It’s pretty easy to see how three-dimensional space might arise from Wolfram Physics.
The hypergraph kinda looks like space, and, for some rules, it kinda looks like it’s three-dimensional:
But our universe isn’t just empty three-dimensional space.
It’s mostly empty space, but there are also particles moving through that space.
There are photons moving at the speed of light.
There are neutrinos, electrons and quarks moving at less than the speed of light.
There are anti-matter counterparts of these particles, such as the anti-electron, or positron.
Sometimes, these particles interact, annihilating each other and producing new particles. An electron and a positron, for example, might collide, annihilating each other and producing a photon:
If Wolfram Physics is to be a successful model of our universe, it must, of course, model these elementary particles and their interactions.
So where are the particles in the hypergraph?
What is a particle in Wolfram’s universe?
The case of the missing particles
Here’s the bad news.
I don’t know how a particle might be modelled in Wolfram Physics.
And... nor does anyone else.
So that’s the end of this article.
What is a particle in Wolfram’s universe?
We don’t know.
...
Still here?
OK, OK.
We don’t know how a particle might be modelled in Wolfram Physics, but we can guess.
Whenever I’m asked: “What is a particle in Wolfram Physics?” here’s my answer:
A particle is a persistent tangle of nodes and edges that propagates through the hypergraph.
Let me explain.
That’s life
I’d love to show you a persistent tangle propagating through the hypergraph, but I can’t... because I’ve never seen one.
What I can show you is something very similar.
In 2020, Stephen Wolfram devised a computational system in which simple rules are applied to a simple structure, yielding complex behaviour. The structure was a hypergraph of nodes and edges. The system came to be known as Wolfram Physics.
50 years previously, in 1970, John Horton Conway, too, devised a computational system in which simple rules are applied to a simple structure, yielding complex behaviour. The structure was a two-dimensional grid of cells. The system came to be known as Conway’s Game of Life.
That’s a terrible name for what Conway created. Conway’s Game of Life is not a game, in the usual sense of the word, and it bears little resemblance to life, as we know it.
But that’s the name we’re stuck with.
Here’s Conway’s Game of Life in action:
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Whereas Wolfram Physics might prove to be a true model of our universe, Conway’s Game of Life has no pretensions to being a model of our universe... not least because it’s two-dimensional, while our universe is three-dimensional.
In many ways, though, Wolfram Physics and Conway’s Game of Life are very similar.
Let me show you how particles arise in Conway’s Game of Life, as a way to illustrate how they might also arise in Wolfram Physics.
The birds and the bees
Conway’s system is simple.
Each cell in his two-dimensional grid is either dead or alive.
For example, here’s a live cell, shown as a circle, surrounded by eight dead cells, shown as dots:
Just like Wolfram’s hypergraph, Conway’s two-dimensional grid of cells evolves through the application of rules.
Four rules are applied to each cell to determine whether it’ll be alive at the next iteration, depending on how many of the eight cells that surround it are alive at the current iteration.
Rule #1: A live cell that’s surrounded by fewer than two other live cells dies
So a live cell that’s not surrounded by any live cells dies:
And a live cell for which only one of the eight surrounding cells is alive also dies:
Rule #2: A live cell that’s surrounded by two or three other live cells lives
So a live cell that’s surrounded by two live cells lives:
And a live cell that’s surrounded by three live cells also lives:
Rule #3: A live cell that’s surrounded by more than three other live cells dies
So a live cell that’s surrounded by four live cells dies:
And a live cell that’s surrounded by eight live cells definitely dies:
Rule #4: A dead cell that’s surrounding by exactly three live cells comes to life
Now, I’ve only been showing you what happens to that one cell in the middle, but these four rules are applied to every cell in the two-dimensional grid at each iteration.
Just like the simple rules of Wolfram Physics, these four simple rules of Conway’s Game of Life can yield surprisingly complex behaviour.
I need only draw an arbitrary pattern of live cells, apply the rules over and over, and watch to see whether the culture of cells lives or dies:
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Looks familiar
You don’t have to watch Conway’s Game of Life for long before you start to see patterns.
There are certain arrangements of live cells that, as long as they remain isolated, don’t change, no matter how many times the rules are applied.
Here are a few of these stable patterns...
a block:
a beehive:
a loaf:
a tub:
a barge:
a boat:
a ship:
a carrier:
a mango:
a pond:
a snake:
and an eater:
There are certain other patterns of live cells that oscillate between two or more different states, again, no matter how many times the rules are applied.
Here are a few of these oscillators...
a blinker:
a beacon:
a toad:
and a clock:
And... you don’t have to watch Conway’s Game of Life for long before you come across one very particular pattern, the glider:
Gliders crawl across the two-dimensional grid, progressing one cell diagonally every four iterations:
Different gliders crawl in different directions. Here are the four glider varieties, one for each of the four diagonal directions:
In Conway’s universe, the glider is the equivalent of what, in our universe, we call an elementary particle.
Just like an elementary particle, it moves through space.
Just like an elementary particle, it remains intact, as long as the space it moves through is empty.
And just like an elementary particle, when it collides with another elementary particle, interesting things happen.
When gliders collide
Take a look at this glancing collision between two gliders:
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Before the collision, one glider is moving northeast and the other southeast.
After the collision, there’s only one glider, moving southwest.
