Build your ownuniverse Part 2 · Time

05What is time if there's no clock?

In Part 1 we grew a universe by pressing Next generation over and over, and it felt like time was passing. But if you look at what we actually did, we applied events, and a generation was just my way of batching them up.

In this model there's no clock ticking away outside the universe. Time comes from the events themselves, and from which events depend on which. The order you happen to click in is one possible history, not a clock, and nothing makes it the official one. Let's look at that properly, starting with the sprout rule from Part 1. Click a relation to apply the rule, then click one of the relations that event just made.

Interactive demo: the sprout rule {{x,y}} → {{x,y},{y,z}} on a tiny network. Each event you apply gets a plum dot; when an event uses a relation that an earlier event made, a plum arrow joins them.

The second event used something the first event made, so it couldn't have happened without it. That plum arrow means "caused", and it's what most of this part is about.

The trouble is that on a big network the arrows turn into spaghetti really quickly. So let me shrink the universe down to something we can actually watch: a universe with one dimension, a row of letters. The rule is BA → AB: wherever a B sits just before an A, they swap. I've numbered the letters so we can tell them apart as they move.

Click any highlighted pair to apply the rule there.

Interactive demo: the string B1 B2 B3 A1 A2 A3 with the rule BA → AB. Clicking a highlighted BA pair swaps it. Each click is an event, and the history of the string builds up underneath.

Every click is an event, and it always stops at AAABBB once every A is to the left of every B. What's interesting is what happened along the way.

Can you skip ahead?

One more thing about time. If the universe is just running a rule, could you skip ahead and find out what it looks like a billion events from now, without doing all the work? Let's try with rule 30 from Part 1.

Interactive demo: rule 30 grows row by row. Pick a row and try to predict the colour of its middle cell; the page finds out by computing every row up to it.

For some rules there's a shortcut. Rule 90 makes a nested pattern, and there's a neat formula for any cell in it. For rule 30 nobody knows one. The page above gets to row t by working out every cell on the way, roughly t × t updates, and Wolfram's conjecture is that there's no fundamentally faster way. It's only a conjecture: nobody has even proved that it needs as many as t operations, let alone t × t, and he's offered prizes to anyone who settles it. So the demo illustrates the idea rather than proving it. Wolfram calls this computational irreducibility, and his suggestion is that the universe is like this too: even the universe can't skip ahead, so time is the computation actually happening proposed.

06Does the order matter?

When you sorted those letters, you picked the order. Somebody else would have picked a different one. So does the order matter?

Let's find out properly. On the left you sort the letters however you like. On the right the computer sorts the same string in a random order. Underneath each, every event becomes a plum dot, placed by which two letters it swapped, so "B3 passed A1" always lands in the same spot. Arrows show which events caused which.

Interactive demo: two copies of B1 B2 B3 A1 A2 A3. You sort one by clicking; the computer sorts the other in a random order. Each event lands as a dot in a 3 by 3 grid with arrows for cause and effect; both grids come out identical.

You get the same dots and the same arrows every time. You took one of 42 possible routes, and I've had the computer check all 42: every one has the same 9 events with the same causes in the demo.

This is called causal invariance. Whatever order the events happen in, the network of cause and effect, the causal graph, comes out the same. Strictly, I've only checked it for this one starting string, not proved it for the rule.

Not every rule is like this. Here's the same sorting rule with one extra rule bolted on, BB → B, so two Bs that end up side by side merge into one. Start from BBAA and try a couple of different orders.

Interactive demo: the rule BA → AB plus BB → B, starting from BBAA. Every order ends at AAB, but different orders make different events and three different causal graphs.
Isn't causal invariance just "you always end up in the same place"?

The demo above already says no: every order ends at AAB, but the causal graphs differ. It's tempting to think the two go together, and for a while the project itself described it that way. But in November 2020 Max Piskunov showed they're different things. The rule {{x},{x,y}} → {{x,y},{y}} always ends in the same final state, yet different orders give different causal graphs. And {{x,y},{y,x}} → {{x}} gives the same causal graph every time, yet can end in different states. The "same ending" idea has a proper name, confluence: whenever two orders take you to two different states, you can always carry on from both until they meet again. For a rule that eventually stops, like the one above, that comes down to every complete order finishing in the same place, so the four orders that all end at AAB show it's confluent from BBAA, and only from that start. Causal invariance is about the events and what caused what. Both examples are checked in this site's tests in the demo.

So why care? Wolfram's bet is that the rule for our universe is causally invariant proposed. If it is, then everybody inside the universe agrees about what caused what, however the events happened to get "scheduled".

And it isn't just a string thing. Here's the showcase rule from Part 1 again, with its causal graph growing alongside it.

Interactive demo: the showcase rule growing generation by generation on the left, and its causal graph of events and plum arrows growing on the right.

