Now that I’ve introduced you to the different kinds of edges that might make up a hypergraph – unary, binary and ternary edges, as well as loops and self-loops – we can have some fun.
Some of rules in the Wolfram model give rise to fascinating universes.
Today, I’m going to show you a few rules that seem to fabricate space itself in much the same way as knitting needles might fabricate a blanket.
And if you think that knitting is a far-fetched analogy, just wait until you see my animations!
The fabric of reality
Take a look at this rule:
{{x, y, y}, {z, x, u}} → {{y, v, y}, {y, z, v}, {u, v, v}}
Seems overcomplicated, doesn’t it?
To apply the rule, we find a hyperedge from a node × to a node y self-looping back to node y, and another hyperedge from a node z to node × to a node u. Then we delete both these hyperedges and create three more: one from node y to a new node v looping back to node y; another from node y to node z to node v; and a third from node u to node v self-looping back to node v.
Let’s apply this overcomplicated rule, starting with two self-loops from the same node.
For the first few iterations, this universe seems pretty random:
But after a dozen applications of the rule, something magical starts to happen:
If you watch that single self-loop that persists from each iteration to the next, you can see that it seems to act like a knitting needle, generating a hypergraph that looks like fabric:
Maybe it’s just me, but this seems highly suggestive.
I’ve been telling you for a while now that the hypergraph is space.
I’ve also been telling you that the rules of Wolfram Physics might generate hypergraphs that look like the space in which we live.
But now, at last, I’m showing you a rule that generates a lattice of nodes and edges that really does resemble a uniform space, and goes on generating that lattice indefinitely.
This is not our universe
Of course, this knitted fabric is not our universe.
It’s clearly two-dimensional, whereas the space in which we live is three-dimensional.
And it’s forever uniform. Run this rule indefinitely, and it’ll generate an ever-larger lattice without a single imperfection.
A knitter would be proud of such perfection.
But the space in which we live is not so perfect.
Sure, empty space might look like this, but the space in which we live is scattered with photons, neutrinos and quarks.
We can speculate that these particles might be persistent knots propagating through the hypergraph through successive applications of the rules in the Wolfram model.
The universe I just showed you is empty, two-dimensional and kinda triangular, whereas our universe is scattered with particles, three-dimensional and kinda spherical.
So if we’re looking for a single rule that generates our universe, this is clearly not it.
But again, it seems highly suggestive. If there’s a rule like this one that generates an empty, two-dimensional space, might there not be other rules that generate other spaces more like our own?
Cushion universe
It turns out there are.
Take this rule:
{{x, y, z}, {u, y, v}} → {{w, z, x}, {z, w, u}, {x, y, w}}
It matches a hyperedge from a node × to a node y to a node z, and another hyperedge from node u to node y to a node v. Then it deletes both those hyperedges, and creates three more: one from a new node w to node z to node x; another from node z to node w to node u; and a third from node × to node y to node w.
Again, at first, this rule doesn’t look like much:
But let it run a while, and order emerges from the chaos:
After a couple of hundred iterations, you can see that the rule is knitting three two-dimensional fabrics, sewn together along the edges.
The rule seems to run along the unsewn edges of each of the three fabrics, knitting another line of each in turn.
Again, this is not our universe. It might look like a three-dimensional cushion, but really it’s just three two-dimensional surfaces.
But again, it’s highly suggestive.
Jester universe
Or take this rule:
{{x, y, z}, {x, u, v}} → {{z, z, w}, {w, w, v}, {u, v, w}}
To apply it, we find a hyperedge from a node × to a node y to a node z, and another hyperedge from node × to a node u to a node v. Then we delete both these hyperedges and create three more: one from node z self-looping back to node z then on to a new node w; another from node w self-looping back to node w then on to node v; and a third from node u to node v to node w.
This rule knits a three-pronged universe that’s reminiscent of a jester’s hat:
Again, three two-dimensional surfaces sewn together at the edges.
But again, highly suggestive.
Girder universe
The knitting rules I’ve shown you so far all yield two-dimensional universes.
There are also plenty of rules that knit one-dimensional universes.
Here’s a fun one:
{{x, y, z}, {u, y, u}} → {{v, y, z}, {v, z, v}, {v, u, v}}
It generates what looks like a latticework girder:
Over time, the girder snakes back on itself:
It’s important to remember that this snaking is just a consequence of the way I draw the hypergraph. From the point of view of an observer inside this universe, it’s not curved, it’s just a long, narrow, one-dimensional girder.
Feels like home
How about our universe?
Is our universe a three-dimensional lattice knitted by a single rule?
Is our universe as uniform as the one-dimensional girders and two-dimensional fabrics I’ve been showing you?
(Except, of course, for the occasional persistent knot propagating through the hypergraph, in other words, the occasional photon, neutrino or quark moving through space?)
I don’t think so.
I think it’s more complicated than that.
It’s just a hunch, but I suspect that no single rule can generate our complex universe.
It’s just a hunch, but I suspect that our universe isn’t a regular lattice. I imagine instead that the hypergraph is a highly irregular chaos of nodes and edges.
And it’s just a hunch, but I suspect that our universe isn’t uniformly three-dimensional. I imagine instead that the dimensionality of the hypergraph varies wildly on a small scale, approximating to three dimensions only at a large scale.
And that raises a question.
Sure, there are rules in the Wolfram model that can knit simple, uniform hypergraphs.
But are there rules that can knit more complex hypergraphs, more irregular hypergraphs?
Well, first I should confess that I cheated a little to generate the universes I just showed you. Whenever the rules matched more than one set of edges, I didn’t choose one of those sets at random. Instead, I chose the set that included the oldest node. This consistent but arbitrary way of choosing was what led to those simple, uniform one- and two-dimensional lattices.
Choosing a match at random, on the other hand, would have yielded a more complex, more irregular hypergraph. In other words, choosing a match at random would have yielded a universe more like our own.
And satisfying as it might be to generate these beautiful lattices through these arbitrary ways of choosing, there are other rules that, regardless of how we choose between matches, knit more chaotic lattices of nodes and edges.
Take this rule:
{{x, y, z}, {x, u, x}} → {{x, u, v}, {z, v, z}, {v, z, y}}
It’s just a single rule, so there’s none of the complexity of the multiway graph here, but it generates a lattice that’s more complex than any we’ve seen so far:
And yet, for all the irregularity of this hypergraph on a small scale, it feels like this rule will knit an ever-expanding universe that’s regular on a large scale:
I can’t help but think that the space in which we live might look a little like this: irregular on a small scale, in a way that supports the propagation of persistent knots through the hypergraph, in other words, the movement of photons, neutrinos and quarks through space, but regular on a large scale.
To put it another way, I can’t help but feel that this universe looks a little like home.