I’ve introduced you to edges between two nodes:
and I’ve shown you graphs made up of these binary edges:
I’ve also introduced you to hyperedges between three nodes:
and I’ve shown you hypergraphs made up of these ternary edges:
I’ve drawn all these edges and hyperedges as straight lines.
Some edges and hyperedges, however, can’t be drawn as straight lines, because they loop back on themselves.
These loops and self-loops might seem unassuming additions to our repertoire of edges and hyperedges.
But don’t let their curvy humility fool you.
They can play a crucial role in the evolution of graphs and hypergraphs... which means that they might play a crucial role in the evolution of the universe itself.
Drawn out
Let’s start with a binary edge from node 1 to node 2.
We can represent it visually as an arrow from a dot with the number 1 on it, representing node 1, to a dot with the number 2 on it, representing node 2:
Or we can represent it numerically as the number 1 followed by a comma followed by the number 2, enclosed in curly brackets:
OK, let’s add another binary edge, from node 2 to node 3.
Here’s the visual representation, an arrow from a dot with the number 2 on it to a dot with the number 3 on it:
And here’s the numerical representation, the number 2 followed by a comma followed by the number 3 in curly brackets:
Now let’s show both edges.
Here they are represented visually, as an arrow from node 1 to node 2 and another arrow from node 2 to node 3:
And here they are represented numerically, as {1, 2}, {2, 3}:
Double edged
You may think I’m drawing this out, in more ways than one, but here’s the point.
I’ve just shown you that {1, 2}, {2, 3} is a perfectly valid numerical representation of a graph with two edges: a binary edge from node 1 to node 2 and a binary edge from node 2 to node 3.
So why not {1, 2}, {1, 2}?
Isn’t that, too, a perfectly valid numerical representation of a graph with two edges: a binary edge from node 1 to node 2 and another binary edge also from node 1 to node 2?
It’s not hard to imagine how we might represent these edges visually.
That binary edge from node 1 to node 2? Well, obviously, we’re going to represent it as an arrow from a dot with the number 1 on it, representing node 1, to a dot with the number 2 on it, representing node 2.
And that other binary edge also from node 1 to node 2? Well, obviously, we’re going to represent it as another arrow from the dot with the number 1 on it, representing node 1, to the dot with the number 2 on it, representing node 2.
Of course, if we drew both edges as straight lines, they’d be on top of each other, and we wouldn’t be able to see that there are two edges between node 1 and node 2.
So instead, we’re going to draw both edges as curved lines:
We’ve never seen two edges between the same two nodes before, but when you represent it numerically as {1, 2}, {1, 2}, it doesn’t seem so strange, and when you represent it visually as two curved arrows between two nodes, well, that doesn’t seem so strange either, does it?
And we needn’t stop at two edges between the same nodes.
How about three?
Or four?
Or five?
Double back
Let’s go one step further.
If we’re going to allow {1, 2}, {1, 2}, why not {1, 2}, {2, 1}?
This, too, seems a perfectly valid numerical representation of a graph with two edges: a binary edge from node 1 to node 2 and another binary edge from node 2 to node 1:
So now we have our first loop: two binary edges looping from one node to another and back again.
Self seeking
But there’s actually a simpler loop we can construct.
Let’s go back to the first binary edges I mentioned.
If we can have a binary edge {1, 2} from node 1 to node 2:
and a binary edge {2, 3} from node 2 to node 3:
then it doesn’t seem unreasonable to imagine that we can have any two numbers in the curly brackets, representing a binary edge between any nodes.
So how about a binary edge {1, 1}?
There’s nothing in our numerical representation to prevent such a binary edge from node 1 to node 1.
Again, it’s not hard to imagine how we might represent this edge visually.
We’ll just need to bend an arrow back on itself, so that it loops from the dot with the number 1 on it, representing node 1, back to the same dot with the number 1 on it, representing the same node 1:
A binary edge that loops from a node back to the same node is called a self-loop.
Hyperloops
So now I’ve shown you loops and self-loops in graphs of binary edges.
It won’t surprise you that we can also have loops and self-loops in hypergraphs of ternary edges.
Here’s a ternary edge between three nodes:
Remember that for ternary edges, we draw a web between the two arrows to show that this is not a pair of binary edges, one from node 1 to node 2 and another from node 2 to node 3, but a single ternary edge, from node 1 to node 2 to node 3.
You can probably guess where I’m going next.
If we can have a ternary edge {1, 2, 3}, then why not a ternary edge {1, 2, 1}?
Again, we draw a web between the two arrows to show that this is a single ternary edge, from node 1 to node 2 and back to node 1.
And, while we’re at it, why not a ternary edge {1, 1, 2}?
This edge goes from node 1, loops back to node 1, then goes to node 2.
And why not a ternary edge {1, 2, 2}?
This edge goes from node 1 to node 2, then loops back to node 2.
How about a ternary edge {1, 1, 1}?
This one goes from node 1, loops back to node 1, then loops back to node 1 again.
In each case, we draw a web between the two arrows, whether they’re straight or curved into a loop or a self-loop, to show that this is a single ternary edge, not a pair of binary edges.
You can see how this works: any sequence of numbers representing nodes represents a valid hyperedge between those nodes, whether it’s {1}, {1, 2}, {1, 2, 3}, {17, 98, 59, 22} or the double self-loop {1, 1, 1}.
Loops rule
We can apply rules to loops and self-loops in the same way as to any other edges.
Take this simple rule:
To apply the rule, we find a self-loop on one node, delete that self-loop, and create a new edge from that node to a new node, as well as two new self-loops on that new node.
Here’s how it plays out:
Since any self-loop can generate a new edge to a new node with two new self-loops, the rule gives a branching, tree-like universe.
Why loops matter
I’ve been wanting to introduce you to loops and self-loops for a long time now.
So many of the most complex and most promising graphs and hypergraphs of Wolfram Physics involve loops and self-loops.
I’m looking forward to showing you some of these beautiful universes in future articles.
But that’s not why loops matter.
The reason loops matter is that they exist, so if we exclude them from our models, then our models will be incomplete.
One of the most compelling aspects of Wolfram Physics is that it doesn’t make too many assumptions about our universe.
The Wolfram model doesn’t assume that only binary edges exist in our universe; it encompasses all edges, including unary, binary, ternary and so on up.
The Wolfram model doesn’t assume that only edges between different nodes exist in our universe; it encompasses all edges, including loops and self-loops between the same nodes.
The Wolfram model doesn’t even assume that there’s one rule that applies to our universe; it encompasses all rules in the ruliad.
If edges can be represented as sequences of numbers representing nodes, then the most complete model of the universe as a hypergraph must allow any sequence of numbers representing nodes, even if one or more of those numbers is repeated.
Including loops and self-loops in the hypergraph reduces the number of arbitrary assumptions we need to make in Wolfram Physics, making it more complete.