07Whose "now"?

So far I've drawn events in rows, one row per generation. That's the Next generation button from Part 1 again. But is that the only way to slice events into moments?

Here's the same sorting rule on a longer string, BABABA…. Every dot is a real event the engine computed, with position along the string going across and generations going up the page.

Interactive demo: the causal graph of BA → AB on a long alternating string, drawn as a diamond lattice. A slider tilts the slices of "now"; the string as seen on each observer's slice is shown underneath, and clicking an event shows its light cone.

A slice is a choice of which events count as done. The rule is that every arrow has to cross your slices going forwards: you can't count an event as done before the events that caused it. Alice's slices go straight across. Bob's are tilted. They disagree about which events happened "at the same time", and they read different strings at their "now". Neither of them is wrong, and they still agree on every single arrow.

If one event caused another, every valid slicing puts them in the same order. If neither caused the other, there are slicings where either one comes first. That's the relativity of simultaneity, straight out of the causal graph in the demo, and it's very much a real feature of our universe observed.

So how far can Bob tilt? Influence spreads through this causal graph at most one letter per generation, and any tilt less than that keeps every arrow crossing his slices forwards. At exactly one letter per generation his slices lie along the arrows instead, calling an event and its cause simultaneous. That's the light cone, the edge rather than a now anyone can have, and past it arrows run backwards. Wolfram's suggestion is that this maximum speed is what we call the speed of light proposed.

In 2020 Jonathan Gorard showed that for causally invariant rules whose causal graphs behave like smooth spacetime when you zoom out, switching between these slicings works like the Lorentz transformations of special relativity maths, if…. The "if" is doing real work there. Toy rules have been shown to have the discrete version of this (Gorard works one through for AB → BA, our sorting rule's mirror image, on a repeating string), but no candidate rule has been shown to reproduce the spacetime and matter of our universe. And the demo above only shows the simultaneity part.

What about gravity?

Back in Part 1 we measured dimension by counting how fast balls grow. Once you've allowed for the dimension, the graininess and the edges, balls that grow a bit slower than they would on a flat sheet mean the space is curved like a sphere. Wolfram and Gorard go further. They treat energy and momentum as how densely causal arrows cross through slices, and show that the vacuum version of Einstein's equations (gravity with no matter around) follows if you assume causal invariance, a dimension that settles down, a smooth limit when you zoom out, and a few more technical conditions maths, if…. Matter gets added separately. It's a really interesting result, but it's a conditional one, and nobody has shown that any specific rule meets those conditions.

08What if we never choose?

Every time you clicked a BA pair you picked one history out of many. What if we don't pick? What if we keep every possible choice at once?

This is every string you can reach from BBBAAA, with an arrow for every event between them. If you sorted the letters in chapter 05 or 06, your route is lit up in yellow.

Interactive demo: the multiway graph of BA → AB from BBBAAA: all 20 strings, with arrows for every possible event. The route you clicked earlier is highlighted as one path among 42.

This is called a multiway graph. Notice how routes split apart and then join back up: two different events can lead to the same string, so different histories keep meeting again.

And this is where chapter 06 comes back in. The computer checked all 42 of these routes there, and every one of them gives the same causal graph in the demo. So there are lots of different histories here, but they all agree about what caused what.

Here's one that never stops. The rule is A → AB, B → A, starting from a single A. I'll call one event along every history a step, so step 1 is everything one event away from A, step 2 is everything two events away, and so on. Click a string to see where it can go next, or grow a whole step at once.

Interactive demo: the multiway graph of A → AB, B → A from A. Expand strings by clicking them; identical strings merge into one node. Counters compare the number of different strings with the number of different histories.

Keep pressing Next step and watch the table. By step 8 there are 34 different strings, but 7,936 different histories lead to them in the demo. Merging is what keeps the drawing small: without it, every history would need its own copy of the string.

Wolfram's suggestion is that the universe doesn't pick one history. The multiway graph has all of them proposed. I'd be careful with the "parallel universes" picture people sometimes reach for here, though. Branches in a multiway graph keep merging back together, and a branch on its own isn't a separate world. It's just one possible history.

09Where's the observer?

If every history is in there, why do we only ever experience one? Wolfram's answer starts by noticing that the branches themselves have a shape.

Take one slice of the multiway graph, every string at the same step, and join two strings whenever they share at least one parent one step back. That gives you a branchial graph: a map of how closely related the different states are. Each string can already be reached by lots of different histories, so it's states that get joined here, not histories.

Interactive demo: the multiway graph of A → AB, B → A with a slider choosing a step. The strings at that step are laid out on the right and joined when they share a parent; hovering a link shows the shared parent, and picking two strings traces them back to their nearest common ancestor and draws the shortest route between them.

