m talking about a set of nodes and edges:

And that’s it. That’s all there is to our universe. Just three nodes and two edges. Simple.

It doesn’t stay that way

This universe doesn’t stay simple for long.

There’s one last concept I want to bring in here. I’ve talked about nodes, edges and graphs. Now I want to talk about rules.

Rules are like the laws of physics. They determine how the universe changes over time.

Actually, let me restate that. If Stephen Wolfram is right, then rules aren’t just like the laws of physics, rules are the laws of physics.

Let’s apply a simple rule to our universe.

The rule has two steps.

Step 1: match edges

Step 1 is to find edges that match the rule.

In the rule I’m going to apply, that means finding two edges that come from the same node.

In the case of our simple universe, matching the edges is easy. There are only two edges, and they do come from the same node. I’ll highlight them in blue to show that we’ve found our match:

Step 2: evolve edges

Step 2 is to evolve the matched edges according the rule.

In the rule I’m going to apply, that means deleting one of the two edges, creating a new node, and creating a new edge from each of the three existing nodes to the new node.

OK, let me run through that again more slowly.

First, we need to delete one of the two edges. I’ve chosen the lower of the two matched edges, and I’ve highlighted it in red, to indicate that it’s about to be deleted.

(Nothing’s going to happen to the other matched edge, so I’ve turned it white again.)

Next, we need to create a new node. I could have put it anywhere, but for now, I’ve placed it at the centre of the three existing nodes.

Finally, we need to create a new edge from each of the three existing nodes to the new node. I’ve highlighted these new edges in green. As ever with edges, their direction matters: they’re from the existing nodes to the new node, so the green arrows all point towards the new node.

The universe evolves

Here’s what the graph looks like after the rule has been applied:

I’ve deleted the red edge, and I’ve turned the green edges white.

We now have four nodes and four edges. Still pretty simple.

It really doesn’t matter where in space I position the nodes and the edges.

(Why not? Because these nodes and edges aren’t in space, these nodes and edges are space. I’ll say more about this in a future post: prepare for your mind to be blown.)

For now, I’m going to reposition the nodes so that they’re more spread out, which will make the graph easier on the eyes:

That’s it. Now you know everything you need to know to work out how this universe evolves: three nodes, two edges, one rule.

Play it again, Sam

Let’s apply the rule again.

Step 1, remember, is to find two edges that come from the same node.

There’s still only one pair of edges that match these criteria, the ones highlighted in blue:

Step 2 is to delete one of the two matched edges, highlighted in red, create a new node, and create new edges, highlighted in green, from the three matched nodes to the new node:

Which leaves the universe looking like this:

Play it again and again and again, Sam

If we apply the rule over and over, something curious happens.

Our simple universe starts to become complex:

The more times we apply the rule, the more complex the universe becomes:

This seems strange.

After only a couple of hundred iterations, the universe already has structure: clusters of tightly connected nodes that are only loosely connected with each other.

How can such a complex universe arise from such a simple rule?

And here’s the crucial question:

Might a universe as complex as our own arise from a similarly simple rule?

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