And look at what happens when these four gliders collide:
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There’s a lot going on here, but here’s what happens at the end of the day.
Before the collision, there are four gliders, two moving southeast, the other two moving southwest and northwest.
After the collision, there are five gliders, two moving southwest, the other three moving southeast, northeast and northwest.
These collisions in Conway’s universe aren’t exactly the same as the collisions of elementary particles in our universe.
But a collision between two gliders, annihilating each other and producing another glider, is pretty similar to a collision between an electron and a positron, annihilating each other and producing a photon.
To boldly go
Talking of photons, electrons and positrons, there are many elementary particles in our universe, from massless photons, through miniscule neutrinos, through electrons thousands of times heavier, to quarks thousands of times heavier still.
And the same seems to be true in Conway’s universe.
Here’s an elementary particle that’s a little larger than a glider:
Like a glider, this spaceship crawls across the two-dimensional grid, progressing one cell every four iterations, albeit orthogonally rather than diagonally.
Here’s a slightly larger spaceship:
Here’s one that’s slightly larger again:
Here’s a spaceship called a copperhead, moving north:
Here’s a dragon, moving west:
Here’s a big glider, which is a bit like a glider, only bigger, moving north-west:
Here’s a lobster, also moving north-west:
Each of these spaceships crawls across the two-dimensional grid, each at its own speed, each in its own direction.
It seems there’s no limit to the size of a spaceship. Here’s a seahorse, moving west of north:
And here’s a greyship, moving west:
A ship this size looks pretty impressive as it moves through space:
Just as there are many elementary particles in our universe, so there are many elementary particles in Conway’s universe, from minimal gliders, through lightweight, middleweight and heavyweight spaceships, through dragons, lobsters and seahorses, to who knows what undiscovered gargantuan galactic craft.
Knotty
Here’s the point.
In Conway’s Game of Life, we’ve discovered a rich taxonomy of patterns of cells that crawl, intact, across the two-dimensional grid.
It’s not hard to imagine that in Wolfram Physics, too, we’ll discover a rich taxonomy of patterns of nodes and edges that propagate, intact, through the hypergraph.
Think of these patterns of nodes and edges as persistent tangles in the hypergraph.
Picture a fishing net with a knot in it.
Picture the knot being repeatedly untied, then retied one square further along the fishing net.
Just as the rules of Conway’s Game of Life, applied to the pattern of cells that make up the glider, cause it to crawl across the two-dimensional grid, so you can imagine the rules of Wolfram Physics, applied to the pattern of nodes and edges that make up a persistent tangle, cause it to propagate through the hypergraph.
And just as gliders take on a series of different arrangements of cells from iteration to iteration, so you can imagine these tangles taking on a series of different arrangements of nodes and edges from iteration to iteration.
If we’re lucky, these persistent tangles in the hypergraph will correspond to the photons, neutrinos, electrons and quarks that move, intact, through the three-dimensional space of our universe.
If we’re really lucky, we’ll be able to compare these patterns of nodes and edges, just as we can compare the gliders and spaceships of Conway’s universe.
And if we’re really, really lucky, the relative sizes of these persistent tangles will correspond precisely to the relative masses of photons, neutrinos, electrons and quarks, proving that the hypergraph might be a true model of our universe.
The big reveal
So here’s where I show you a persistent tangle propagating through the hypergraph, the equivalent in Wolfram’s universe of a glider in Conway’s universe, the equivalent, indeed, of a particle in our universe.
Drum roll, please...
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Truth is, I have nothing to show.
We’re not there yet.
Here’s the thing.
Conway’s computational system, in which simple rules are applied to a simple structure, yielding complex behaviour, doesn’t take much in the way of computational power.
When I simulate Conway’s Game of Life on my low-powered laptop, I can execute thousands iterations on millions of cells in just a few seconds.
Wolfram’s computational system, on the other hand, demands some serious computational power.
When it comes to simulating Wolfram Physics on my low-powered laptop, I have no idea how long it would take to execute thousands of iterations on millions of nodes and edges, because my code slows to a crawl after a mere hundred iterations on a mere hundred nodes and edges.
And that’s just applying a single rule to trace a single path through the multiway graph. Trace every possible path for every possible rule, and what’s already an impossibly large simulation becomes unimaginably large.
A good particle these days is hard to find
The elementary particles in Conway’s universe were discovered by running simulations and stumbling upon these curiosities.
The glider was discovered in 1970.
The big glider wasn’t discovered until 1989.
The dragon wasn’t discovered until 2000.
The lobster wasn’t discovered until 2011.
The copperhead wasn’t discovered until 2016.
You get the picture.
It took decades to discover all these gliders and spaceships.
No matter how small and simple the patterns of cells, Conway’s Game of Life is so complex that these equivalents of elementary particles are hard to find, even using automated search algorithms.
With Wolfram Physics, it’s worse.
Way, way worse.
No matter how small and simple the patterns of nodes and edges, the Wolfram model is so much more complex that persistent tangles in the hypergraph may be impossible to find.
We can imagine that in Wolfram Physics, there might be a rich taxonomy of patterns of nodes and edges that propagate, intact, through the hypergraph, corresponding precisely to the elementary particles that move, intact, through the three-dimensional space of our universe.
The question is, will we ever find them?