Pick two strings and the demo traces them back to their nearest common ancestor: the most recent earlier string both can be reached from (sometimes there are two). Neighbours in the branchial graph share a parent one step back, and strings further apart share an ancestor longer ago. For this particular rule, the number of links between two strings matched the number of steps back to their nearest common ancestor for every pair I checked, up to step 7 in the demo. That isn't true of branchial graphs in general, but here it's a nice way to see the graph as a map of how related the states are.

None of this is just a string thing. You can build a multiway graph of whole hypergraphs in exactly the same way, and slice it into branchial graphs the same way too. I've stuck to strings because you can actually read them.

Wolfram proposes that this branchial space is where quantum mechanics lives proposed. Being in a superposition would be like being spread across nearby branches, and entanglement would be about how close things are in branchial space.

I want to be really clear about one thing though: branching on its own isn't quantum mechanics. Real quantum mechanics needs amplitudes that can cancel each other out, and a rule for turning them into probabilities. Gorard's 2020 paper builds those in, but it needs some big extra assumptions to do it maths, if…. The demo above has none of that.

So where's the observer? In Wolfram's picture you're not outside the multiway graph looking in, you're part of it. And observers like us can't track every branch, so we lump lots of different states together and experience what looks like a single thread of history. Here's a toy version of that lumping.

Interactive demo: choose what an observer can tell apart (the whole string, its length, its last letter, or its first letter). States the observer can't distinguish are grouped, and the tangled multiway graph turns into something much simpler. The original graph stays visible underneath.

That's only an illustration of one ingredient proposed. In this toy, the only observer who sees a single thread is one who can't see anything about the strings at all, so lumping on its own clearly isn't the whole story. Wolfram's claim is about observers like us in a vastly bigger system, and it leans on causal invariance too. His and Gorard's actual account of measurement also involves how an observer slices the multiway graph and some extra rules that merge branches, and nothing in the multiway graph actually disappears when we group things.

10So, is it true?

I've been fascinated by this stuff for years, so I want to be fair about it. Here's my honest attempt at a ledger of what we've seen.

ClaimStatusThe catch
A tiny rule can grow an intricate networkin the demoNone, you grew one.
Dimension can emerge and be measured from insidein the demoEstimates need big networks and still drift as they grow.
From BBBAAA, all 42 complete orders give the same causal graphin the demoOne rule and one starting string, checked exhaustively. Not a proof for every start.
Relativity of simultaneity from slicing a causal graphin the demoOnly the simultaneity part, not the full Lorentz transformations.
Special relativity from causal invariancemaths, if…Needs causal invariance and a smooth spacetime-like limit (Gorard 2020).
Einstein's equations of gravitymaths, if…The vacuum equations, assuming causal invariance, a settled dimension, a smooth limit and more; matter is added separately. No rule shown to meet the conditions.
Quantum mechanics from multiway systemsproposedThe derivations add a lot of structure by hand, and the probability rule is defined, not derived.
Our universe is a hypergraph running one ruleproposedNo rule has been found that gives our universe.
A new prediction confirmed by experimentnone yetNo candidate rule has produced a sharp, quantified prediction to test.

The critics' main points are fair ones. Without a specific rule there are no specific predictions, so there isn't an experiment yet that could decide it one way or the other. Lots of very different rules also seem to give similar large-scale behaviour (you saw a bit of that in Part 1's zoo, where quite different rules all grew into two-dimensional surfaces), which makes it hard to know which details mean anything. Physicists like Scott Aaronson have also raised specific doubts about the quantum side.

Wolfram himself has moved on to an even bigger idea since 2021, which he calls the ruliad: the result of running every possible rule in every possible way. In that picture there's no single rule for our universe to find, and the laws of physics come from the kind of observers we are proposed. I'm not going to pretend I can show you that one in a demo.

Before the verdict, here's the picture from the very start of Part 1 again. Hopefully it reads a bit differently now. The dots are elements and the lines are relations. The two loops it started from were just given; after that, every relation was made by an event, and every event after the first used things earlier events made. If you measure it, the tangle comes out a bit more than two-dimensional. I drew it with one particular updating order, and to be honest, nobody knows yet whether this rule gives the same causal graph for every order.

Interactive demo: the network from the start of Part 1, grown from one rule. Drag dots, click them, or grow it.

So is it true? I honestly don't know. What would change my mind? Someone finding a rule that gives three dimensions and the particles we actually see, or a prediction that can be checked and turns out right. On the other side, if decades go by with neither, that would tell us something too.

What I do know is that it changed how I think about what a law of physics could be. Space you measure by counting hops, and time that's just the rule running, isn't what I pictured a law of physics looking like. Whether or not it's how our universe works, I think it's one of the most fun ideas I've come across, and I hope poking at it was fun for you too.

If you want to keep going, the playground has everything from both parts. Go and find a rule nobody has looked at yet. Cheerio!

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